For rings of integers in an extension of number fields there are c lassical methods for detecting ramification and for identifying ramificati on as being tame or wild. Noether's theorem characterizes tame ramificatio n in terms of a normal basis and tame ramification can also be detected vi a the surjectivity of the norm map. We take the latter fact and use the Ta te cohomology spectrum to detect wild ramification in the context of commu tative ring spectra. I will discuss several examples in the context of top ological K-theory and modular forms.
\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Du Pei (Harvard) DTSTART:20210526T130000Z DTEND:20210526T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/19 DESCRIPTION:Title: Hidden algebraic structures in geometry from fivebranes \nby Du Pei (Harvard) as part of Sheffield Pure Maths Colloquia\n\n\nA bstract\nThe existence of quantum field theories in higher dimensions pred icts many hidden algebraic structures in geometry and topology. In this ta lk\, I will survey some recent developments where such algebraic structure s lead to new insights into 1) the quantization of moduli spaces of Higgs bundles\, 2) the categorification of quantum invariants of 3-manifolds\, a nd 3) novel types of TQFTs in four dimensions.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Kurinczuk (University of Sheffield) DTSTART:20211103T140000Z DTEND:20211103T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/20 DESCRIPTION:Title: Local Langlands in families for classical groups\nb y Robert Kurinczuk (University of Sheffield) as part of Sheffield Pure Mat hs Colloquia\n\n\nAbstract\nThe conjectural local Langlands correspondence connects representations of p-adic groups to certain representations of G alois groups of local fields called Langlands parameters. In recent joint work with Dat\, Helm\, and Moss\, we have constructed moduli spaces of La nglands parameters over Z[1/p] and studied their geometry. We expect this geometry is reflected in the representation theory of the p-adic group. Our main conjecture “local Langlands in families” describes the GIT qu otient of the moduli space of Langlands parameters in terms of the centre of the category of representations of the p-adic group generalising a theo rem of Helm-Moss for GL(n). I will give an introduction to this picture a nd explain how after inverting the "non-banal primes" one can prove this c onjecture for the local Langlands correspondence for classical groups of A rthur and others.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Ananyo Dan (University of Sheffield) DTSTART:20211110T140000Z DTEND:20211110T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/21 DESCRIPTION:Title: McKay correspondence for isolated Q-Gorenstein singular ities\nby Ananyo Dan (University of Sheffield) as part of Sheffield Pu re Maths Colloquia\n\n\nAbstract\nThe McKay correspondence is a (natural) correspondence between the (non-trivial) irreducible representations of a finite subgroup G of $SL(2\,\\C)$ and the irreducible components of the ex ceptional divisor of a minimal resolution of the associated quotient singu larity $\\C^2//G$. A geometric construction for this correspondence was gi ven by González-Sprinberg and Verdier\, who showed that the two sets also correspond bijectively to the set of indecomposable reflexive modules on the quotient singularity. This was generalized to higher dimensional quoti ent singularities (i.e.\, quotient of $\\C^n$ by a finite subgroup of $SL( n\,\\C)$) by Ito-Reid\, where the above sets were substituted by certain s maller subsets. It was further generalized to more general quotient singul arities by Bridgeland-King-Reid\, Iyama-Wemyss and others\, using the lang uage of derived categories. In this talk\, I will survey past results and discuss what happens for the isolated Q-Gorenstein singularities case (not necessarily a quotient singularity). If time permits\, I will discuss app lications to Matrix factorization. This is joint work in progress with J. F. de Bobadilla and A. Romano-Velazquez.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Oscar Randal-Williams (University of Cambridge) DTSTART:20211124T140000Z DTEND:20211124T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/22 DESCRIPTION:Title: Homeomorphisms of $\\R^d$\nby Oscar Randal-Williams (University of Cambridge) as part of Sheffield Pure Maths Colloquia\n\n\n Abstract\nThe group Top(d) of homeomorphisms of d-dimensional Euclidean sp ace is a basic object in geometric topology\, with its quotient Top(d)/O(d ) by the subgroup of linear isometries completely controlling the differen ce between smooth and topological manifolds in all dimensions (except 4). I will explain some of the classical methods for studying the topology of this group\, and report on some recent advances.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Sven Raum (Stockholm University) DTSTART:20220511T123000Z DTEND:20220511T133000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/23 DESCRIPTION:Title: Simple operator algebras associated with groups and gro up-like structures\nby Sven Raum (Stockholm University) as part of She ffield Pure Maths Colloquia\n\n\nAbstract\nOne of the original motivations of Murray and von Neumann introducing operator algebras was to study the unitary representation theory of groups. This naturally leads to the quest ion of studying building blocks of representation theory\, that is simple operator algebras associated with groups. From a modern point of view\, no t only groups but also other group-like structures such as groupoids shoul d be investigated. This talk introduces the audience to group and groupoid operator algebras and tells the story of how our point of view on their s implicity changed dramatically over the past 10 years. At the end of the t alk\, I will present some results on simple groupoid C*-algebras that were obtained in joint work with Kennedy\, Kim\, Li and Ursu.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Oliver Lorscheid (IMPA / University of Groeningen) DTSTART:20211215T140000Z DTEND:20211215T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/24 DESCRIPTION:Title: The moduli space of matroids\nby Oliver Lorscheid ( IMPA / University of Groeningen) as part of Sheffield Pure Maths Colloquia \n\n\nAbstract\nMatroids are combinatorial gadgets that reflect properties of linear algebra in situations where this latter theory is not available . This analogy prescribes that the moduli space of matroids should be a Gr assmannian over a suitable base object\, which cannot be a field or a ring \; in consequence usual algebraic geometry does not provide a suitable fra mework. In joint work with Matt Baker\, we use algebraic geometry over F1\ , the so-called field with one element\, to construct such moduli spaces. As an application\, we streamline various results of matroid theory and fi nd simplified proofs of classical theorems\, such as the fact that a matro id is regular if and only if it is binary and orientable.\n\nWe will dedic ate the first half of this talk to an introduction of matroids and their g eneralizations. Then we will outline how to use F1-geometry to construct t he moduli space of matroids. In a last part\, we will explain why this the ory is useful to simplify classical results in matroid theory.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Shahn Majid (Queen Mary University of London) DTSTART:20220518T130000Z DTEND:20220518T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/25 DESCRIPTION:Title: Quantum Riemannian geometry of the $A_n$ graph\, jets a nd geodesics\nby Shahn Majid (Queen Mary University of London) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nWe describe recent resul ts in quantum or noncommutative Riemannian geometry based on bimodule conn ections. Here\nthe coordinate algebra can be any unital algebra A equipped with a differential structure expressed as a \nbimodule $\\Omega^1$ of $1 $-forms as part of a differential graded algebra with $A$ in degree $0$. T he simplest case\nis $A$ the commutative algebra of functions on the verti ces of a directed graph with $\\Omega^1$ spanned by the arrows. \nWe show in this framework that the intrinsic quantum Riemannian geometry of the $A _n$ graph $\\bullet-\\bullet- …-\\bullet$ of $n$ vertices is necessarily $q$-deformed with $q^{2(n+1)}=1$. Its $q\\to1$ limit is the intrinsic qua ntum Riemannian geometry of the natural numbers viewed as a half-line grap h. We then discuss more generally how solutions of the Yang-Baxter or brai d relations arise naturally from noncommutative differential geometry and relate both to quantum jet bundles and\nto the notion of a quantum geodesi c.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Umut Varolgunes (Bogazici University) DTSTART:20220525T130000Z DTEND:20220525T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/26 DESCRIPTION:Title: Local-to-global methods in relative symplectic cohomolo gy\nby Umut Varolgunes (Bogazici University) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nIn my thesis\, I introduced a Floer theore tic invariant for compact subsets of symplectic manifolds called relative symplectic cohomology. This invariant has already proved to be very useful in symplectic rigidity questions and also opened the way to a fruitful re interpretation of mirror symmetry. Most of these applications rely on an a nalogue of Mayer-Vietoris property from topology that holds for relative s ymplectic cohomology under well-understood geometric assumptions. I will b riefly introduce the invariant\, discuss the Mayer-Vietoris property and p resent some computations relevant to mirror symmetry. I will try to make t he talk accessible to a more diverse audience by mainly sticking to dimens ion two\, where a symplectic form is nothing but an area form.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Vesna Stojanoska (University of Illinois Urbana-Champaign) DTSTART:20220309T150000Z DTEND:20220309T160000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/27 DESCRIPTION:Title: Duality for some Galois groups in stable homotopy theor y\nby Vesna Stojanoska (University of Illinois Urbana-Champaign) as pa rt of Sheffield Pure Maths Colloquia\n\n\nAbstract\nIn classical algebra\, the integer primes p help decompose objects as well as problems into thei r p-primary parts\, which may be easier to study. The same is true in homo topy theory\, but the situation is more interesting since for each integer prime p\, there are infinitely many nested homotopical primes. For each o f those homotopical primes\, there is an (unramified) Galois group that go verns the local story and encodes the symmetries of chromatic homotopy the ory. These Galois groups turn out to be particularly nice profinite groups \, known as compact p-adic analytic. Such groups and their fascinating dua lity properties within algebra were studied by Lazard. I will try to expla in a newer result\, which shows that their homotopical duality properties are even better\, giving powerful implications for the chromatic Galois ex tensions that they govern.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Hossein Movasati (IMPA) DTSTART:20220316T140000Z DTEND:20220316T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/28 DESCRIPTION:Title: Periods of families of curves in threefolds\nby Hos sein Movasati (IMPA) as part of Sheffield Pure Maths Colloquia\n\n\nAbstra ct\nClemens' conjecture states that the the number of rational curve in a generic quintic threefold is finite. If it is false we prove that certain periods of rational curves in such a quintic threefold must vanish. Our me thod is based on a generalization of a proof of Max Noether's theorem usi ng infinitesimal variation of Hodge structures and its reformulation in te rms of integrals and Gauss-Manin connection.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Bachir Bekka (Université de Rennes 1) DTSTART:20220608T130000Z DTEND:20220608T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/29 DESCRIPTION:Title: The spectral gap property for group actions\nby Bac hir Bekka (Université de Rennes 1) as part of Sheffield Pure Maths Colloq uia\n\n\nAbstract\nA measure preserving action of a group G on a measure s pace X gives rise to a unitary representation of G on the Hilbert space $L ^2(X)$. This action may or may not have the spectral gap property which is a very strong form of ergodicity. For instance\, groups with Kazhdan's property T always have this property. We will survey the importance of t he spectral gap property in various problems arising in graph theory\, dy namical systems or operator algebras. In the case where X is a homogeneous space arising from an algebraic group\,\nwe will show that the absence o f the spectral gap property is often related to amenability.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Wushi Goldring (Stockholm University) DTSTART:20220427T130000Z DTEND:20220427T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/31 DESCRIPTION:Title: Propagating algebraicity of automorphic representations via functoriality\nby Wushi Goldring (Stockholm University) as part o f Sheffield Pure Maths Colloquia\n\n\nAbstract\nAutomorphic representation s are some of the richest and most mysterious mathematical objects discove red to-date. They simultaneously generalize (i) infinite-dimensional repre sentations of real Lie groups\, (ii) modular forms and (iii) the Hecke cha racters of class field theory. As such\, automorphic representations incor porate representation theory\, analysis and arithmetic. \n\nIn the late 19 60's\, Robert Langlands laid out a program to unravel much of the seemingl y hidden structure of automorphic representations. To begin to understand the Langlands program\, it is useful -- at least at first -- to distinguis h two kinds of conjectures: Roughly\, Langlands' Functoriality Principle c an be seen as intrinsic to automorphic representations -- revealing a myri ad of relations between different automorphic representations of different groups. By contrast\, the extrinsic Langlands correspondence explains how certain automorphic representations should be related to Galois theory an d algebraic geometry. Every automorphic representation has associated nume rical invariants called Hecke eigenvalues -- these are complex numbers. On e of the most interesting aspects of the Langlands program is that some au tomorphic representations have Hecke eigenvalues which are algebraic numbe rs\, while for others they are transcendental. At this time\, we seem to l ack a conceptual understanding for why this dichotomy exists. While the La nglands correspondence suggests that certain automorphic representations s hould have algebraic Hecke eigenvalues\, it remains unclear -- even at the level of conjectures -- wherein lies the watershed line between algebraic and transcendental. \n\nI will spend most of my talk introducing automorp hic representations\, their Hecke eigenvalues\, functoriality and the corr espondence. The end goal of my talk is then to explain what can be said ab out the algebraicity of Hecke eigenvalues by combining (1) Previously know n cases of algebraicity and (2) Langlands functoriality. On the one hand\, I will explain why the algebraicity of Hecke eigenvalues does propagate f rom some cases to others via functoriality -- this gives new theorems and conjectures on algebraicity of Hecke eigenvalues. On the other hand\, I wi ll explain why most cases -- including Maass forms -- are not reducible to known ones via functoriality.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Dan Ciubotaru (University of Oxford) DTSTART:20221116T140000Z DTEND:20221116T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/32 DESCRIPTION:Title: Unipotent conjugacy classes and group representations\nby Dan Ciubotaru (University of Oxford) as part of Sheffield Pure Math s Colloquia\n\n\nAbstract\nOne of the first theorems that we prove in the complex representation theory of a finite group is that the number of irre ducible representations (up to isomorphism) equals the number of conjugacy classes in the group. For example\, for the group of permutations of the set {1\,2\,...\,n}\, the conjugacy classes are parametrised by partitions of n (the cycle decomposition) and so are the complex irreducible represen tations (via Young's construction from the 1890s). But this is not a natur al bijection\, just like there is not a natural isomorphism between a fini te vector space and its dual in general. However\, for certain classes of groups\, that come with extra structure (like the ones appearing in Lie th eory)\, one expects natural relations between the irreducible representati ons of the group\, on one hand\, and conjugacy classes in a *dual* group\, on the other. This happens for example\, when the group in question is a finite reflection crystallographic group\, or a connected algebraic group over a finite or local field. In these correspondences\, a particularly in teresting role is played by the unipotent conjugacy classes in the dual gr oup. I will give a survey of some of these connections and then emphasise the case of (infinite-dimensional) representations of reductive algebraic groups (like the general linear group of n by n matrices) with coefficient s in a local field\, where I'll explain what the unipotent classes tell us about the growth of characters and the parametrisation of such representa tions. The new results in the talk are joint with Lucas Mason-Brown and Em ile Okada.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Luis Garcia (University College London) DTSTART:20221123T140000Z DTEND:20221123T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/33 DESCRIPTION:Title: Modular symbols\, linking numbers and the Euler class\nby Luis Garcia (University College London) as part of Sheffield Pure M aths Colloquia\n\nLecture held in J-11 Hicks Building.\n\nAbstract\nModula r symbols are a fundamental tool for the computation of the homology of ce rtain linear groups. It has been observed that\, surprisingly\, they also control the relations among certain trigonometric and elliptic functions. After introducing modular symbols and their elementary properties I will e xplain why this is the case and give some arithmetic applications.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Fraser (Wichita State University) DTSTART:20230524T130000Z DTEND:20230524T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/34 DESCRIPTION:Title: (ONLINE) Two strategies for Fourier decay of measures i n Diophantine approximation\nby Robert Fraser (Wichita State Universit y) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nIn 1980 and 19 81\, Kaufman constructed measures with polynomial Fourier decay on the set of badly-approximable numbers and the set of well-approximable numbers. T he strategy for the badly-approximable numbers uses the continued fraction expansion together with a change-of variables\, and the strategy for the well-approximable numbers uses the cancellation of an exponential sum. We will discuss the application of both of these strategies to the set of num bers approximable to exact order introduced by Bugeaud. This talk is based on joint work with Reuben Wheeler.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Cecilia Salgado (University of Groeningen) DTSTART:20221207T140000Z DTEND:20221207T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/35 DESCRIPTION:Title: CANCELLED\nby Cecilia Salgado (University of Groeni ngen) as part of Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Tyler Kelly (University of Birmingham) DTSTART:20230426T130000Z DTEND:20230426T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/36 DESCRIPTION:Title: CANCELLED!\nby Tyler Kelly (University of Birmingha m) as part of Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Markus Szymik (University of Sheffield) DTSTART:20230215T140000Z DTEND:20230215T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/37 DESCRIPTION:Title: CANCELLED DUE TO UCU STRIKE\nby Markus Szymik (Univ ersity of Sheffield) as part of Sheffield Pure Maths Colloquia\n\nAbstract : TBA\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Johannes Girsch (University of Sheffield) DTSTART:20230208T140000Z DTEND:20230208T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/38 DESCRIPTION:Title: Introduction to the representation theory of $p$-adic g roups\nby Johannes Girsch (University of Sheffield) as part of Sheffie ld Pure Maths Colloquia\n\n\nAbstract\nThe Langlands program is a set of w ide-reaching conjectures with great importance to many aspects of number t heory. One kind of objects that are being studied in this setting are repr esentations of $p$-adic groups. I will explain what these groups are and m ention some rather strange properties they possess. Then I will mention so me aspects of their representation theory and mention some recent results. \n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Evgenios Kakariadis (University of Newcastle) DTSTART:20230301T140000Z DTEND:20230301T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/39 DESCRIPTION:Title: Morita equivalence for operator systems\nby Evgenio s Kakariadis (University of Newcastle) as part of Sheffield Pure Maths Col loquia\n\n\nAbstract\nIn ring theory\, Morita equivalence preserves many p roperties of the objects\, and generalizes the isomorphism equivalence bet ween commutative rings. A strong Morita equivalence for selfadjoint operat or algebras was introduced by Rieffel in the 60s\, and works as a correspo ndence between their representations. In the past 30 years there has been an interest to develop a similar theory for nonselfadjoint operator algebr as and operator spaces with much success. Taking motivation from recent wo rk of Connes and van Suijlekom\, we will present a Morita theory for opera tor systems. We will give equivalent characterizations of Morita equivalen ce via Morita contexts\, bihomomoprhisms and stable isomorphism\, while we will highlight properties that are preserved in this context. Time permit ted we will provide applications to rigid systems\, function systems and n on-commutative graphs. This is joint work with George Eleftherakis and Iva n Todorov.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Samuele Anni (Aix-Marseille University) DTSTART:20230308T140000Z DTEND:20230308T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/40 DESCRIPTION:Title: (ONLINE) Isomorphisms of modular Galois representations and graphs\nby Samuele Anni (Aix-Marseille University) as part of She ffield Pure Maths Colloquia\n\n\nAbstract\nIn this talk\, I will explain h ow to test efficiently and effectively whether two odd modular Galois repr esentations of the absolute Galois group of the rational numbers are isomo rphic. In particular\, I will present new optimal bounds on the number of traces to be checked (joint work with Peter Bruin\, University of Leiden). I will also briefly discuss graphs of isomorphisms associated to such obj ects\, related results on Hecke algebras\, and a database of modular repre sentations.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Min Lee (moved to Term 2) (University of Bristol) DTSTART:20221109T140000Z DTEND:20221109T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/41 DESCRIPTION:by Min Lee (moved to Term 2) (University of Bristol) as part o f Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Min Lee (University of Bristol) DTSTART:20230517T130000Z DTEND:20230517T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/42 DESCRIPTION:Title: Frobenius numbers and further - equidistribution of rat ional points on the expanding horospheres\nby Min Lee (University of B ristol) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nFix a fin ite set R of positive integers bigger than one with no common factors. The Frobenius number for R is the largest number that cannot be written as a linear combination of the integers in R with non-negative integral coeffic ients. \n\n\nIn general\, Frobenius numbers fluctuate. To study such thing s\, we search for structures. Here\, the given set of positive integers R can be a point in the lattices studied in the dynamics and number theory c rossover. We study the behaviour of these rational points on expanding clo sed horospheres in the space of lattices. The equidistribution of these ra tional points is proved by Einsiedler\, Mozes\, Shah and Shapira (2016). T heir proof uses techniques from homogeneous dynamics and relies particular ly on measure-classification theorems\, due to Ratner. We pursue an altern ative strategy based on Fourier analysis\, Weil's bound for Kloosterman su ms\, recently proved bounds (by M. Erdélyi and Á. Tóth) for matrix Kloo sterman sums\, Roger's formula\, and the spectral theory of automorphic fu nctions.\n\n\nThis is a joint work with D. El-Baz\, B. Huang\, J. Marklof and A. Strömbergsson.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Eleonore Faber (University of Leeds) DTSTART:20221102T140000Z DTEND:20221102T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/43 DESCRIPTION:Title: From the magic square of rotations and reflections to t he McKay correspondence\nby Eleonore Faber (University of Leeds) as pa rt of Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Celine Maistret (University of Bristol) DTSTART:20221026T130000Z DTEND:20221026T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/44 DESCRIPTION:Title: The Birch and Swinnerton-Dyer conjecture and the Parity conjecture\nby Celine Maistret (University of Bristol) as part of She ffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Vlad Bavula (University of Sheffield) DTSTART:20221019T130000Z DTEND:20221019T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/45 DESCRIPTION:Title: Holonomic modules and 1-generation in the Jacobian Conj ecture\nby Vlad Bavula (University of Sheffield) as part of Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Andre Henriques (University of Oxford) DTSTART:20221012T130000Z DTEND:20221012T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/46 DESCRIPTION:Title: 2d QFTs as objects of mathematics\nby Andre Henriqu es (University of Oxford) as part of Sheffield Pure Maths Colloquia\n\nAbs tract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Haluk Sengun (University of Sheffield) DTSTART:20221005T130000Z DTEND:20221005T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/47 DESCRIPTION:Title: Local theta correspondence via $C^*$-algebras\nby H aluk Sengun (University of Sheffield) as part of Sheffield Pure Maths Coll oquia\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Neil Strickland (University of Sheffield) DTSTART:20230503T130000Z DTEND:20230503T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/48 DESCRIPTION:by Neil Strickland (University of Sheffield) as part of Sheffi eld Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Polyxeni Spilioti (University of Göttingen) DTSTART:20230329T140000Z DTEND:20230329T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/49 DESCRIPTION:Title: (ONLINE) On the spectrum of twisted Laplacians and the Teichmüller representation\nby Polyxeni Spilioti (University of Gött ingen) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nIn this ta lk\, we will present some results concerning the spectrum of Laplacians wi th non unitary twists acting on sections of flat vector bundles over compa ct hyperbolic surfaces. These non self-adjoint Laplacians have discrete sp ectrum inside a parabola in the complex plane. For representations of the fundamental group of the base surface which are of Teichmüller type\, we investigate the high energy limit and give a precise description of the bu lk of the spectrum where Weyl’s law is satisfied in terms of critical ex ponents of the representation which are completely determined by the Manha ttan curve associated to the Teichmüller deformation. This is joint work with Frédéric Naud.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Tyler Kelly (University of Birmingham) DTSTART:20231004T130000Z DTEND:20231004T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/50 DESCRIPTION:Title: Moduli of genus-zero higher spin curves and their invar iants\nby Tyler Kelly (University of Birmingham) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nIn mathematics\, we like classifying o bjects. A moduli space is a space where each point represents a(n isomorph ism class of a) space satisfying certain criteria\, giving a geometric ans wer to a classification problem. Often the geometry of such spaces are int eresting in our own right and their corresponding enumerative information has rich structure. We will study the case of genus-zero n-pointed curves and a generalisation where they are further equipped with an r-spin struct ure. Enumerative invariants built from their characteristic classes have r ich structure due to generalisations of predictions of Witten confirmed by Kontsevich. We will explain approaches to understanding these invariants on a very concrete level through combinatorial structures like recursion a nd tropical geometry.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/50/ END:VEVENT BEGIN:VEVENT SUMMARY:Markus Szymik (University of Sheffield) DTSTART:20231011T130000Z DTEND:20231011T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/51 DESCRIPTION:Title: From rings to algebraic K-theory and back\nby Marku s Szymik (University of Sheffield) as part of Sheffield Pure Maths Colloqu ia\n\n\nAbstract\nAlgebraic K-theory is a conceptual tool for the classifi cation of mathematical objects. A typical scenario comes from linear algeb ra: the classification of vector spaces and\, more generally\, modules ove r a given ring. In this colloquium talk\, I will advertise this tool and i ts use in non-linear algebra. The focus will be on examples\, and I will d iscuss groups\, rings\, and many other more or less exotic algebraic struc tures.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/51/ END:VEVENT BEGIN:VEVENT SUMMARY:Lassina Dembélé (King's College London) DTSTART:20231018T130000Z DTEND:20231018T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/52 DESCRIPTION:Title: (Black History Month Special Colloquium) How mentorship could help fight underrepresentation in STEM\nby Lassina Dembélé (K ing's College London) as part of Sheffield Pure Maths Colloquia\n\n\nAbstr act\nThere is no denial that certain visible minorities are severely under represented in STEM. I hear people often say that the best way to fight un derrepresentation is to have more role models from those minority groups. That is true\, perhaps. However\, I believe that there needs to be an inte rmediate solution until we reach that point when we have enough role model s to have an impact. Based on my own personal experience\, I want to expla in how an innovative approach to mentorship can help fight underrepresenta tion.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Reem Yassawi (Queen Mary University of London) DTSTART:20231025T130000Z DTEND:20231025T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/53 DESCRIPTION:Title: Automatic sequences in dynamics and number theory\n by Reem Yassawi (Queen Mary University of London) as part of Sheffield Pur e Maths Colloquia\n\n\nAbstract\nAn infinite sequence $a = (a_n)_{n\\geq 0 }$ is $q$-automatic if an is a finite-state function of the base-$q$ expan sion of $n$. This means that there exists a deterministic finite automaton that takes the base-q expansion of n as input and produces the symbol an as output for each $n \\in \\mathbb{N}$.\n\nAutomatic sequences appear in diverse fields of mathematics\, such as algebra\, logic\, number theory\, and topological dynamics. They have the advantage of lend- ing themselves to computation\, so that in each area there arise specific problems concer ning automatic sequences\, and much of the time\, constructive solutions.\ n\nI will give a background of their characterisations in algebra and dyna mics\, via Furstenberg’s\, Cobham’s and Christol’s theorems. I will then talk about joint work with Eric Rowland and Manon Stipulanti\, concer ning automatic sequences in number theory\, and also about joint work with Johannes Kellendonk\, concerning automatic sequences in topological dynam ics\, ending with a topological invariant which seems to defy computation. \n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/53/ END:VEVENT BEGIN:VEVENT SUMMARY:Eli Hawkins (University of York) DTSTART:20231101T110000Z DTEND:20231101T120000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/54 DESCRIPTION:Title: Quantization of Multiply Connected Manifolds\nby El i Hawkins (University of York) as part of Sheffield Pure Maths Colloquia\n \n\nAbstract\nGiven a compact Kähler manifold satisfying an integrality c ondition\, the Berezin-Toeplitz geometric quantization construction produc es matrix algebras\; these fit together into a fundamental example of stri ct deformation quantization. The integrality condition can be circumvented by passing to the universal covering space\, if the lift of the symplecti c form is exact\; in this case\, the symplectic form determines a $2$-cocy cle of the fundamental group. The key to analyzing this construction is to use Hilbert $C^*$-modules\, which generalize Hilbert spaces. The resultin g algebras are more interesting than matrix algebras and are partially det ermined by index theorems. The simplest example is the noncommutative toru s\, and this gives higher-genus noncommutative Riemann surfaces as well.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/54/ END:VEVENT BEGIN:VEVENT SUMMARY:Veronique Fischer (University of Bath) DTSTART:20231129T140000Z DTEND:20231129T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/55 DESCRIPTION:Title: Sub-Riemannian quantum limits\nby Veronique Fischer (University of Bath) as part of Sheffield Pure Maths Colloquia\n\n\nAbstr act\nWe will start with a short discussion on semi-classical analysis to i ntroduce the concept of quantum limits. We will present an overview of sub -Riemannian geometry and the recent developments of spectral geometry in t his context\, especially quantum limits on nilpotent Lie groups.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/55/ END:VEVENT BEGIN:VEVENT SUMMARY:Paul Johnson (University of Sheffield) DTSTART:20231122T140000Z DTEND:20231122T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/57 DESCRIPTION:Title: From Orbifold Hilbert schemes to Sec(x)\nby Paul Jo hnson (University of Sheffield) as part of Sheffield Pure Maths Colloquia\ n\n\nAbstract\nThe Hilbert Scheme of points of n points in the plane is a smooth algebraic variety with a rich topology connected to partitions and representation theory. If G acts on a C^2\, it also acts on the Hilbert s cheme of points. The question of when certain G fixed point sets are none mpty winds up having a connection to zig-zag permutations\, which are coun ted by the Taylor series coefficients of Tan(x) and Sec(x).\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/57/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Graves (University of Leeds) DTSTART:20231108T140000Z DTEND:20231108T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/58 DESCRIPTION:Title: Homology of diagram algebras\nby Daniel Graves (Uni versity of Leeds) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\ nDiagram algebras\, such as the Brauer algebras and Temperley-Lieb algebra s\, have been studied for many years. They appear in wide-ranging places s uch as statistical mechanics\, knot theory and representation theory. Howe ver\, the study of the homology of these algebras is a very young field in deed\, having emerged over the course of last decade. In this talk I will give an introduction to these diagram algebras\, their homology and their connection to group homology and homological stability. Time permitting\, I will discuss some recent generalizations of these algebras.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/58/ END:VEVENT BEGIN:VEVENT SUMMARY:Ozgur Bayindir (City University of London) DTSTART:20240228T140000Z DTEND:20240228T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/59 DESCRIPTION:Title: Algebraic K-theory and chromatic redshift\nby Ozgur Bayindir (City University of London) as part of Sheffield Pure Maths Coll oquia\n\n\nAbstract\nI will begin with an introduction to algebraic K-theo ry\, ring spectra and the chromatic redshift conjecture. After this\, I wi ll talk about our new proof of the redshift conjecture for Lubin-Tate spec tra and our algebraic K-theory computations.\n\nThis work is partially joi nt with Christian Ausoni and Tasos Moulinos.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/59/ END:VEVENT BEGIN:VEVENT SUMMARY:Simon Willerton (University of Sheffield) DTSTART:20240306T140000Z DTEND:20240306T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/60 DESCRIPTION:Title: Instantaneous dimension of metric spaces via spread and magnitude\nby Simon Willerton (University of Sheffield) as part of Sh effield Pure Maths Colloquia\n\n\nAbstract\nSome spaces seem to have diffe rent dimensions at different\nscales. A long thin strip might appear one- dimensional at a distance\,\nthen two-dimensional when zoomed in on\, but when zoomed in on even\ncloser it is seen to be made of a finite array of points\, so at that\nscale it seems zero-dimensional. I will present a wa y of quantifying\nthis phenomenon using a couple of measures of the size o f metric spaces\,\nnamely magnitude and spread. I will show lots of exam ples for finite\nmetric spaces.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/60/ END:VEVENT BEGIN:VEVENT SUMMARY:Evgeny Shinder (University of Sheffield) DTSTART:20240313T140000Z DTEND:20240313T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/61 DESCRIPTION:Title: Gromov's cancellation question in birational algebraic geometry\nby Evgeny Shinder (University of Sheffield) as part of Sheff ield Pure Maths Colloquia\n\n\nAbstract\nI explain some cancellation and n on-cancellation phenomena in algebraic geometry and relate them to the str ucture of the Grothendieck ring of varieties and to the groups of biration al self-maps of algebraic varieties\, in particular the Cremona groups.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/61/ END:VEVENT BEGIN:VEVENT SUMMARY:Emine Yildirim (University of Leeds) DTSTART:20240320T140000Z DTEND:20240320T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/62 DESCRIPTION:Title: Why the Return to Pictures in Algebra\nby Emine Yil dirim (University of Leeds) as part of Sheffield Pure Maths Colloquia\n\n\ nAbstract\nIn ancient Greece\, geometry was about points\, lines\, circles \, and communicated through pictures. The 17th Century marked a transforma tive shift\, connecting geometry with algebra\, and lead to working with e quations over visual representations. Algebraic geometry emerged as a magi cal blend of geometric intuition and algebraic methods. Commutative algebr a\, mainly the study of polynomial rings and their ideals\, dominated the field for an extensive period. Then with the emergence of noncommutative a lgebras\, such as matrix algebras\, our unstoppable geometric intuition hi t an immovable wall. The solution? A return to pictures as representations . In this expository talk\, I will introduce a visual perspective on algeb ras\, exploring path algebras and their captivating connections to differe nt fields.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/62/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Samuel (University of Birmingham) DTSTART:20240417T130000Z DTEND:20240417T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/63 DESCRIPTION:by Tony Samuel (University of Birmingham) as part of Sheffield Pure Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/63/ END:VEVENT BEGIN:VEVENT SUMMARY:Catherine Meusburger (University of Erlangen) DTSTART:20240424T130000Z DTEND:20240424T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/64 DESCRIPTION:Title: Dijkgraaf-Witten theory with defects\nby Catherine Meusburger (University of Erlangen) as part of Sheffield Pure Maths Colloq uia\n\n\nAbstract\nWe use 3d defect TQFTs to give a gauge theoretical form ulation of (untwisted) Dijkgraaf-Witten TQFT with defects. This leads to a simple description in terms of embedding quivers\, groupoids and their re presentations. Defect Dijkgraaf-Witten TQFTs is then formulated in terms o f spans of groupoids and representations of spans. This is work in progres s with João Faría-Martins\, University of Leeds.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/64/ END:VEVENT BEGIN:VEVENT SUMMARY:David Corfield (University of Kent) DTSTART:20240501T130000Z DTEND:20240501T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/65 DESCRIPTION:by David Corfield (University of Kent) as part of Sheffield Pu re Maths Colloquia\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/65/ END:VEVENT BEGIN:VEVENT SUMMARY:Rohini Ramadas (University of Warwick) DTSTART:20240508T130000Z DTEND:20240508T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/67 DESCRIPTION:Title: Complex dynamics via algebraic geometry (MOVED TO FALL) \nby Rohini Ramadas (University of Warwick) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nComplex dynamics began in the early 1900s w ith the study of iterating polynomial functions with complex coefficients. This simple idea gives rise to beautiful fractal pictures such as the Man delbrot set\, as well as interesting mathematical questions of many differ ent flavours (algebraic\, analytic\, topological\, arithmetic\, etc.). The field gained momentum in the 1980s due to work of Thurston\, Douady-Hubba rd\, Sullivan\, and others\, connecting these dynamical questions to surfa ce topology and the theory of 3-manifolds. The last decade has seen many b reakthroughs achieved via new tools from number theory\, measure theory an d algebraic geometry. I will discuss some of these recent developments\, h ighlighting the interplay between topology on one hand and algebraic geome try/number theory on the other hand.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/67/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Magee (Durham University) DTSTART:20241009T130000Z DTEND:20241009T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/68 DESCRIPTION:Title: Convergence of unitary representations of discrete grou ps\nby Michael Magee (Durham University) as part of Sheffield Pure Mat hs Colloquia\n\n\nAbstract\nLet G be an infinite discrete group\; e.g. hyp erbolic 3-manifold group.\nFinite dimensional unitary representations of G of fixed dimension are usually very hard to understand. However\, there a re interesting notions of convergence of such representations as the dimen sion tends to infinity. One notion — strong convergence — is of intere st both from the point of view of G alone but also through recently realiz ed applications to spectral gaps of locally symmetric spaces. For example\ , this notion bypasses (unconditionally) the use of Selberg's Eigenvalue C onjecture in obtaining existence of large area hyperbolic surfaces with ne ar-optimal spectral gaps. \n\nThe talk is a broadly accessible discussion on these themes\, based on joint works with W. Hide\, L. Louder\, D. Puder \, J. Thomas.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/68/ END:VEVENT BEGIN:VEVENT SUMMARY:Herbert Gangl (Durham University) DTSTART:20241016T130000Z DTEND:20241016T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/69 DESCRIPTION:Title: The beauty of Zagier's Polylogarithm Conjecture\nby Herbert Gangl (Durham University) as part of Sheffield Pure Maths Colloqu ia\n\n\nAbstract\nDirichlet related the residue at s=1 of the Dedekind zet a function of a number field F (a slight generalisation of the famous Riem ann zeta function) to two important arithmetical notions: the size of the ideal class group and the `volume' of the unit group in the number ring O_ F of F. Generalising this surprising connection\, the special values of th e Dedekind zeta function of a number field F at integer argument n should\ , according to Zagier's Polylogarithm Conjecture\, be expressed via a dete rminant of F-values of the n-th polylogarithm function. Goncharov laid out a vast program incorporating this conjecture using properties of multiple polylogarithms and the structure of a motivic Lie coalgebra.\nIn this imp ressionist talk I intend to give a rough idea of the developments from the early days on\, avoiding most of the technical bits\, and also hint at a number of recent results for higher weight\, some in joint work with\, or developed by\, S.Charlton\, D.Radchenko as well as D.Rudenko and his coll aborators.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/69/ END:VEVENT BEGIN:VEVENT SUMMARY:Alex Fink (Queen Mary University of London) DTSTART:20241023T130000Z DTEND:20241023T140000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/70 DESCRIPTION:Title: Matroid inequalities from algebraic geometry\nby Al ex Fink (Queen Mary University of London) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nMatroids are combinatorial structures that track ``independence'' relations on a set.\nA key example is linear independence of some linear functions on a vector space.\nNot all matroids come from a vector space\,\nbut those that don't behave in surprising algebraic ways as if they do.\nBreakthroughs of the last decade have opened a kit of tool s\nfrom\, and inspired by\, algebraic geometry to prove inequalities for m atroids\,\namong them the ``matroid Hodge theory'' of June Huh and others. \nI'll start by motivating matroids\,\nand aim to end with enough about my work in progress with Andy Berget\nto show how its central tool is differ ent to matroid Hodge theory.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/70/ END:VEVENT BEGIN:VEVENT SUMMARY:Rohini Ramadas (University of Warwick) DTSTART:20241106T140000Z DTEND:20241106T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/71 DESCRIPTION:Title: Complex dynamics via algebraic geometry\nby Rohini Ramadas (University of Warwick) as part of Sheffield Pure Maths Colloquia\ n\n\nAbstract\nComplex dynamics began in the early 1900s with the study of iterating polynomial functions with complex coefficients. This simple ide a gives rise to beautiful fractal pictures such as the Mandelbrot set\, as well as interesting mathematical questions of many different flavours (al gebraic\, analytic\, topological\, arithmetic\, etc.). The field gained mo mentum in the 1980s due to work of Thurston\, Douady-Hubbard\, Sullivan\, and others\, connecting these dynamical questions to surface topology and the theory of 3-manifolds. The last decade has seen many breakthroughs ach ieved via new tools from number theory\, measure theory and algebraic geom etry. I will discuss some of these recent developments\, highlighting the interplay between topology on one hand and algebraic geometry/number theor y on the other hand.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/71/ END:VEVENT BEGIN:VEVENT SUMMARY:Gavin Brown (University of Warwick) DTSTART:20241113T140000Z DTEND:20241113T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/72 DESCRIPTION:Title: Flops and noncommutative potentials\nby Gavin Brown (University of Warwick) as part of Sheffield Pure Maths Colloquia\n\n\nAb stract\nI give an overview of a project with Michael Wemyss to classify si mple 3-fold flops. This amounts to understanding when a smooth rational cu rve (i.e. the Riemann sphere) inside a complex 3-dimensional manifold can be contracted. (One might think of this as a 3d analogue of shrinking the central axis of a Moebius strip to a point\, and indeed one could do an un necessarily elaborate analysis of that situation by the same methods.) In fact\, this large family of surgery operations is central to 3d complex ge ometry\, but has nevertheless resisted classification\, or even the const ruction of a set of representative examples.\n\nBeing a manifold\, one can describe the situation by glueing together patches - and it is enough to glue together two copies of the affine space $\\mathbb{C}^3$ by a simple f ormula .. but with lots of free parameters\, most of which do not contract . However finding good (i.e. contractible) glue functions (or even classif ying them) seems to be a bit needle-in-a-haystack. Instead\, we translate the problem to one of classifying noncommutative germs $f(x\,y)$ [or equiv alently certain complete local algebras up to isomorphism]\, where the nec essary criteria seem more amenable. That context feels much like the class ical singularity theory of function germs in the style of Arnold (types AD E and all that)\, and we can solve enough of that problem to construct all flops and to provide a classification.\n\nFrom one point of view\, I’d like to give some idea of what Theorems 5.1 and 5.4 of the following expos itory tea-time article mean:\nhttps://arxiv.org/abs/2410.21500\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/72/ END:VEVENT BEGIN:VEVENT SUMMARY:Cheuk Yu Mak (University of Sheffield) DTSTART:20241127T140000Z DTEND:20241127T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/73 DESCRIPTION:Title: From area preserving homeomorphism groups to symplectic Khovanov homology and beyond\nby Cheuk Yu Mak (University of Sheffiel d) as part of Sheffield Pure Maths Colloquia\n\n\nAbstract\nIn the first h alf of the talk\, I will explain some recent breakthroughs in the study of the area preserving homeomorphism groups of surfaces using Floer theory. After that\, I will explain what happens when we try to generalize it to h igher dimensions and the relation to Khovanov homology as well as the Hilb ert schemes of points. No prior knowledge on Floer theory or symplectic ge ometry is assumed.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/73/ END:VEVENT BEGIN:VEVENT SUMMARY:Jason Semeraro (University of Loughborough) DTSTART:20241120T140000Z DTEND:20241120T150000Z DTSTAMP:20260404T110824Z UID:SheffieldPureMaths/76 DESCRIPTION:Title: Fusion-stable representations of finite groups\nby Jason Semeraro (University of Loughborough) as part of Sheffield Pure Math s Colloquia\n\n\nAbstract\nThis is joint work with my PhD student Tom Lawr ence.\n\nFor a prime p\, the p-decomposition matrix D of a finite group G records the way each irreducible ordinary representation of G breaks up in to irreducible p-Brauer characters under reduction modulo p. Multiplying D by its transpose yields the Cartan matrix\, whose determinant is well-kno wn to be a power of p. A representation of a Sylow p-subgroup S of G is fu sion-stable if it is left invariant by the conjugation action of G. After first fixing a basis B of fusion-stable representations of S one can consi der an analogue of D for fusion-stable representations which records how e ach irreducible ordinary representation of G breaks up in B under restrict ion to S. It turns out this matrix has many properties analogous to those of the classical decomposition matrix\, and using them one can show that t he modulus square of the determinant of the fusion-stable character table (columns indexed by G-classes of p-elements\, rows by elements of B) is al ways a particular power of p independent of the choice of B. I conjectured that the same result holds for any saturated fusion system on S and I'll provide some evidence for this by explicitly computing with some infinite families of exotic examples. If time permits I will also explain how this project fits within the larger framework of "exotic representation theory" whose aim to extend results about ordinary representations to the setting s of fusion systems\, spetses and other related structures.\n LOCATION:https://stable.researchseminars.org/talk/SheffieldPureMaths/76/ END:VEVENT END:VCALENDAR