BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Arkadiy Skopenkov (Independent University of Moscow\, and Moscow I nstitute of Physics and Technology) DTSTART:20211008T160000Z DTEND:20211008T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/2 DESCRIPTION:Title: Invariants of graph drawings in the plane\nby Arkadiy Skopenkov (Independent University of Moscow\, and Moscow Institute of Physics and Technology) as part of Selected Topics in Mathematics - On line Edition\n\n\nAbstract\nWe present a simplified exposition of some cla ssical results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results o n the border of topology and combinatorics. In particular\, we define a $\ \bmod 2$−valued self−intersection invariant (i.e. the van Kampen numbe r) and its generalizations. We present elementary formulations and argumen ts accessible to mathematicians not specialized in any of the areas mentio ned. Motivated by algorithmic\, combinatorial and geometric problems\, we introduce starting ideas of algebraic topology.\n\nThis talk is based on w ork in the following paper https://arxiv.org/abs/1805.10237\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /2/ END:VEVENT BEGIN:VEVENT SUMMARY:Polina Vytnova (University of Warwick) DTSTART:20211015T160000Z DTEND:20211015T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/3 DESCRIPTION:Title: Hausdorff dimension of Gauss−Cantor sets and the ir applications to the study of classical Markov spectrum\nby Polina V ytnova (University of Warwick) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe classical Lagrange and Markov spectra a re subsets of the real line which arise in connection with some problems i n theory Diophantine approximation theory. In 1921 O. Perron gave a defini tion in terms of continued fractions\, which allowed to study the Markov a nd Lagrange spectra using limit sets of iterated function schemes.\n\nIn t his talk we will see how an upper bounds on the Hausdorff dimension of the difference of the Markov and Lagrange spectra can be computed by the mean s of estimating Hausdorff dimension of the certain Gauss-Cantor sets.\n\nT he talk is based on a joint work with C. Matheus\, C. G. Moreira and M. Po llicott.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /3/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexey Sossinsky (Independent University of Moscow) DTSTART:20211022T140000Z DTEND:20211022T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/4 DESCRIPTION:Title: Energy Functionals and the Normal Forms of Knots a nd Plane Curves\nby Alexey Sossinsky (Independent University of Moscow ) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract \nFor any $\\varepsilon\\in[0\,1]$\, I will define the notion of $\\vareps ilon$-thin knot\, which for $\\varepsilon\\in[0\,1]$ is a mathematical mod el of a thin flexible rope of fixed length and\, for $\\varepsilon=0$\, is a classical smooth knot of fixed length. The aim is to construct an energ y functional that would classify $\\varepsilon$-thin knots with a fixed po sitive $\\varepsilon$\nby bringing them to “form”\, i.e.\, to a positi on corresponding to a local minimum of the functional.\n\nThe talk will be gin with the demonstration of mechanical experiments with wire knots\, sho wing how they automatically switch from any position to normal form. I wil l then show videos of computer experiments with thin knots (from joint wor k with my former pupils O. Karpenkov and S. Avvakumov)\, showing how they evolve from some initial position to their normal form. In the videos\, we deal with polygonal knots rather than smooth thin knots\, and the functio nal consists of a summand that tries to straighten out the curve locally a nd a summand that forbids self-intersections (and the subsequent crossing of one part of the curve by another). In practice\, the algorithm always t erminates\, which is actually amazing\, because its termination is not a d eterminstic fact (i.e.\, not a theorem) − it is due to probabilistic law s of nature.\n\nAn unexpected result of our study is that the normal form is not always unique − for the eight knot there are two: they depend on the mechanical characteristics of the wire\, and on the initial shape of t he knot.\n\nAfter that I will discuss other possible energy functionals\, in particular involving not only the minimization of curvature\, but also that of torsion (physically\, this means we are dealing with elastic wire knots that resist twisting).\n\nTime permitting\, I will briefly explain o ur work on plane curves\, a by-product of which was our solution of the Eu ler elasticae problem\, and formulate some conjectures.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /4/ END:VEVENT BEGIN:VEVENT SUMMARY:Lenny Fukshansky (Claremont McKenna College) DTSTART:20211029T140000Z DTEND:20211029T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/5 DESCRIPTION:Title: Representing integers by multilinear polynomials\nby Lenny Fukshansky (Claremont McKenna College) as part of Selected To pics in Mathematics - Online Edition\n\n\nAbstract\nGiven a homogeneous mu ltilinear polynomial $F(x)$ in $n$ variables with integer coefficients\, w e obtain some sufficient conditions for it to represent all the integers. Further\, we derive effective results\, establishing bounds on the size of a solution $x$ to the equation $F(x)=b$\, where $b$\nis any integer. For a special class of polynomials coming from determinants of rectangular mat rices we are able to obtain necessary and sufficient conditions for such a n effective representation problem. This result naturally connects to the problem of extending a collection of primitive vectors to a basis in a lat tice\, where we present counting estimates on the number of such extension s.\n\nThe talk is based on joint works with A. Boettcher and with M. Forst .\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /5/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Slilaty (Wright State University) DTSTART:20211105T150000Z DTEND:20211105T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/6 DESCRIPTION:Title: Representing Matroids using Partial Fields\nby Daniel Slilaty (Wright State University) as part of Selected Topics in Ma thematics - Online Edition\n\nAbstract: TBA\n\nThe abstract of the talk is available on the seminar website\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /6/ END:VEVENT BEGIN:VEVENT SUMMARY:Irene Pasquinelli (University of Bristol) DTSTART:20211119T150000Z DTEND:20211119T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/7 DESCRIPTION:Title: Mapping class group orbit closures for non-orienta ble surfaces\nby Irene Pasquinelli (University of Bristol) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe study o f the asymptotic growth of the number of closed geodesics on a hyperbolic surface dates back to Huber (1961) and has implications in various fields of mathematics. In her thesis\, Mirzakhani proved that for an orientable h yperbolic surface of finite area\, the number of simple closed geodesics o f length less than L is asymptotically equivalent to a polynomial in L\, w hose degree only depends on the Euler characteristic. \n\nWhen looking at non-orientable surfaces\, the situation is very different. One of the main differences in this framework is the behaviour of the action of the mappi ng class group on the space of measured laminations. \n\nIn a joint work w ith Erlandsson\, Gendulphe and Souto\, we characterised mapping class grou p orbit closures of measured laminations\, projective measured laminations and points in Teichmueller space.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /7/ END:VEVENT BEGIN:VEVENT SUMMARY:Giovanni Panti (University of Udine) DTSTART:20211126T150000Z DTEND:20211126T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/8 DESCRIPTION:Title: Attractors of dual continued fractions\nby Gio vanni Panti (University of Udine) as part of Selected Topics in Mathematic s - Online Edition\n\n\nAbstract\nWe identify continued fractions with pie cewise-projective Markov maps on some interval\, provided that all branch es are induced by elements in a fixed Hecke triangle group. Ordinary conti nued fractions constitute a very special case\, relative to the (2\,3\,inf inity) group. The branches of the map are expanding on the domain interval \, but contracting on some other part of the real projective line. As such \, they give rise to an IFS whose attractor is relevant\, since it determi nes the natural extension and several properties\, both algebraic and dyna mical\, of the original system. We use an appropriate generalization of th e classical Minkowski Question Mark function to simultaneously linearize a ll maps resulting from the same triangle group\, and to prove that the abo ve IFS satisfies the open set condition. We draw consequences -as well as open problems- from these facts.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /8/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Felikson (Durham University) DTSTART:20211203T150000Z DTEND:20211203T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/9 DESCRIPTION:Title: Friezes for a pair of pants\nby Anna Felikson (Durham University) as part of Selected Topics in Mathematics - Online Edi tion\n\n\nAbstract\nFrieze patterns are numerical arrangements that satisf y a local arithmetic rule. Conway and Coxeter showed that frieze patterns are tightly connected to triangulated polygons. Recently\, friezes were ac tively studied in connection to the theory of cluster algebras\, and the n otion of a frieze obtained a number of generalisations. In particular\, on e can define a frieze associated with a bordered marked surface endowed wi th a decorated hyperbolic metric.\n\nWe will review the construction and w ill show that some nice properties can be extended friezes associated to a pair of pants. This work is joint with Ilke Canakci\, Ana Garcia and Pave l Tumarkin.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /9/ END:VEVENT BEGIN:VEVENT SUMMARY:John Hunton (Durham University) DTSTART:20211210T150000Z DTEND:20211210T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/10 DESCRIPTION:Title: What is the space group of an aperiodic pattern?< /a>\nby John Hunton (Durham University) as part of Selected Topics in Math ematics - Online Edition\n\n\nAbstract\nThe space groups/crystallographic groups/Bieberbach groups (depending on what dimension you are interested i n) have played a classical role in the understanding of periodic patterns in d-dimensional space for many generations: the space group of a periodic pattern $P\\subset\\mathbf{R}^d$ is the subgroup of the full isometry gro up of $\\mathbf{R}^d$ that fixes $P$. In some sense the space group determ ines up to local decoration the pattern itself. \n \nBut what of aperiodic patterns\, such as the Penrose tiling\, or the objects used to model quas icrystals\, or more generally any aperiodic pattern? By their nature they are rather short of symmetries\, but on the other hand\, they have a lot o f approximate symmetries\, structure that can be accessed by topological m ethods. \n \nIn this talk I will sketch some methods developed with Jamie Walton to define the corresponding "space groups" for such objects.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /10/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexey Sossinsky (Independent University of Moscow) DTSTART:20211217T150000Z DTEND:20211217T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/11 DESCRIPTION:Title: Thickened Knots: Energy Functionals and Normal Fo rms\nby Alexey Sossinsky (Independent University of Moscow) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nFor any $ \\varepsilon \\in [0\, 1]$\, I will define the notion of $ \\varepsilon$-t hickened knot\, which for $ \\varepsilon \\in (0\, 1]$ is a mathematical m odel of a thin flexible rope of fixed length with extremities glued togeth er. The aim is to construct an energy functional that would classify $ \\v arepsilon$-thickened knots with a fixed positive $ \\varepsilon$ by bringi ng them to "normal form" by gradient descent\, i.e.\, to a position corres ponding to a local minimum of the functional.\n\nThe talk will begin with the demonstration of mechanical experiments with wire knots\, showing how they automatically switch from any position to normal form. I will then sh ow videos of computer experiments with thickened knots (from joint work wi th my former pupils O. Karpenkov and S. Avvakumov)\, showing how they evol ve from some initial position to their normal form. In the videos\, we dea l with polygonal knots rather than smooth thickened knots\, and the functi onal consists of a summand that tries to straighten out the curve locally and a summand that forbids self-intersections (and the subsequent crossing of one part of the curve by another). In practice\, the algorithm always terminates\, which is actually amazing\, because its termination is not a deterministic fact (i.e.\, not a theorem) - it is due to probabilistic law s of nature.\n\nAn unexpected result of our study is that the normal form is not always unique - for the eight knot there are two: they depend on th e mechanical characteristics of the wire\, and on the initial shape of the knot.\nI will then state some theorems about $ \\varepsilon$-thickened kn ots and formulate a conjecture about their complete and effective classifi cation. Time permitting\, I will briefly explain our work on plane curves\ , a by-product of which was our solution of the Euler elasticae problem\, formulate some conjectures\, and discuss other possible energy functionals \, in particular involving not only the minimization of curvature\, but al so that of writhe (physically\, this means we are dealing with elastic wir e knots that resist twisting).\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /11/ END:VEVENT BEGIN:VEVENT SUMMARY:Ivan Izmestiev (TU Wien) DTSTART:20220204T150000Z DTEND:20220204T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/12 DESCRIPTION:Title: Regge symmetry\nby Ivan Izmestiev (TU Wien) a s part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nT he Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovere d as a consequence of an asymptotic formula in mathematical physics. Here we give an elementary geometric proof of Regge symmetries in Euclidean\, s pherical\, and hyperbolic geometry.\nThe talk is based on a joint work wit h Arseniy Akopyan.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /12/ END:VEVENT BEGIN:VEVENT SUMMARY:Fatemeh Mohammadi (Ghent University) DTSTART:20220304T150000Z DTEND:20220304T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/14 DESCRIPTION:Title: Matroid stratifications of hypergraph varieties a nd their realization spaces\nby Fatemeh Mohammadi (Ghent University) a s part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nI will provide an introductory talk to hypergraph varieties\, focusing on t he combinatorial aspect. The main themes of the talk are (1) connecting th e geometric properties of hypergraphs to their minimal matroids\; (2) redu cing the geometric invariants of these matroids to grid matroids\; and (3) understanding the realizability of these matroids. Finally\, I will menti on the application to conditional independence models in statistics and wi ll present some geometric questions and computational challenges around th is problem. This is based on joint works with Kevin Grace\, Oliver Clarke \, and Harshit J Motwani.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /14/ END:VEVENT BEGIN:VEVENT SUMMARY:Alan Haynes (University of Houston) DTSTART:20220211T150000Z DTEND:20220211T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/15 DESCRIPTION:Title: Gap theorems for linear forms and for rotations o n higher dimensional tori\nby Alan Haynes (University of Houston) as p art of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThis talk is based on joint work with Jens Marklof\, and with Roland Roeder. T he three distance theorem states that\, if $x$ is any real number and $N$ is any positive integer\, the points $x\, 2x\, \\ldots \, Nx \\mod 1$ part ition the unit interval into component intervals having at most 3 distinct lengths. We will present two higher dimensional analogues of this problem . In the first we consider points of the form $mx+ny \\mod 1$\, where $x$ and $y$ are real numbers and $m$ and $n$ are integers taken from an expand ing set in the plane. This version of the problem was previously studied b y Geelen and Simpson\, Chevallier\, Erdős\, and many other people\, and i t is closely related to the Littlewood conjecture in Diophantine approxima tion. The second version of the problem is a straightforward generalizatio n to rotations on higher dimensional tori which\, surprisingly\, has been mostly overlooked in the literature. For the two dimensional torus\, we ar e able to prove a five distance theorem\, which is best possible. In highe r dimensions we also have bounds\, but establishing optimal bounds is an o pen problem.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /15/ END:VEVENT BEGIN:VEVENT SUMMARY:Bernd Schulze (Lancaster University) DTSTART:20220429T140000Z DTEND:20220429T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/16 DESCRIPTION:Title: Group-theoretic methods for the rigidity analysis of discrete structures\nby Bernd Schulze (Lancaster University) as pa rt of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nGeome tric rigidity theory is concerned with the rigidity and flexibility analys is of bar-joint frameworks and related constraint systems of geometric obj ects.\nIn the beginning of this talk\, we will give a brief introduction t o this area\, which has a rich history that can be traced back to classica l work of Euler\, Cauchy and Maxwell on the rigidity of polyhedra and skel etal frames. One of the major recent research directions in this field is to study the impact of symmetry on the rigidity of bar-joint frameworks. W e show how group representation theory can be used to reveal `hidden' infi nitesimal motions and states of self-stress in symmetric frameworks that c annot be detected with Maxwell's basic counting rule from 1864. We then sh ow how this symmetry-adapted counting rule\, which was originally discover ed by the engineer Simon Guest and the chemist Patrick Fowler\, can be use d to derive an efficient new method for constructing symmetric frameworks with a large number of `fully-symmetric' or `anti-symmetric' states of sel f-stress. Maximizing the number of independent states of self-stress of a planar framework\, as well as understanding their symmetry properties\, ha s important practical applications\, for example in the design and constru ction of gridshells. We show the usefulness of our method by applying it t o some practical examples.\n\nThis is joint work with Cameron Millar (SOM) \, Arek Mazurek (Mazurek Consulting) and William Baker (SOM).\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /16/ END:VEVENT BEGIN:VEVENT SUMMARY:Padraig Ó Catháin (Dublin City University) DTSTART:20220325T150000Z DTEND:20220325T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/17 DESCRIPTION:Title: Sequencing of Steiner Triple Systems\nby Padr aig Ó Catháin (Dublin City University) as part of Selected Topics in Mat hematics - Online Edition\n\n\nAbstract\nSteiner triple systems (STSs) are the simplest and best understood infinite family of block designs.\nThey are equivalent to decompositions of complete graphs into edge disjoint tri angles.\nA sequencing of an STS is a bijection of the points with the inte gers $[1\, \\ldots\, v]$.\n\nIn 2019\, Stinson and Kreher introduced L-goo d sequencings\, in which no block is contained in an\ninterval of length L \, that is $[d\, d+1\, \\ldots\, d+L]$. Stinson and Veitch gave an algorit hmic proof\nthat an STS with v points has an L-good sequencing for $v \\ge q L^{6}/16 + O(L^{5})$. This was later improved\nto $v \\geq L^{4}/2 + O(L ^{3})$ by Blackburn and Linial with an explicit greedy algorithm.\n\nUsing the Lovasz Local Lemma\, we prove that every STS with $v \\geq 121L^{2}$ admits an L-good\nsequencing. The exponent $2$ is optimal\, in the sense t hat counterexamples are known without L-good\nsequencings for $v = \\Theta (L^{2-\\epsilon})$ for any $\\epsilon > 0$. In fact\, we prove a more gene ral result on\ndecompositions of general Steiner systems into sufficiently large (as a function of L) disjoint independent sets.\nThis is joint work with Daniel Horsley of Monash University.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /17/ END:VEVENT BEGIN:VEVENT SUMMARY:Krishnendu Gongopadhyay (IISER Mohali) DTSTART:20220311T150000Z DTEND:20220311T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/18 DESCRIPTION:Title: Reversibility of isometries\nby Krishnendu Go ngopadhyay (IISER Mohali) as part of Selected Topics in Mathematics - Onli ne Edition\n\n\nAbstract\nIn this talk\, I shall review recent work on rev ersibility of isometries of Hermitian spaces over the complex numbers and over the quaternions. I shall explain what I mean by reversibility in the talk and how it has been classified in some Lie groups.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /18/ END:VEVENT BEGIN:VEVENT SUMMARY:Sinai Robins (Universidade de São Paulo) DTSTART:20220506T140000Z DTEND:20220506T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/19 DESCRIPTION:by Sinai Robins (Universidade de São Paulo) as part of Select ed Topics in Mathematics - Online Edition\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /19/ END:VEVENT BEGIN:VEVENT SUMMARY:Sabir Gusein-Zade (Lomonosov Moscow State University) DTSTART:20220401T140000Z DTEND:20220401T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/20 DESCRIPTION:Title: Non-commutative analogue of the Berglund–Hübsc h–Henningson duality and symmetries of the orbifold Euler characteristic (I)\nby Sabir Gusein-Zade (Lomonosov Moscow State University) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe Ber glund–Hübsch–Henningson (BHH–) duality is a particular case of the mirror symmetry. It is described as a duality on the set of pairs $(f\, G) $ consisting of an invertible polynomial and a subgroup $G$ of diagonal sy mmetries of $f$. Symmetries of invariants of BHH-dual pairs are related to the mirror symmetry. There is a method to extend the BBH-duality to the s et of pairs $(f\, G')$\, where $G'$ is the semidirect product of a group $ G$ of diagonal symmetries of $f$ and a group $S$ of permutations of the co ordinates preserving $f$. The construction is based on ideas of A. Takahas hi and therefore is called the Berglund-Hübsch-Henningson-Takahashi (BHHT -) duality. Invariants of BHHT-dual pairs have symmetries similar to mirro r ones only under some restrictions on the group $S$: the so-called parity condition (PC). Under the PC-condition it is possible to prove symmetries of the orbifold Euler characteristic and of some other orbifold invariant s for actions on the Milnor fibers of dual pairs. The talk is based on joi nt results with W. Ebeling.\n\nThis is the first half of a two part talk.\ n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /20/ END:VEVENT BEGIN:VEVENT SUMMARY:Sabir Gusein-Zade (Lomonosov Moscow State University) DTSTART:20220408T140000Z DTEND:20220408T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/21 DESCRIPTION:Title: Non-commutative analogue of the Berglund–Hübsc h–Henningson duality and symmetries of the orbifold Euler characteristic (II)\nby Sabir Gusein-Zade (Lomonosov Moscow State University) as par t of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe Be rglund–Hübsch–Henningson (BHH–) duality is a particular case of the mirror symmetry. It is described as a duality on the set of pairs $(f\, G )$ consisting of an invertible polynomial and a subgroup $G$ of diagonal s ymmetries of $f$. Symmetries of invariants of BHH-dual pairs are related t o the mirror symmetry. There is a method to extend the BBH-duality to the set of pairs $(f\, G')$\, where $G'$ is the semidirect product of a group $G$ of diagonal symmetries of $f$ and a group $S$ of permutations of the c oordinates preserving $f$. The construction is based on ideas of A. Takaha shi and therefore is called the Berglund-Hübsch-Henningson-Takahashi (BHH T-) duality. Invariants of BHHT-dual pairs have symmetries similar to mirr or ones only under some restrictions on the group $S$: the so-called parit y condition (PC). Under the PC-condition it is possible to prove symmetrie s of the orbifold Euler characteristic and of some other orbifold invarian ts for actions on the Milnor fibers of dual pairs. The talk is based on jo int results with W. Ebeling.\n\nThis is the second half of a two part talk .\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /21/ END:VEVENT BEGIN:VEVENT SUMMARY:Simon Kristensen (Aarhus University) DTSTART:20220617T140000Z DTEND:20220617T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/22 DESCRIPTION:by Simon Kristensen (Aarhus University) as part of Selected To pics in Mathematics - Online Edition\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /22/ END:VEVENT BEGIN:VEVENT SUMMARY:James Cruickshank (University of Galway) DTSTART:20221019T150000Z DTEND:20221019T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/23 DESCRIPTION:Title: Global Rigidity of Triangulated Manifolds\nby James Cruickshank (University of Galway) as part of Selected Topics in Ma thematics - Online Edition\n\n\nAbstract\nIn this talk I will begin with a n introduction to global rigidity of frameworks and graphs and then descri be recent joint work with Bill Jackson and Shin-ichi Tanigawa in which we prove the following.\n

\n

\n Theorem. For \\(d\\ge q 3\\) the graph of a connected triangulated \\((d-1)\\)-manifold is gener ically globally rigid in \\(\\mathbb R^{d}\\) if and only if the graph is \\((d+1)\\)-connected and\, if \\(d=3\\)\, not planar.\n

\n

\nThis proves and generalises a conjecture of Connelly. I will also discuss some applications of this result and of the techniques we use in the proof. We prove the generic case of a conjecture of Kalai on the reconstructability of a polytope from its space of stresses. We also use our methods to gene ralise parts of the Lower Bound Theorem to a larger class of simplicial co mplexes.\n

\n

\n\n Some context for a general audience.\n\nA graph is said to be globally rigid in \\(\\mathbb R^d\\) if a g eneric embedding of the vertex set in \\(\\mathbb R^d\\) is determined\, u p to isometry of \\(\\mathbb R^d\\)\, by the distances between adjacent ve rtices. There is a weaker local version of rigidity in which the embeddin g is only determined within some neighbourhood. More detail\, and examples \, will be given in the talk. \n

\n

\nJackson and Jordán\, foll owing earlier work of Connelly\, have characterised graphs that are global ly rigid in \\(\\mathbb R^2\\) in terms of the 2-dimensional rigidity matr oid. However extending this characterisation to higher dimensions is a ve ry challenging open problem. Indeed there are very few examples known of naturally interesting infinite families of graphs for which the global rig idity problem in \\(\\mathbb R^d\\)\, for \\(d\\geq 3\\)\, has been settle d. \n

\n

\nOne interesting family of graphs in this context are tho se arising as graphs of triangulations of manifolds. Fogelsanger showed t hat the graph of a triangulated \\(d\\)-manifold is locally rigid in \\(\\ mathbb R^{d+1}\\)\, but the global rigidity problem for such graphs remain ed open. Connelly conjectured that \\(4\\)-connected triangulations of non -spherical surfaces are globally rigid in \\(\\mathbb R^3\\). In higher d imensions\, even the global rigidity of graphs of simplicial polytopes rem ained an open question. Kalai\, Tay and others have used the local rigidi ty theory of graphs to prove important results concerning face numbers of pseudomanifolds\, but global rigidity theory remains relatively unexplored in that context.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /23/ END:VEVENT BEGIN:VEVENT SUMMARY:Igor Potapov (University of Liverpool) DTSTART:20221026T150000Z DTEND:20221026T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/24 DESCRIPTION:Title: Reachability Problems in Matrix Semigroups\nb y Igor Potapov (University of Liverpool) as part of Selected Topics in Mat hematics - Online Edition\n\n\nAbstract\nMatrices and matrix products play a crucial role in the representation and analysis of various computationa l processes\, The central decision problem in matrix semigroups is the mem bership problem: "Decide whether a given matrix $M$ belongs to a finitely generated matrix semigroup". By restricting $M$ to be the identity (zero) matrix the problem is known as the identity (mortality) problem. \n\nUnfor tunately\, many simply formulated and elementary problems for matrices are inherently difficult to solve even in dimension two\, and most of these p roblems become undecidable in general starting from dimension three or fou r. For example\, the identity problem for $3\\times 3$ matrices of integer s is the long-standing open problem.\n\nIn this talk I will provide an ove rview about various decision problems in matrix semigroups such as members hip\, vector reachability\, freeness\, scalar reachability\, etc. Also\, I will focus on the number of state-of-the-art theoretical computer science techniques as well as decidability\, undecidability and computational com plexity results.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /24/ END:VEVENT BEGIN:VEVENT SUMMARY:Anton Lukyanenko (George Mason University) DTSTART:20221102T160000Z DTEND:20221102T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/25 DESCRIPTION:Title: Heisenberg continued fractions: overview and rece nt results\nby Anton Lukyanenko (George Mason University) as part of S elected Topics in Mathematics - Online Edition\n\n\nAbstract\nContinued fr action theory over the real numbers has a long connection to real hyperbol ic geometry.\nAbout 10 years ago\, Joseph Vandehey and I proposed a new CF algorithm over the non-commutative\nHeisenberg group\, which is designed to take advantage of complex hyperbolic theory\,\nand connects directly to the work of Falbel-Francsics-Lax-Parker\, Hersonsky-Paulin\, Series\, \nK atok-Ugarkovici\, Nakada\, Hensley\, and others.\n\nWe have since connecte d the theory to Diophantine approximation\, established ergodicity\nof the Gauss-type map (for a folded variant of the CF)\, and developed a broader framework of\nIwasawa CFs\, which include many real\, higher-dimensional\ , and non-commutative CF algorithms.\nMore recently\, we returned to the E uclidean setting to explore the dynamics of CFs over the complex \nnumbers \, quaternions\, octonions\, as well as defining new CF algorithms in \\(\ \mathbb{R}^3\\).\n\nIn this talk\, I will start by discussing the A. Hurwi tz complex CFs as a motivating higher-dimensional algorithm\,\nthen discus s the Heisenberg group and Heisenberg CFs\, and then provide an overview o f my work with\nVandehey\, finishing with this year's results in Euclidean space.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /25/ END:VEVENT BEGIN:VEVENT SUMMARY:Valentin Ovsienko (University of Reims Champagne-Ardenne) DTSTART:20221116T160000Z DTEND:20221116T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/26 DESCRIPTION:Title: Shadows of numbers: supergeometry with a human fa ce\nby Valentin Ovsienko (University of Reims Champagne-Ardenne) as pa rt of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nIn th is elementary and accessible to everybody talk\, I will explain an attempt to apply supersymmetry and supergeometry to arithmetic. The following gen eral idea looks crazy. What if every integer sequence has another integer sequence that follows it like a shadow? I will demonstrate that this is in deed the case\, though perhaps not for every integer sequence\, but for ma ny of them. The main examples are those of the Markov numbers and Somos se quences.\n\nIn the second part of the talk\, I will discuss the notions of supersymmetric continued fractions and the modular group\, and arrive at yet a more crazy idea that every rational and every irrational has its own shadow. The second part of the talk is a joint work with Charles Conley.\ n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /26/ END:VEVENT BEGIN:VEVENT SUMMARY:Karin Baur (University of Leeds) DTSTART:20221123T160000Z DTEND:20221123T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/27 DESCRIPTION:Title: Frieze patterns and cluster theory\nby Karin Baur (University of Leeds) as part of Selected Topics in Mathematics - Onl ine Edition\n\n\nAbstract\nCluster categories and cluster algebras can be described via triangulations of surfaces or via Postnikov diagrams. \nIn type A\, such triangulations lead to frieze patterns or SL\\(_2\\)-friezes in the sense of Conway and Coxeter. \nWe explain how infinite frieze pat terns arise and how Grassmannians or Plücker coordinates give rise to SL\\(_k\\)-friezes.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /27/ END:VEVENT BEGIN:VEVENT SUMMARY:Matteo Raffaelli (TU Wien) DTSTART:20221207T160000Z DTEND:20221207T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/28 DESCRIPTION:Title: Curvature-adapted submanifolds of semi-Riemannian groups\nby Matteo Raffaelli (TU Wien) as part of Selected Topics in M athematics - Online Edition\n\n\nAbstract\nGiven a semi-Riemannian hypersu rface $M$ of a semi-Riemannian manifold $Q$\, one says that $M$ is $\\text it{curvature-adapted}$ if\, for each $p \\in M$\, the normal Jacobi operat or and the shape operator of $M$ at $p$ commute. The first operator measur es the curvature of the ambient manifold along the normal vector of $M$\, whereas the second describes the curvature of $M$ as a submanifold of $Q$. This condition can be generalized to submanifolds of arbitrary codimensio n.\n\nIn this talk I will present joint work with Margarida Camarinha addr essing the case where the ambient manifold is a Lie group equipped with a bi-invariant metric. In particular\, we will see that\, if the normal bund le of $M$ is $\\textit{closed under the Lie bracket}$ (i.e.\, if each norm al space corresponds\, under the group's left action\, to a Lie subalgebra )\, then curvature adaptedness can be understood geometrically\, in terms of left translations. Incidentally\, our analysis offers a new case-indepe ndent proof of a well-known fact: every three-dimensional Lie group equipp ed with a bi-invariant semi-Riemannian metric has constant curvature.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /28/ END:VEVENT BEGIN:VEVENT SUMMARY:John Blackman (University of Liverpool) DTSTART:20221214T160000Z DTEND:20221214T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/29 DESCRIPTION:Title: Cutting Sequences and the p-adic Littlewood Conje cture\nby John Blackman (University of Liverpool) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nOne of the main them es of Diophantine approximation is the study of how well real numbers can be approximated by rational numbers. Classically\, a real number is define d to be well-approximable if the Markov constant is 0\, i.e. \\(M(x):=\\li m \\inf {q||qx||}=0\\). Otherwise\, the number is badly approximable\, wit h larger values of \\(M(x)\\) indicating worse rates of approximation. As a slight twist on this notion of approximability\, the \\(p\\)-adic Little wood Conjecture asks if -- given a prime \\(p\\) and a badly approximable number \\(x\\) -- one can always find a subsequence of \\(xp^k\\) such tha t the Markov constant of this sequence tends to \\(0\\)\, i.e. if \\(\\lim \\inf {M(xp^k)} =0\\).\n\nIn this talk\, I will outline a brief history o f the \\(p\\)-adic Littlewood Conjecture and discuss how hyperbolic geomet ry can be used to help understand the problem further. In particular\, I w ill discuss how one can represent integer multiplication of continued frac tions by replacing one triangulation of the hyperbolic plane with an alter native triangulation. Finally\, I will give a reformulation of pLC using i nfinite loops -- a family of objects that arise from this setting.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /29/ END:VEVENT BEGIN:VEVENT SUMMARY:Andy Hone (University of Kent) DTSTART:20230208T160000Z DTEND:20230208T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/30 DESCRIPTION:Title: Continued fractions from hyperelliptic curves \nby Andy Hone (University of Kent) as part of Selected Topics in Mathemat ics - Online Edition\n\n\nAbstract\nWe consider a family of nonlinear maps that are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus \\(g\\)\, as originally described by van d er Poorten. Using the connection with the classical theory of \\(J\\)-frac tions and orthogonal polynomials\, we show that in the simplest case \\(g= 1\\) this provides a straightforward derivation of Hankel determinant form ulae for the terms of a general Somos-\\(4\\) sequence\, which were found in particular cases by Chang\, Hu\, and Xin. We extend these formulae to t he higher genus case\, and prove that generic Hankel determinants in genus \\(2\\) satisfy a Somos-\\(8\\) relation. Moreover\, for all \\(g\\) we s how that the iteration for the continued fraction expansion is equivalent to a discrete Lax pair with a natural Poisson structure\, and the associat ed nonlinear map is a discrete integrable system\, connected with solution s of the infinite Toda lattice. If time permits\, we will also mention the link to (Stieltjes) \\(S\\)-fractions via contraction\, and a family of m aps associated with the Volterra lattice\, described in current joint work with John Roberts and Pol Vanhaecke.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /30/ END:VEVENT BEGIN:VEVENT SUMMARY:Iskander Aliev (Cardiff University) DTSTART:20230301T160000Z DTEND:20230301T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/31 DESCRIPTION:Title: Sparse integer points in rational polyhedra: boun ds for the integer Caratheodory rank\nby Iskander Aliev (Cardiff Unive rsity) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbs tract\nWe will give an overview of the recent results on sparse integer po ints (that is\, the integer points with a relatively large number of zero coordinates) in the rational polyhedra of the form \\(\\{x: Ax=b\, x\\geq 0\\}\\)\, where \\(A\\) is an integer matrix\, and \\(b\\) is an integer v ector. In particular\, we will discuss the bounds on the Integer Caratheod ory rank in various settings and proximity/sparsity transference results.\ n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /31/ END:VEVENT BEGIN:VEVENT SUMMARY:Nikolay Moshchevitin (Israel Institute of Technology (Technion)) DTSTART:20230308T160000Z DTEND:20230308T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/32 DESCRIPTION:Title: Geometry of Diophantine Approximation\nby Nik olay Moshchevitin (Israel Institute of Technology (Technion)) as part of S elected Topics in Mathematics - Online Edition\n\n\nAbstract\nWe discuss s ome classical and modern results related to the geometry of Diophantine Ap proximation\, in particular some multidimensional generalizations of conti nued fractions algorithm related to patterns of the best approximations. I mportant tools for the study of the properties of approximations are relat ed to irrationality measure functions. We will give a brief introduction i nto the theory and explain a recent conjecture by Schmidt and Summerer and its solution.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /32/ END:VEVENT BEGIN:VEVENT SUMMARY:Bob Connelly (Cornell University) DTSTART:20230315T160000Z DTEND:20230315T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/33 DESCRIPTION:Title: Global Rigidity of Braced Convex Polygons\nby Bob Connelly (Cornell University) as part of Selected Topics in Mathemati cs - Online Edition\n\n\nAbstract\nA framework in the plane is globally ri gid if any other realization of the framework with corresponding edges the same length is congruent. For example\, a collection of triangles placed end-to-end without overlap such that a bar connecting the first triangle to the last\, intersecting the interior of each triangle\, is globally ri gid. We would like to tell when a convex polygon with braces inside conne cting the vertices so that for \\(n\\) vertices there are \\(n-2\\) intern al braces\, then this framework is always globally rigid. But we can’t do that yet. However\, we have some interesting classes of braced convex polygonal frameworks that are always globally rigid.\n\n \n\nThis is join t work with Bob Connelly\, Bill Jackson\, Shin-ichi Tanagawa\, and Albert Zhen\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /33/ END:VEVENT BEGIN:VEVENT SUMMARY:Thomas Garrity (Williams College) DTSTART:20230322T160000Z DTEND:20230322T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/34 DESCRIPTION:Title: On Partition Numbers and Multi-dimensional Contin ued Fractions\nby Thomas Garrity (Williams College) as part of Selecte d Topics in Mathematics - Online Edition\n\n\nAbstract\nThis talk will lin k partition numbers from combinatorics with a certain multi-dimensional co ntinued fraction algorithm from number theory and dynamical systems.\n\nAn drew and Eriksson's Introduction to Integer Partitions starts with discuss ing Euler's identity\, Every number has as many integer partitions int o odd parts as into distinct parts. As they state\, this is quite sur prising if you have never seen it before. There are\, though\, many other equally if not more surprising partition identities. For all there are two basic questions. First\, how to even guess the existence of any potential partition identities. Then\, once a possible potential identity is conjec tured\, how to prove it.\n\nIn joint work with Bonanno\, Del Vigna and Iso la\, there was developed a link between traditional continued fractions an d the slow triangle map (a type of multi-dimensional continued fraction al gorithm) with integer partitions of numbers into two or three distinct par ts\, with multiplicity. These maps were initially introduced for number th eoretic reasons but have over the years exhibited many interesting dynamic al properties. In work with Wael Baalbaki\, we will see that the slow tria ngle map\, when extended to higher dimensions\, will provide a natural map (an almost internal symmetry) from the set of integer partitions to itsel f.\n\nThus we will allow us to create a new technique for generating any n umber of partition identities.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /34/ END:VEVENT BEGIN:VEVENT SUMMARY:Pavel Tumarkin (Durham University) DTSTART:20230329T150000Z DTEND:20230329T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/35 DESCRIPTION:Title: Farey graph and ideal tetrahedra\nby Pavel Tu markin (Durham University) as part of Selected Topics in Mathematics - Onl ine Edition\n\n\nAbstract\nWe construct a 3-dimensional analog of the Fare y tesselation and show that it inherits many properties of the usual 2-dim ensional Farey graph. As a by-product\, we get a 3-dimensional counterpart of the Ptolemy relation. The talk is based on an ongoing work joint with Anna Felikson\, Oleg Karpenkov and Khrystyna Serhiyenko.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /35/ END:VEVENT BEGIN:VEVENT SUMMARY:Egon Schulte (Northeastern University) DTSTART:20230405T150000Z DTEND:20230405T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/36 DESCRIPTION:Title: Skeletal Polyhedral Geometry and Symmetry\nby Egon Schulte (Northeastern University) as part of Selected Topics in Math ematics - Online Edition\n\n\nAbstract\nThe study of highly symmetric stru ctures in Euclidean \\(3\\)-space has a long and fascinating history traci ng back to the early days of geometry. With the passage of time\, various notions of polyhedral structures have attracted attention and have brought to light new exciting figures intimately related to finite or infinite gr oups of isometries. A radically different\, skeletal approach to polyhedra was pioneered by Grunbaum in the 1970's building on Coxeter's work. A pol yhedron is viewed not as a solid but rather as a finite or infinite period ic geometric edge graph in space equipped with additional polyhedral super -structure imposed by the faces. Since the mid 1970's there has been a lot of activity in this area. Much work has focused on classifying skeletal p olyhedra and complexes by symmetry\, with the degree of symmetry defined v ia distinguished transitivity properties of the geometric symmetry groups. These skeletal figures exhibit fascinating geometric\, combinatorial\, an d algebraic properties and include many new finite and infinite structures .\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /36/ END:VEVENT BEGIN:VEVENT SUMMARY:Valérie Berthé (IRIF) DTSTART:20230412T150000Z DTEND:20230412T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/37 DESCRIPTION:Title: Balanced words and symbolic dynamical systems \nby Valérie Berthé (IRIF) as part of Selected Topics in Mathematics - O nline Edition\n\n\nAbstract\nThe chairman assignment problem can be stated as follows: \\(k\\) states are assumed to form a union and each year a un ion chairman must be selected so that at any time the cumulative number of chairmen of each state is proportional to its weight.\nIt is closely rela ted to the (discrete) apportionment problem\, which has its origins in the question of allocating seats in the house of representatives in the Unite d States\, in a proportional way to the population of each state.\nThe ric hness of this problem lies in the fact that it can be reformulated both as a sequencing problem in operations research for optimal routing and sched uling\, and as a symbolic discrepancy problem\, in the field of word combi natorics\, where the discrepancy measures the difference between the numbe r of occurrences of a letter in a prefix of an infinite word and the expec ted value in terms of frequency of occurrence of this letter. \nWe will se e in this lecture how to construct infinite words with values in a finite alphabet having the smallest possible discrepancy\, by revisiting a constr uction due to R. Tijdeman in terms of dynamical systems.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /37/ END:VEVENT BEGIN:VEVENT SUMMARY:Jörg Thuswaldner (Montanuniversität Leoben) DTSTART:20230419T150000Z DTEND:20230419T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/38 DESCRIPTION:Title: Multidimensional continued fractions and symbolic codings of toral translations\nby Jörg Thuswaldner (Montanuniversit ät Leoben) as part of Selected Topics in Mathematics - Online Edition\n\n \nAbstract\nThe aim of this lecture is to find symbolic codings for transl ations on the $d$-dimensional torus that enjoy many of the well-known prop erties of Sturmian sequences (like low complexity\, balance of factors\, b ounded remainder sets of any scale). Inspired by the approach of G. Rauzy we construct such codings by the use of multidimensional continued fractio n algorithms that are realized by sequences of substitutions. This is join t work with V. Berthé and W. Steiner.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /38/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Gekhtman (University of Notre Dame) DTSTART:20230510T150000Z DTEND:20230510T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/40 DESCRIPTION:Title: Unified approach to exotic cluster structures in simple Lie groups\nby Michael Gekhtman (University of Notre Dame) as p art of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nWe p resent a construction for cluster charts in simple Lie groups compatible w ith Poisson structures in the Belavin-Drinfeld classification. The key ing redient is a birational Poisson map from the group to itself that transfor m a Poisson bracket associated with a nontrivial Belavin-Drinfeld data int o the standard one. It allows us to obtain a cluster chart as a pull-back of the Berenstein-Fomin-Zelevinsky cluster coordinates on the open double Bruhat cell. This is a joint work with M. Shapiro and A. Vainshtein.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /40/ END:VEVENT BEGIN:VEVENT SUMMARY:Nathan Uricchio (WPI) DTSTART:20230426T150000Z DTEND:20230426T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/41 DESCRIPTION:by Nathan Uricchio (WPI) as part of Selected Topics in Mathema tics - Online Edition\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /41/ END:VEVENT BEGIN:VEVENT SUMMARY:Ekaterina Shemyakova (University of Toledo) DTSTART:20231018T150000Z DTEND:20231018T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/42 DESCRIPTION:Title: On super cluster algebras based on super Plü \;cker and super Ptolemy relations\nby Ekaterina Shemyakova (Universit y of Toledo) as part of Selected Topics in Mathematics - Online Edition\n\ n\nAbstract\nI will speak about super exterior powers and our results on s uper analogs of Plü\;cker embedding for the Grassmann manifold.\n\nTh e problem was motivated by the search for the definition of super cluster algebras. Based on the obtained super Plü\;cker relations (which we h ave for the general case)\, we propose a super cluster structure for supe r Grassmannians \\(\\mathrm{Gr}_{2|0}(n|1)\\). The exchange graph structu re is now understood. \n\nWe show how to simplify the super Plü\;cke r relations for \\(\\mathrm{Gr}_{r|1}(n|1)\\)\, which can be seen as dual to $\\mathrm{Gr}_{n-r|0}(n|1)$. We also present how super Ptolemy relatio ns of Penner-Zeitlin for decorated super Teichmü\;ller space --- the basis of Musiker-Ovenhouse-Zhang's super cluster algebra definition --- ca n be re-written as super Plü\;cker relations.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /42/ END:VEVENT BEGIN:VEVENT SUMMARY:Derek Kitson (Mary Immaculate College) DTSTART:20231206T160000Z DTEND:20231206T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/43 DESCRIPTION:Title: Rigid graphs in dimension 3\nby Derek Kitson (Mary Immaculate College) as part of Selected Topics in Mathematics - Onli ne Edition\n\n\nAbstract\nA graph is rigid in \\(d\\)-\;dimensional Euc lidean space if there is an embedding of the vertices which admits no non& #45\;trivial edge-\;length preserving continuous motion. Rigid graphs i n dimensions \\(1\\) and \\(2\\) are characterised by simple counting rule s\, but currently no such rules are available in higher dimensional Euclid ean spaces. We will provide a gentle introduction to graph rigidity and re port on recent progress in characterising rigid graphs for a class of cyli ndrical normed spaces of dimension \\(3\\). This is joint work with Sean D ewar (University of Bristol).\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /43/ END:VEVENT BEGIN:VEVENT SUMMARY:Sophie Morier-Genoud (University of Reims) DTSTART:20231011T150000Z DTEND:20231011T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/44 DESCRIPTION:Title: q-Analogs of rational numbers and the Burau repre sentation of the braid group B3.\nby Sophie Morier-Genoud (University of Reims) as part of Selected Topics in Mathematics - Online Edition\n\n\n Abstract\nThe most popular \\(q\\)-\;analogs of numbers are certainly t he \\(q\\)-\;integers and the \\(q\\)-\;binomial coefficients of Gau ss which both appear in various areas of mathematics and physics. Classica l sequences of integers often have interesting \\(q\\)-\;analogs. With Valentin Ovsienko we recently suggested a notion of \\(q\\)-\;analogs f or rational numbers. Our approach is based on combinatorial properties and continued fraction expansions of the rationals. The subject can be develo ped in connections with various topics such as enumerative combinatorics\, cluster algebras\, homological algebra\, knots invariants... I will give an overview of the theory and present an application to the problem of cl assification of faithful specialisations of the Burau representation of B3 . This last part is joint work with V. Ovsienko and A. Veselov.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /44/ END:VEVENT BEGIN:VEVENT SUMMARY:Fran Burstall (University of Bath) DTSTART:20231129T160000Z DTEND:20231129T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/45 DESCRIPTION:Title: Isothermic surfaces and Noether's theorem\nby Fran Burstall (University of Bath) as part of Selected Topics in Mathemat ics - Online Edition\n\n\nAbstract\nIsothermic surfaces were intensively s tudied in the late 19th century and have seen a recent revival of interest due to links with soliton theory. In this talk\, I will descri be this classical theory and the modern integrable systems approach via a pencil of flat connections. I will explain how this connections arise from a variational characterisation of isothermic surfaces\, due to Bohle-Pete rs-Pinkall\, together with Noether's theorem. This gives a puzzling link t o the conservations laws for CMC surfaces discovered by Korevaar-Kusner-So lomon.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /45/ END:VEVENT BEGIN:VEVENT SUMMARY:Vladlen Timorin (Higher School of Economics) DTSTART:20231213T160000Z DTEND:20231213T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/46 DESCRIPTION:Title: Aperiodic points for outer billiards\nby Vlad len Timorin (Higher School of Economics) as part of Selected Topics in Mat hematics - Online Edition\n\n\nAbstract\nThis is a joint project with A. K anel-\;Belov\, Ph. Rukhovich\, and V. Zgurskii. A Euclidean outer billi ard on a convex figure in the plane is the map sending a point outside the figure to the other endpoint of a segment touching the figure at the midd le. Iterating such a process was suggested by J. Moser as a crude model of planetary motion. Polygonal outer billiards are arguably the principal ex amples of Euclidean piecewise rotations\, which serve as a natural general ization of interval exchange maps. They also found applications in electri cal engineering. Previously known rigorous results on outer billiards on r egular \\(N\\)-\;polygons are\, apart from trivial cases of \\(N =3\,4\,6\\)\, based on dynamical self-\;similarities (this approach was originated by S. Tabachnikov). Dynamical self-\;similarities have been found so far only for \\(N=5\,7\,8\,9\,10\,12\\). In his ICM 2022 address \, R. Schwartz asked whether outer billiard on the regular \\(N\\)-\ ;gon has an aperiodic orbit if \\(N\\) is not \\(3\\)\, \\(4\\)\, \\(6\\)< /q>. We answer this question in affirmative for \\(N\\) not divisible by \ \(4\\). Our methods are not based on self-\;similarity. Rather\, scisso r congruence invariants (including that of Sah-\;Arnoux-\;Fathi) pla y a key role.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /46/ END:VEVENT BEGIN:VEVENT SUMMARY:Victor Kleptsyn (Université de Rennes) DTSTART:20231108T160000Z DTEND:20231108T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/47 DESCRIPTION:Title: From the percolation theory to Fuchsian equations and Riemann-\;Hilbert problem\nby Victor Kleptsyn (Université de Rennes) as part of Selected Topics in Mathematics - Online Edition\n\n\nAb stract\nConsider the critical percolation problem on the hexagonal lattice : each of (tiny) hexagons is independently declared «\; open »\; or «\; closed »\; with probability (\\(1/2\\)) &mdash\; by a fa ir coin tossing. Assume that on the boundary of a simply connected domain four points A\,B\,C\,D are marked. Then either there exists an «\; op en »\; path\, joining AB and CD\, or there is a «\; closed &raqu o\; path\, joining AD and BC (one can recall the famous «\; Hex &raqu o\; game here).\n\n\n\n

\n

\n\nCardy's formula\, rigorously proved by S. Smirnov\, gives an explicit value of the limit of such percolation p robability\, when the same smooth domain is put onto lattices with smaller and smaller mesh. Though\, a next natural question is: what if more than four points are marked? And thus that there are more possible configuratio ns of open/closed paths joining the arcs? \n\n\n

\n

\n\nIn our join t work with M. Khristoforov we obtain the answer as an explicit integral f or the case of six marked points on the boundary\, passing through Fuchsia n differential equations\, Riemann surfaces\, and Riemann-\;Hilbert pro blem. We also obtain a generalisation of this answer to the case when one of the marked points is inside the domain (and not on the boundary).\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /47/ END:VEVENT BEGIN:VEVENT SUMMARY:Iván Rasskin (Aix-Marseille University) DTSTART:20231115T160000Z DTEND:20231115T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/48 DESCRIPTION:Title: On the arithmetic and geometric properties of reg ular polytopal sphere packings and their connection to knot theory\nby Iván Rasskin (Aix-Marseille University) as part of Selected Topics in Ma thematics - Online Edition\n\n\nAbstract\nhe Apollonian Circle Packing (AC P) is a classic geometric fractal with diverse applications across various domains\, particularly in number theory. This is due to its ability to be realized as an integral packing\, where the curvatures of all the circles are integers. The ACP is constructed iteratively\, beginning with an init ial packing whose combinatorial structure is encoded by a tetrahedron. By changing the initial configuration\, the ACP can be generalized for any po lyhedron. However\, not every polyhedron is integral in the sense that it can generate an integral packing. Moreover\, in higher dimensions\, not ev ery polytope is crystallographic\, meaning that it can generate an Apollon ian-like sphere packing. In this talk\, we will study the case of regular polytopes in any dimension to determine whether they are integral and crys tallographic. Additionally\, we will explore how the symmetry inherent in the polytope can be leveraged to extract special cross-sections of the pac kings. Furthermore\, we will demonstrate how a specific section of an orth oplicial/hyperoctahedral Apollonian sphere packing can be utilized as a ge ometric framework to establish an upper bound on a knot invariant for rati onal links.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /48/ END:VEVENT BEGIN:VEVENT SUMMARY:Herman Servatius (Worcester Polytechnic Institute) DTSTART:20231101T160000Z DTEND:20231101T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/49 DESCRIPTION:Title: Rigidity and movability of configurations in the projective plane\nby Herman Servatius (Worcester Polytechnic Institute ) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract \nConfigurations of points and lines in the plane have a long history. The Theorem of Pappus from the fourth century begins a classical theory that has been advanced by Desargues\, Pascal\, Cayley\, Steinitz\, Grassman and many others. The study of such objects and their generalizations has deep roots in algebra\, geometry\, topology and combinatorics.\n\nIn this talk we discuss recent work which is the result of regarding these classical s tructures as geometric constraint systems. The objects of interest then be come the topology\, geometry\, and parameterizations of the space of reali zations of a configuration. Some of the tools derive from those developed by civil and mechanical engineers in the analysis of the statics of struct ures and the kinematics of linkages.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /49/ END:VEVENT BEGIN:VEVENT SUMMARY:Pär Kurlberg (KTH) DTSTART:20231004T150000Z DTEND:20231004T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/50 DESCRIPTION:Title: Repulsion in number theory and physics\nby P är Kurlberg (KTH) as part of Selected Topics in Mathematics - Online Edit ion\n\n\nAbstract\nZeros of the Riemann zeta function and eigenvalues of q uantized chaotic Hamiltonians appears to have something in common. Namely \, they both seem to be ruled by random matrix theory and consequently sho uld exhibit "repulsion" in the sense that small gaps between elements are very rare. More mysteriously\, while zeros of different L-\;functions (i.e.\, generalizations of the Riemann zeta function) are "mostly independ ent" they also exhibit subtle repulsion effects on zeros of other L -\;functions.\n\nWe will give a survey of the above phenomena. Time pe rmitting we will also discuss repulsion between eigenvalues of "arithmetic Seba billiards"\, a certain singular perturbation of the Laplacian on the 3D torus \\(R^3/Z^3\\). The perturbation is weak enough to allow for ari thmetic features from the unperturbed system to be brought into play\, yet strong enough to probably induce repulsion.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /50/ END:VEVENT BEGIN:VEVENT SUMMARY:Jeffrey Shallit (University of Waterloo) DTSTART:20231122T160000Z DTEND:20231122T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/51 DESCRIPTION:Title: Proving results in combinatorics on words and num ber theory using a decidable logic theory\nby Jeffrey Shallit (Univers ity of Waterloo) as part of Selected Topics in Mathematics - Online Editio n\n\n\nAbstract\nDavid Hilbert's dream\, of a deterministic finite procedu re that could decide if a given theorem statement is true or false\, was k illed off by Gä\;del and Turing. Nevertheless\, there are some logical theories\, such as Presburger arithmetic\, that are decidable.\n\nIn this talk I will discuss one such theory\, Bü\;chi arithmetic\, and its im plementation in a computer system called Walnut. With this free software\, one can prove non-trivial theorems in combinatorics on words and number t heory\; it suffices to state the desired result in first-order logic\, typ e it into the system\, and wait. So far\, it has been used in over 70 pape rs in the literature\, and has even detected errors in some published pape rs.\n\nMy recent book\, The Logical Approach to Automatic Sequences\, published by Cambridge University Press\, has more details.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /51/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Esterov (London Institute for Mathematical Sciences) DTSTART:20231025T150000Z DTEND:20231025T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/52 DESCRIPTION:by Alexander Esterov (London Institute for Mathematical Scienc es) as part of Selected Topics in Mathematics - Online Edition\n\nAbstract : TBA\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /52/ END:VEVENT BEGIN:VEVENT SUMMARY:Nicholas Ovenhouse (Yale University) DTSTART:20240214T150000Z DTEND:20240214T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/53 DESCRIPTION:Title: Higher Continued Fractions from Dimer Models and Plane Partitions\nby Nicholas Ovenhouse (Yale University) as part of S elected Topics in Mathematics - Online Edition\n\n\nAbstract\nThere is a w ell-known relation between ordinary continued fractions and certain matrix products in \\(\\text{SL}(2\,\\mathbb{Z})\\). There is also a theorem of Schiffler and Canakci that the entries of these matrix products count the perfect matchings on certain planar graphs called ''snake graphs". Togethe r with Musiker\, Schiffler\, and Zhang\, we studied the enumeration of ''\ \(m\\)-\;dimer covers" on these snake graphs (these are combinatorial g eneralizations of perfect matchings)\, and obtained formulas in terms of p roducts of \\(\\text{SL}(m+1\,\\mathbb{Z})\\) matrices. This led to a defi nition of ''higher continued fractions". I will discuss these higher conti nued fractions\, their properties\, and their combinatorial interpretation s (including perfect matchings\, lattice paths\, plane partitions\, and mo re). Time permitting\, I will mention work-\;in-\;progress about \\( q\\)-\;analogs of these notions.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /53/ END:VEVENT BEGIN:VEVENT SUMMARY:Dmitry Badziahin (Univesity of Sydney) DTSTART:20240221T120000Z DTEND:20240221T130000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/54 DESCRIPTION:Title: Simultaneous Diophantine approximation on the Ver onese curve\nby Dmitry Badziahin (Univesity of Sydney) as part of Sele cted Topics in Mathematics - Online Edition\n\n\nAbstract\nMeasuring the s et of simultaneously well approximable points on manifolds is one of the m ost intricate problems in metric theory of Diophantine approximation. Unli ke the dual case of well approximable linear forms\, the results here are known to depend on a manifold. For example\, some of the manifolds do not contain simultaneously very well approximable points at all\, while for th e others the set of such points always has positive Hausdorff dimension. I n this talk\, we will closely look at the Veronese curve \\(\\{x\, x^2\, x ^3\, \\ldots\, x^n\\}\\)\, discuss what is known about the sets of simulta neously well approximable points on it and provide several new results. In particular\, for \\(n=3\\) we provide the Hausdorff dimension of the set of \\(x\\) such that \\(\\lambda_3(x) \\le \\lambda\\) where \\(\\lambda\\ le \\frac25\\) or \\(\\lambda\\ge \\frac79\\).\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /54/ END:VEVENT BEGIN:VEVENT SUMMARY:Eleonore Faber (University of Leeds) DTSTART:20240228T190000Z DTEND:20240228T200000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/55 DESCRIPTION:Title: Friezes and resolutions of plane curve singularit ies\nby Eleonore Faber (University of Leeds) as part of Selected Topic s in Mathematics - Online Edition\n\n\nAbstract\nConway‐\;Coxeter fri ezes are arrays of positive integers satisfying a determinantal condition\ , the so‐\;called diamond rule. \n Recently\, these combinatorial obj ects have been of considerable interest in representation theory\, since t hey encode cluster combinatorics of type A.\n\nIn this talk I will discuss a new connection between Conway‐\;Coxeter friezes and the combinator ics of a resolution of a complex curve singularity: via the beautiful rela tion between friezes and triangulations of polygons one can relate each fr ieze to the so‐\;called lotus of a curve singularity\, which was intr oduced by Popescu‐\;Pampu. \nThis allows to interpret the entries in the frieze in terms of invariants of the curve singularity\, and on the ot her hand\, we can see cluster mutations in terms of the desingularization of the curve. \nThis is joint work with Bernd Schober.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /55/ END:VEVENT BEGIN:VEVENT SUMMARY:Martin Henk (TU Berlin) DTSTART:20240306T150000Z DTEND:20240306T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/56 DESCRIPTION:Title: Polynomial bounds in Koldobsky's discrete slicing problem\nby Martin Henk (TU Berlin) as part of Selected Topics in Mat hematics - Online Edition\n\n\nAbstract\nIn 2013\, Koldobsky posed the pro blem to find a constant \\(d_n\\)\,\n depending only on the dimension \\(n \\)\, such that for any\n origin-symmetric convex body \\(K\\subset\\mathb b{R}^n\\) there exists an\n \\((n-1)\\)-dimensional linear subspace \\(H\\ subset\\mathbb{R}^n\\) with\n \\[\n |K\\cap\\mathbb{Z}^n| \\leq d_n\\\,|K \\cap H\\cap \\mathbb{Z}^n|\\\,\\text{vol}(K)^{\\frac 1n}.\n \\]\nIn this article we show that \\(d_n\\) is bounded from above by\n\\(c\\\,n^2 \\\,\\omega(n)/\\log(n)\\)\, where \\(c\\) is an absolute constant and \\( \\omega(n)\\) is\nthe flatness constant. Due to the recent best known uppe r bound on\n\\(\\omega(n)\\) we get a \\({c\\\,n^3\\log(n)^2}\\) bound on \\(d_n\\). This improves on former bounds. \n\n\n (Based on joi nt works with Ansgar Freyer.)\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /56/ END:VEVENT BEGIN:VEVENT SUMMARY:Will Traves (United States Naval Academy) DTSTART:20240313T150000Z DTEND:20240313T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/57 DESCRIPTION:Title: Incidence results defining plane curves\nby W ill Traves (United States Naval Academy) as part of Selected Topics in Mat hematics - Online Edition\n\n\nAbstract\nI'll explain Hermann Grassmann's approach to the geometry of curves. In the mid-1800's\, he characterized p oints on cubics using a clever incidence construction. I'll discuss ways t o extend Grassmann's results. In particular\, I will explain how to use a straightedge to find the ninth point of intersection of two cubics\, given just \\(8\\) points common to the two curves and one extra point on each cubic.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /57/ END:VEVENT BEGIN:VEVENT SUMMARY:Ilya Bogaevskii (University of Liverpool) DTSTART:20240320T150000Z DTEND:20240320T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/58 DESCRIPTION:Title: Discontinuous gradient ODEs\, trajectories in the minimal action problem\, and massive points in one cosmological model \nby Ilya Bogaevskii (University of Liverpool) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe gradient of a concave f unction is discontinuous vector field but has well-\;defined trajectori es. We formulate an existence and forward-\;uniqueness theorem and its generalisation for non-\;stationary case. Using the latter we construct trajectories in the minimal action problem and investigate how massive po ints appear. Their formation simulates the large-\;scale matter distrib ution in one of the simplest cosmological models based on the Burgers equa tion.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /58/ END:VEVENT BEGIN:VEVENT SUMMARY:Jarosław Kędra (University of Aberdeen) DTSTART:20241004T140000Z DTEND:20241004T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/59 DESCRIPTION:Title: Bi-invariant metric on groups\nby Jarosław K ędra (University of Aberdeen) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nA bi-invariant metric on a group \\(G\\) is a metric such that both the right and the left action of \\(G\\) on itsel f is by isometries. Examples of such metrics include the Hofer metric on t he group of Hamiltonian diffeomorphisms of a symplectic manifold\, the ref lection length on a Coxeter group\, the commutator length and many others mostly in group theory and dynamics. A particularly interesting example is the cancellation length on free groups\, which was first discovered by bi ologists investigating RNA folding.\n\nIn the talk\, I will discuss variou s examples\, present a sample of results and open problems.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /59/ END:VEVENT BEGIN:VEVENT SUMMARY:Boris Springborn (TU Berlin) DTSTART:20241018T140000Z DTEND:20241018T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/60 DESCRIPTION:Title: The hyperbolic geometry of numbers\nby Boris Springborn (TU Berlin) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nIn his 1880 thesis\, Markov classified the unimodul ar indefinite binary quadratic forms whose values on the integer lattice ( minus the origin) stay farthest away from zero. \nThis is closely linked t o the classification of the worst approximable irrational numbers. \nThe p rimary tool in this theory have always been continued fractions.\nWell-kno wn connections to hyperbolic geometry are based on the fact that continued fractions describe the symbolic dynamics of geodesics in the Farey triang ulation of the hyperbolic plane. \nThis talk\, however\, will be about a n ew geometric approach to the Markov theory that eliminates the complicated symbolic dynamics of continued fractions by considering the set of all id eal triangulations of the modular torus and not just the simplest and most symmetric one. \nIn the end\, the problem boils down to the question: How far can a geodesic that crosses a triangle stay away from the vertices? \ nThis geometric approach can also be used to classify the worst approximab le rational (sic!) numbers.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /60/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Miranda (UCLA) DTSTART:20241101T150000Z DTEND:20241101T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/61 DESCRIPTION:Title: Fully Flexible Periodic Polyhedral Surfaces\n by Robert Miranda (UCLA) as part of Selected Topics in Mathematics - Onlin e Edition\n\n\nAbstract\nA polyhedral surface has an $n$-dimensional flex if there exists a continuous family of realizations $\\{Q_t : t \\in [0\,1 ]^n\\}$ which are pairwise nonisomorphic. Gaifullin and Gaifullin showed t hat if a 2-periodic polyhedral surface in is homeomorphic to a plane\, the n it can have at most a 1-dimensional flex which preserves periodicity. Gl azyrin and Pak later found an example of a 2-periodic polyhedral surface\, not homeomorphic to a plane\, which has a full 3-dimensional flex which p reserves periodicity. In this talk\, I will present a new construction of a fully flexible 3-periodic polyhedral surface using the universality of p olyhedral linkages and discuss generalizations to general periodic polyhed ral surfaces and higher dimensions.\n\nThis talk is based on joint work wi th Alexey Glazyrin (University of Texas Rio Grande Valley) and Igor Pak (U niversity of California Los Angeles)\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /61/ END:VEVENT BEGIN:VEVENT SUMMARY:Esther Banaian (UC Riverside) DTSTART:20241115T150000Z DTEND:20241115T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/62 DESCRIPTION:Title: Skein relations and bases for cluster algebras fr om punctured surfaces\nby Esther Banaian (UC Riverside) as part of Sel ected Topics in Mathematics - Online Edition\n\n\nAbstract\nCluster algebr as are commutative rings with a set of recursively-defined generators. Man y cluster algebras with desirable properties arise from a surface with mar ked points (S\,M) in the sense that they can be realized geometrically thr ough the decorated Teichmüller space of (S\,M). Kantarcı Oğuz-Yıldır ım and Pilaud-Reading-Schroll have recently exhibited a method to use the set of order ideals of a poset to give a direct formula for any generator of a cluster algebra of surface type. We use this construction to give "s kein relations"\, which will be multiplication formulas for elements of th e cluster algebra which arise from resolving intersections of the correspo nding curves on the surface. By working with surfaces which could have int ernal marked points\, called "punctures"\, we generalize previously known skein relations from Musiker-Schiffler-Williams and Canakci-Schiffler\, wh o largely only work in unpunctured surfaces. A corollary of our results is the ability to construct bases for cluster algebras from punctured surfa ces. This talk is based on joint work with Wonwoo Kang and Elizabeth Kelle y.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /62/ END:VEVENT BEGIN:VEVENT SUMMARY:Khrystyna Serhiyenko (University of Kentucky) DTSTART:20241122T150000Z DTEND:20241122T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/63 DESCRIPTION:Title: SL\\(_k\\) tilings and paths in \\(\\mathbb{Z}^k\ \)\nby Khrystyna Serhiyenko (University of Kentucky) as part of Select ed Topics in Mathematics - Online Edition\n\n\nAbstract\nAn SL\\(_k\\) fr ieze is a bi-\;infinite array of integers where adjacent entries satisf y a certain diamond rule. \nSL\\(_2\\) friezes were introduced and studie d by Conway and Coxeter. \nLater\, these were generalized to infinite matr ix-\;like structures called tilings as well as higher values of \\(k\\) . \nA recent paper by Short showed a bijection between bi-\;infinite pa ths of reduced rationals in the Farey graph and SL\\(_2\\) tilings. \nWe e xtend this result to higher \\(k\\) by constructing a bijection between SL \\(_k\\) tilings and certain pairs of bi-\;infinite strips of vectors i n \\(Z^k\\) called paths. \nThe key ingredient in the proof is the relatio n to Pluecker friezes and Grassmannian cluster algebras. \nAs an applicati on\, we obtain results about periodicity\, duality\, and positivity for ti lings.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /63/ END:VEVENT BEGIN:VEVENT SUMMARY:Bernd Schulze (Lancaster University) DTSTART:20241206T150000Z DTEND:20241206T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/64 DESCRIPTION:Title: A differential approach to Maxwell-Cremona liftin gs\nby Bernd Schulze (Lancaster University) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nIn 1864\, James Clerk Maxw ell introduced a link between self-stressed frameworks in the plane and pi ecewise linear liftings to 3-space. This connection has found numerous app lications in areas such as rigidity theory\, discrete and computational ge ometry\, control theory and structural engineering. While there are some g eneralisations of this theory to liftings of d-complexes in d-space\, exte nsions for liftings of frameworks in d-space for d at least 3 have been mi ssing. In this talk we introduce differential liftings on general graphs u sing differential forms associated with the elements of the homotopy group s of the complements to the frameworks. Such liftings play the role of int egrands for the classical notion of liftings for planar frameworks. These differential liftings have a natural extension to self-stressed frameworks in higher dimensions. As a result we generalise the notion of classical l iftings to both graphs and multidimensional k-complexes in d-space (k=2\, …\,d). This is joint work with Oleg Karpenkov\, Fatemeh Mohammadi and C hristian Mueller.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /64/ END:VEVENT BEGIN:VEVENT SUMMARY:Jeffrey Vaaler (University of Texas at Austin) DTSTART:20250213T150000Z DTEND:20250213T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/65 DESCRIPTION:Title: Sums of small fractional parts and a problem of L ittlewood\nby Jeffrey Vaaler (University of Texas at Austin) as part o f Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nWe prove estimates for certain partial sums of which the simplest nontrivial exampl es are\n\\[\n\\sum\\limits_{n=1}^N \\frac{1}{||\\alpha n||\\\,||\\beta n|| }\,\\ \\text{ and }\\ \\sum\\limits_{m=1}^M\\sum\\limits_{n=1}^N \\frac{1} {||\\alpha m+\\beta n||}.\n\\]\nHere $\\alpha$ and $\\beta$ are real numbe rs\, and $||x||$ is the distance from the real number $x$ to the nearest i nteger. Such estimates are somewhat related to a notorious open problem of Littlewood: is it true that for all pairs of real numbers $\\alpha$ and $ \\beta$ we have\n\\[\n\\lim\\limits_{n\\to\\infty}\\inf n||\\alpha n||\\\, ||\\beta n|| =0\\\,?\n\\]\nWe also consider estimates for more general sum s that contain products of many linear forms in many variables. If time pe rmits we also discuss an analogue of such problems in function fields. Thi s is joint work with Thá\;i Hoà\;ng Lê\;.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /65/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam Evans (Loughborough University) DTSTART:20241025T140000Z DTEND:20241025T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/66 DESCRIPTION:Title: Arithmetic and geometry of Markov polynomials \nby Sam Evans (Loughborough University) as part of Selected Topics in Mat hematics - Online Edition\n\n\nAbstract\nMarkov polynomials are the Lauren t-polynomial solutions of the Markov equation\n$$X^2+Y^2+Z^2=aXYZ\,$$\nwhi ch are the results of the cluster mutations applied to the initial triple $(x\,y\,z)$.\n\nThey were first discussed by Propp\, who proved that their coefficients are non-negative integers. This result relates to an interes ting combinatorial interpretation of Markov numbers\, related to perfect m atchings on snake graphs.\n\nWe present more results and formulate new con jectures about the specific coefficients that appear in these Markov polyn omials. Whilst many of the results remain conjecture\, based on observatio ns from numerical investigations\, in specific cases the results can be pr oven. In particular\, in subsets of Markov polynomials such as the Fibona cci polynomials\; in the sense of Caldero and Zelevinsky.\n\nThe talk is b ased on the ongoing joint work with Alexander Veselov and Brian Winn\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /66/ END:VEVENT BEGIN:VEVENT SUMMARY:Miloslav Torda (University of Liverpool) DTSTART:20241129T150000Z DTEND:20241129T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/67 DESCRIPTION:Title: Geometric Perspectives on the Crystallization of Molecular Crystals: Crystallographic Symmetry Group Packings\, Uniform Tes sellations\, and Molecular Frameworks\nby Miloslav Torda (University o f Liverpool) as part of Selected Topics in Mathematics - Online Edition\n\ n\nAbstract\nThe mathematical theory of crystallization is still in its ea rly stages\, even for simple mono-\;atomic systems. Molecular systems p resent additional challenges due to their intermolecular interactions. How ever\, by focusing on molecules with inherent symmetries\, the problem of periodic ground state formation in the thermodynamic limit becomes more tr actable. In this talk\, we consider an additive Hamiltonian at zero temper ature\, involving pairwise atom-\;atom interaction potentials and cryst allographic symmetry groups&ndash\;discrete isometry groups of Euclidean s pace that include a lattice subgroup. We begin by investigating two-\;d imensional molecular systems where the pairwise interaction potential cons ists of repulsive and dispersion attraction terms. By examining the denses t packings of regular polygons within wallpaper groups\, Archimedean circl e packings\, and their corresponding dual frameworks\, we formulate a conj ecture regarding the ground state configurations of molecules with six- \;fold rotational symmetry. We then consider two-\;dimensional molecula r salt frameworks composed of negatively charged nodes connected by positi vely charged triangular linkers. For these purely electrostatic systems\, we enumerate possible ground states via sums of integer reciprocals. Build ing on these observations\, we extend our considerations to tetrahedral sa lt frameworks and propose that the ground states of these systems can be r epresented by an absolutely symmetric quadratic form. This work is in prog ress jointly with Roland Pú\;č\;ek (University of Jena) and Andrew I Cooper (University of Liverpool).\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /67/ END:VEVENT BEGIN:VEVENT SUMMARY:Anthony Nixon (Lancaster University) DTSTART:20250227T150000Z DTEND:20250227T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/68 DESCRIPTION:Title: Stable cuts\, NAC-colourings and flexible realisa tions of graphs\nby Anthony Nixon (Lancaster University) as part of Se lected Topics in Mathematics - Online Edition\n\n\nAbstract\nA (2-\;dim ensional) realisation of a graph G is a pair \\((G\,p)\\)\, where \\(p\\) maps the vertices of \\(G\\) to \\(\\mathbb{R}^2\\). A realisation is flex ible if it can be continuously deformed while keeping the edge lengths fix ed\, and rigid otherwise. Similarly\, a graph is flexible if its generic r ealisations are flexible\, and rigid otherwise. We show that a minimally r igid graph has a flexible realisation with positive edge lengths if and on ly if it is not a 2-\;tree. This confirms a conjecture of Grasegger\, L egersky and Schicho. Our proof is based on a characterisation of graphs wi th \\(n\\) vertices and \\(2n-3\\) edges and without stable cuts due to Le and Pfender. We also strengthen a result of Chen and Yu\, who proved that every graph with at most \\(2n-4\\) edges has a stable cut\, by showing t hat every flexible graph has a stable cut. Additionally\, we investigate t he number of NAC-\;colourings in various graphs. A NAC-\;colouring i s a type of edge colouring introduced by Grasegger\, Legersky and Schicho\ , who showed that the existence of such a colouring characterises the exis tence of a flexible realisation with positive edge lengths. \n\nThis is jo int work with Clinch\, Garamvolgyi\, Haslegrave\, Huynh and Legersky.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /68/ END:VEVENT BEGIN:VEVENT SUMMARY:Maxim Arnold (UT Dallas) DTSTART:20250313T150000Z DTEND:20250313T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/69 DESCRIPTION:Title: Circle patterns and ideal polygon folding\nby Maxim Arnold (UT Dallas) as part of Selected Topics in Mathematics - Onli ne Edition\n\n\nAbstract\nFolding of the ideal polygon in its \\(j\\)&ndas h\;th vertex reflects the vertex in the corresponding short diagonal. We s how that compositions of such foldings along any Coxeter element provide L iouville integrable system on the moduli space of ideal polygons. This res ult also provides integrability for Shramm circle patterns. This is joint work with Anton Izosimov.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /69/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikhail Gabdullin (University of Illinois) DTSTART:20250320T150000Z DTEND:20250320T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/70 DESCRIPTION:Title: Primes with small primitive roots\nby Mikhail Gabdullin (University of Illinois) as part of Selected Topics in Mathemat ics - Online Edition\n\n\nAbstract\nLet \\(\\delta(p)\\) tend to zero arbi trarily slowly as \\(p\\to\\infty\\). We exhibit an explicit set \\(\\math cal{S}\\) of primes \\(p\\)\, defined in terms of simple functions of the prime factors of \\(p-1\\)\, for which the least primitive root of \\(p\\) is at most \\( p^{1/4-\\delta(p)}\\) for all \\(p\\in \\mathcal{S}\\)\, w here \\(\\#\\{p\\leq x: p\\in \\mathcal{S}\\} \\sim \\pi(x)\\) as \\(x\\to \\infty\\). This is a joint work with Kevin Ford and Andrew Granville.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /70/ END:VEVENT BEGIN:VEVENT SUMMARY:Amanda Burcroff (Harvard University) DTSTART:20250410T140000Z DTEND:20250410T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/71 DESCRIPTION:Title: Combining Combinatorics and Mirror Symmetry in Cl uster Algebra Positivity\nby Amanda Burcroff (Harvard University) as p art of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nThe theory of cluster algebras gives a combinatorial framework for understandi ng the previously opaque nature of certain algebraic and geometric spaces. Cluster algebras are celebrated for their intriguing positivity propertie s\, which unify positivity phenomena in many areas of math and physics. T wo distinct proofs of this positivity have emerged\, one combinatorial and the other using scattering diagrams from mirror symmetry. Combining thes e approaches\, we give a directly computable\, manifestly positive\, and e lementary (yet highly nontrivial) formula describing generalized cluster s cattering diagrams in rank \\(2\\). Using this\, we prove the Laurent pos itivity of generalized cluster algebras of all ranks\, resolving a conject ure of Chekhov and Shapiro from 2014. This is joint work with Kyungyong L ee and Lang Mou.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /71/ END:VEVENT BEGIN:VEVENT SUMMARY:Shin-ichi Tanigawa (University of Tokyo) DTSTART:20250417T140000Z DTEND:20250417T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/72 DESCRIPTION:Title: Identifying Generic Points from Non-Generic Measu rements\nby Shin-ichi Tanigawa (University of Tokyo) as part of Select ed Topics in Mathematics - Online Edition\n\n\nAbstract\nThe generic globa l rigidity characterization by Gortler\, Healy\, and Thurston is one of th e most significant results in graph rigidity theory. In particular\, this characterization implies that global rigidity is a generic property of gra phs: either every generic realization of a graph in \\(d\\)-\;space is globally rigid\, or none of them are. Although a few variations are known\ , our understanding of its extendability to other rigidity models remains limited.\n\nIn this talk\, we examine the generic rigidity problem within the framework of the point identifiability problem\, by Cruickshank\, Moha mmadi\, Nixon\, and Tanigawa. We present several successful examples in \\ (\\mathscr{l}_p\\)-\;rigidity and tensor completion and discuss unsolve d problems.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /72/ END:VEVENT BEGIN:VEVENT SUMMARY:Luke Jeffreys (University of Bristol) DTSTART:20250327T150000Z DTEND:20250327T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/73 DESCRIPTION:Title: On the complement of the Lagrange spectrum in the Markov spectrum\nby Luke Jeffreys (University of Bristol) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nInitially s tudied by Markov around \\(1880\\)\, the Lagrange and Markov spectra are c omplicated subsets of the real line that play a crucial role in the study of Diophantine approximation and the study of binary quadratic forms. In t he \\(1920\\)s\, Perron gave an amazingly useful description of the spectr a in terms of continued fractions and\, in the \\(1960\\)s\, Freiman demon strated that the Lagrange spectrum is a strict subset of the Markov spectr um. It still remains a difficult task to find points in the complement of the Lagrange spectrum within the Markov spectrum and modern research is fo cussed on further developing our understanding of this complement.\n\nIn t his talk\, I will introduce these spectra\, discussing the historical resu lts above\, and speak about recent works with Harold Erazo\, Carlos Matheu s and Carlos Gustavo Moreira finding new points in the complement and obta ining better lower bounds for its Hausdorff dimension.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /73/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Pratoussevitch (University of Liverpool) DTSTART:20250424T140000Z DTEND:20250424T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/74 DESCRIPTION:Title: Farey Bryophylla\nby Anna Pratoussevitch (Uni versity of Liverpool) as part of Selected Topics in Mathematics - Online E dition\n\n\nAbstract\nThe construction of the Farey tessellation in the hy perbolic plane starts with a finitely generated group of symmetries of an ideal triangle and induces a remarkable fractal structure on the boundary of the hyperbolic plane\, encoding every element by the continued fraction related to the structure of the tessellation. The problem of finding a ge neralisation of this construction to the higher dimensional hyperbolic spa ces has remained open for many years. In this paper we make the first step s towards a generalisation in the three-\;dimensional case. We introduc e conformal bryophylla\, a class of subsets of the boundary of the hyperbo lic \\(3\\)-\;space which possess fractal properties similar to the Far ey tessellation. We classify all conformal bryophylla and study the proper ties of their limiting sets. This is joint work with Oleg Karpenkov.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /74/ END:VEVENT BEGIN:VEVENT SUMMARY:Michail Zhitomirskii DTSTART:20250926T160000Z DTEND:20250926T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/75 DESCRIPTION:Title: Local Classification Problems with Functional Mod uli\nby Michail Zhitomirskii as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nI will discuss the class of local classi fication problems\, including classification of vector distributions\, Rie mannian metrics\, and real hypersurfaces in \\(\\mathbb{C}^n\\)\, where th e functional dimension of the space of objects is bigger than that of the transformation group\, unlike the classification problem of singularity th eory where it is not so. I will explain that combining a coordinate-free a pproach with normal forms gives a nice explanation of known results and ma ny new results.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /75/ END:VEVENT BEGIN:VEVENT SUMMARY:Rebecca Sheppard (University of Liverpool) DTSTART:20251008T150000Z DTEND:20251008T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/76 DESCRIPTION:Title: Not your Usual Circle: Geometry on the Integer Gr id\nby Rebecca Sheppard (University of Liverpool) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nInteger geometry exp lores objects whose vertices lie on the integer lattice \\(\\mathbb{Z}^2\\ )\, with congruence defined by lattice-preserving affine transformations. In this project\, I introduced remarkable geometric objects called integer circles: discrete analogues of Euclidean circle. These objects challenge our geometric intuition regarding circles. Unlike their classical counterp arts\, integer circles are unbounded\, exhibit nontrivial arithmetic struc ture\, and possess positive density in the plane.\n\nIn this talk\, I will define integer circles\, illustrate their unusual behaviour\, and demonst rate how to rigorously compute their densities and intersection patterns.\ n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /76/ END:VEVENT BEGIN:VEVENT SUMMARY:James Dolan (University of Liverpool) DTSTART:20251029T160000Z DTEND:20251029T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/77 DESCRIPTION:Title: Integer Angles of Integer Polygons\nby James Dolan (University of Liverpool) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nIn 2008\, the first formula expressing con ditions on the geometric continued fractions for lattice angles of triangl es was derived\, while the cases of n-gons for \\(n > 3\\) remained unreso lved. In this talk\, we introduce an integer geometric analogue to the cla ssical sum of interior angles of a polygon theorem that will act as an extension to the above result in the \\(n>3\\) cases. I first will fra me historical contributions in this area by drawing comparison to their Eu clidean counterparts. This will provide background for a simplified overvi ew of the main results for the \\(n>3\\) case\, introducing novel notions in integer geometry such as chord curvature. Finally\, I will briefly touc h on the consequences of this work within the field of toric singularities .\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /77/ END:VEVENT BEGIN:VEVENT SUMMARY:Thọ Nguyễn Phước (University of Ostrava) DTSTART:20251105T160000Z DTEND:20251105T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/78 DESCRIPTION:Title: On a Theorem of Nathanson on Diophantine Approxim ation\nby Thọ Nguyễn Phước (University of Ostrava) as part of S elected Topics in Mathematics - Online Edition\n\n\nAbstract\nIn 1974\, M. B. Nathanson proved that every irrational number \\(\\alpha\\) represente d by a simple continued fraction with infinitely many elements greater tha n or equal to \\(k\\) is approximable by an infinite number of rational nu mbers \\(p/q\\) satisfying \\(|\\alpha-p/q| < 1/(\\sqrt{k^2 + 4}q^2) \\). In this talk we refine this result.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /78/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrei Zabolotskii (The Open University) DTSTART:20251119T160000Z DTEND:20251119T170000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/79 DESCRIPTION:Title: Friezes\, Cluster Algebras\, and the Poly Phenome non\nby Andrei Zabolotskii (The Open University) as part of Selected T opics in Mathematics - Online Edition\n\n\nAbstract\nCoxeter friezes are r elated to closed paths in the Farey graph and triangulated polygons\, as w ell as to the most basic and important examples of cluster algebras. Clust er algebras are certain algebras of rational functions\, which actually tu rn out to consist only of Laurent polynomials ‒\; a surprising fact known as the Laurent phenomenon. We will introduce all these objects usin g an interactive d emonstration\, outline the connections between them and introduce a su rprising phenomenon in cluster algebras: specialising variables in a speci fic controlled way turns Laurent polynomials into (non-Laurent) polynomial s.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /79/ END:VEVENT BEGIN:VEVENT SUMMARY:Konstantin Khanin (University of Toronto) DTSTART:20251211T130000Z DTEND:20251211T140000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/80 DESCRIPTION:Title: Renormalization and Rigidity in Dynamical Systems \nby Konstantin Khanin (University of Toronto) as part of Selected Top ics in Mathematics - Online Edition\n\n\nAbstract\nRenormalization ideas w ere introduced in dynamics in the late 1970s. By now\, renormalization is one of the most important methods of asymptotic analysis in the theory of dynamical systems. This talk serves as an introduction to dynamical renorm alization. I'll also discuss closely connected rigidity theory and formula te some open problems. No previous knowledge of renormalization will be as sumed.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /80/ END:VEVENT BEGIN:VEVENT SUMMARY:Jane Coons (Worcester Polytechnic Institute) DTSTART:20260213T150000Z DTEND:20260213T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/81 DESCRIPTION:Title: Likelihood Geometry of Brownian Motion Tree Model s\nby Jane Coons (Worcester Polytechnic Institute) as part of Selected Topics in Mathematics - Online Edition\n\n\nAbstract\nBrownian motion tre e models are used to describe the evolution of a continuous trait along a phylogenetic tree under genetic drift. Such a model is obtained by placing linear constraints on a mean-zero multivariate Gaussian distribution acco rding to the topology of the underlying tree. We investigate the enumerati ve geometry of the standard and dual maximum likelihood estimation problem s in these models. In particular\, we study the number of complex critical points of the log-likelihood and dual log-likelihood functions\, known as the ML-degree and dML-degree\, respectively. We use the toric geometry of Brownian motion tree models to give a formula for the dML-degree for all trees. We also prove a formula for the ML-degree of a star tree and show t hat for general trees\, the ML-degree does not depend on the location of t he root.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /81/ END:VEVENT BEGIN:VEVENT SUMMARY:Dirk Siersma (Utrecht University) DTSTART:20260220T150000Z DTEND:20260220T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/82 DESCRIPTION:Title: Distance and critical points on PL-manifolds\ nby Dirk Siersma (Utrecht University) as part of Selected Topics in Mathem atics - Online Edition\n\n\nAbstract\nThere is long history in the relatio n between the critical points of distance function and concur- rent normal s to a submanifold in Euclidean space. The study of caustics and counting the number of normals play a important role. In this talk we will give a g eneral approach to the study of crit- ical points of the distance function to a PL submanifold X. Examples are: polygons in the plane and in space a nd polygonal surfaces in 3-space (not necessarily convex)\, etc. What is t he relation between normals and critical points ? Are generic singularitie s Morse and if so what is the index ? We will discuss the bifurcation set and show that for a knotted closed PL-curve there are at least 10 concurre nt normals. Also for a convex simple polytope there is a point at least 10 concurrent normals. What can be said about the ED-degree?\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /82/ END:VEVENT BEGIN:VEVENT SUMMARY:Camilla Hollanti (Aalto University) DTSTART:20260313T150000Z DTEND:20260313T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/83 DESCRIPTION:Title: Well-rounded lattices and applications to securit y\nby Camilla Hollanti (Aalto University) as part of Selected Topics i n Mathematics - Online Edition\n\n\nAbstract\nI will give a brief introduc tion to well-rounded lattices and to their utility in (post-quantum) secur ity. We will see how the lattice theta series naturally arises in these co ntexts and discuss its connections to well-rounded lattices.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /83/ END:VEVENT BEGIN:VEVENT SUMMARY:Lev Birbrair (Universidade Federal do Ceará) DTSTART:20260417T140000Z DTEND:20260417T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/84 DESCRIPTION:by Lev Birbrair (Universidade Federal do Ceará) as part of Se lected Topics in Mathematics - Online Edition\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /84/ END:VEVENT BEGIN:VEVENT SUMMARY:Sagar Kalane (The Institute of Mathematical Sciences (IMSc)) DTSTART:20260508T140000Z DTEND:20260508T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/85 DESCRIPTION:by Sagar Kalane (The Institute of Mathematical Sciences (IMSc) ) as part of Selected Topics in Mathematics - Online Edition\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /85/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexey Ustinov (HSE University) DTSTART:20260320T150000Z DTEND:20260320T160000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/86 DESCRIPTION:Title: On the periodicity of Somos sequences\nby Ale xey Ustinov (HSE University) as part of Selected Topics in Mathematics - O nline Edition\n\n\nAbstract\nFor integer \\(k\\geq4 \\) Somos--\\(k \\) se quence is a sequence generated by quadratic recurrence relation of the for m $$s_{n+k}s_n=\\sum_{j=1}^{[k/2]}\\alpha_js_{n+k-j}s_{n+j}\,$$ where \\(\ \alpha_j \\) are constants and \\(s_0 \, \\dots\, s_{k-1} \\) are initial data. Among them exist an important class of sequences with many propertie s. This class consists of\nfinite rank sequences. \nThe sequence \ \(\\{s_n\\}_{n=-\\infty}^\\infty \\) has a (finite) rank \\(r \\) if max imal rank of two infinite matices\n$$\\left.\\vphantom{\\sum}(s_{m+n}s_{m- n})\\right|_{m\,n=-\\infty}^\\infty\,\\qquad \\left.\\vphantom{\\sum}(s_{m +n+1}s_{m-n})\\right|_{m\,n=-\\infty}^\\infty$$\nis \\(r \\). If \\(r=2 \\ ) then general term of Somos sequence can be expressed in terms of ellipti c function. One can consider a general finite rank sequence as a sequence admitting more complicated addition theorem.\n\nPresumably the following p roperties are more or less equivalent: finitness of the rank\, Laurent phe nomenon\, periodicity \\(\\pmod N \\)\, solvability in theta-functions. Th e talk will be devoted to periodicity \\(\\pmod N \\) of general integer f inite rank sequences.\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /86/ END:VEVENT BEGIN:VEVENT SUMMARY:Yefei Ma (Université de Montpellie) DTSTART:20260501T140000Z DTEND:20260501T150000Z DTSTAMP:20260404T110825Z UID:SelectedTopics-Liverpool/87 DESCRIPTION:by Yefei Ma (Université de Montpellie) as part of Selected To pics in Mathematics - Online Edition\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SelectedTopics-Liverpool /87/ END:VEVENT END:VCALENDAR