Well-behaved simple C*-algebr as (very simple axioms)
\n\n\n\n\n\nbehaved simple case (some ev idence!)\, and the non-simple
\n\n\n\ncase\, of course well behaved to begin with! There are
\n< !-- /wp:paragraph -->\n\n\nalready a number of res ults in the well-behaved non-simple
\n\n\n\ncase.
\n\n LOCATION:https://stable.researchseminars.org/talk/MathematicalPhysics/64/ END:VEVENT BEGIN:VEVENT SUMMARY:Christopher Schafhauser (Lincoln\, Nebraska) DTSTART:20221004T153000Z DTEND:20221004T170000Z DTSTAMP:20260404T110744Z UID:MathematicalPhysics/65 DESCRIPTION:Title: Subalgebras of simple AF-algebras\nby Christopher Schafhauser (Lincoln\, Nebraska) as part of Wales MPPM Mathematical Physic s\n\n\nAbstract\nA long-standing open question\, formalized by Blackadar a nd Kirchberg in the mid ‘90s\, asks for an abstract characterization of C*-subalgebras of AF-algebras. I will discuss some recent progress on thi s question: every separable\, exact C*-algebra which satisfies the UCT and admits a faithful\, amenable trace embeds into an AF-algebra. Moreover\, the AF-algebra may be chosen to be simple and unital with unique trace an d the embedding may be taken to be trace-preserving. Modulo the UCT\, thi s characterizes C*-subalgebras of simple\, unital AF-algebras. As an appl ication\, for any countable\, discrete\, amenable group G\, the reduced C* -algebra of G embeds into a UHF-algebra. I'll also discuss how these tech niques have played a role in a new proof to the classification theorem of separable\, simple\, nuclear\, Z-stable\, UCT C*-algebras\, which is based on joint work with Jos\\'e Carri\\'on\, James Gabe\, Aaron Tikuisis\, and Stuart White.\n LOCATION:https://stable.researchseminars.org/talk/MathematicalPhysics/65/ END:VEVENT BEGIN:VEVENT SUMMARY:William Slofstra (Waterloo) DTSTART:20221101T163000Z DTEND:20221101T180000Z DTSTAMP:20260404T110744Z UID:MathematicalPhysics/66 DESCRIPTION:Title: Bell inequalities and decision problems in C*-algebras \nby William Slofstra (Waterloo) as part of Wales MPPM Mathematical Ph ysics\n\n\nAbstract\nBell inequalities have been in the news lately with t he recent physics Nobel prize. On the mathematics side\, given a Bell ineq uality\, we'd like to be able to determine the maximum possible violation allowed in quantum mechanics. This simple problem has led to a lot of deve lopments in the last few years. In particular\, Ji\, Natarajan\, Vidick\, Wright\, and Yuen have shown that it's impossible to calculate this violat ion even approximately\, thereby resolving the Connes embedding problem in operator algebras. In this talk\, I'll give an overview of these developm ents\, starting with physics\, but with the aim of showing that there are lots of other interesting decision problems in operator algebras. I'll inc lude work with Honghao Fu and Carl Miller on the membership problem for qu antum correlations\, and work in progress with Arthur Mehta and Yuming Zha o on deciding positivity.\n LOCATION:https://stable.researchseminars.org/talk/MathematicalPhysics/66/ END:VEVENT BEGIN:VEVENT SUMMARY:Gabor Szabo (KU Leueven) DTSTART:20221108T163000Z DTEND:20221108T180000Z DTSTAMP:20260404T110744Z UID:MathematicalPhysics/67 DESCRIPTION:Title: The dynamical Kirchberg-Phillips theorem\nby Gabor Szabo (KU Leueven) as part of Wales MPPM Mathematical Physics\n\n\nAbstra ct\nIn this talk I will present the main results of a joint work with Jame s Gabe: Given a countable discrete group G\, two amenable and outer G-acti ons on stable Kirchberg algebras are cocycle conjugate precisely when they are equivariantly KK-equivalent. In fact we have a suitable version of su ch a theorem for actions of locally compact groups. After discussing some of the core concepts\, I will state the classification results and\, depen ding on the available time\, discuss some interesting corollaries.\n LOCATION:https://stable.researchseminars.org/talk/MathematicalPhysics/67/ END:VEVENT BEGIN:VEVENT SUMMARY:Tatiana Gateva-Ivanova (American University in Bulgaria) DTSTART:20221115T163000Z DTEND:20221115T180000Z DTSTAMP:20260404T110744Z UID:MathematicalPhysics/68 DESCRIPTION:Title: Set-theoretic solutions of the Yang-Baxter equation an d their quadratic algebras\nby Tatiana Gateva-Ivanova (American Univer sity in Bulgaria) as part of Wales MPPM Mathematical Physics\n\n\nAbstract \nSet-theoretic solutions of the Yang--Baxter equation (YBE) form a meetin g-ground\nof mathematical physics\, algebra and combinatorics. Such a solu tion $(X\,r)$\nconsists of a non-empty set $X$\nand a bijective map $r:X\\ times X\\to X\\times X$ which satisfies the braid\nrelation\n\\[\nr^{12}r^ {23}r^{12} = r^{23}r^{12}r^{23}\n\\]\nin $X\\times X\\times X$\, where $r^ {12}=r\\times id_X\,\\\;\\\; r^{23}= id_X\\times r$.\nIn this case $(X\,r) $ is called \\emph{a braided set}.\nA braided set $(X\,r)$ with $r$ inv olutive is called a\nsymmetric set.\n\n\n\nSet theoretic solutions exten d to special linear\n solutions of YBE\, but more important than this is t hat set-theoretic\nsolutions lead to their own remarkable algebraic and co mbinatoric\nstructures\, only somewhat analogous to quantum group\nconstru ctions. They also lead to\n a great deal of combinatorics\, group action on $X$\, \nmatched pairs of groups\, matched pairs of monoids\, braided gr oups and braided monoids\;\nsemigroups of I type with a structure of distr ibutive lattice\;\nquadratic algebras with very nice\n algebraic and homol ogical properties such as being:\n(i) Artin-Schelter\n regular algebras\;\ n(ii) Koszul algebras\;\n(iii) Noetherian domains\nwith a $\\textbf{k}$-ba sis of Poincar\\'{e}-Birkhoff-Witt type\;\n(iv)\nwith good computational p roperties -the theory of\nnoncommutative Groebner bases is applicable.\n\n \n\nAssociated to each set-theoretic solution are several algebraic constr uctions:\n the braided group $G(X\,r)$\, the braided monoid $S(X\,r)\,$ th e quadratic $\\textbf{k}$ algebra $A(\\textbf{k}\, X\, r) \\cong\n\\textbf {k} S(X\,r)$ over a field $\\textbf{k}$ called \\emph{an Yang-Baxter algeb ra}. Each of these structures is generated by $X$ and has quadratic relati ons $xy=\\cdot r(x\,y)$\,\nwhere "$\\cdot$" denotes product in the free mo noid\, resp. free group.\n\nIn this talk we shall discuss \\emph{the pur ely\nmathematical theory} of the Yang-Baxter algebras $A(\\textbf{k}\, X\ , r)$\, their remarkable algebraic properties and some applications\nto no ncommutative algebraic geometry- Artin-Schelter regularity\, analogues of Veronese subalgebras\, and Segre products\, Veronese morphisms and Segre morphisms for Yang-Baxter algebras.\n LOCATION:https://stable.researchseminars.org/talk/MathematicalPhysics/68/ END:VEVENT BEGIN:VEVENT SUMMARY:Gregory Moore (Rutgers) DTSTART:20221122T163000Z DTEND:20221122T180000Z DTSTAMP:20260404T110744Z UID:MathematicalPhysics/69 DESCRIPTION:Title: Summing over bordisms in TQFT\nby Gregory Moore (R utgers) as part of Wales MPPM Mathematical Physics\n\n\nAbstract\nI will r eview the contents of the paper I wrote with Anindya Banerjee 2201.00903\, \nincluding a few updates. We consider a new construction in TQFT which w as motivated by recent developments in the quantum gravity community. Spec ifically\, we aim to give mathematical precision to a model proposed by D. Marolf and H. Maxfield 2002.08950. Given a TQFT one can consider - forma lly - the sum over all the amplitudes associated to all bordisms between f ixed ingoing and outgoing spatial slices. We discuss when this constructio n makes sense and a curious splitting property satisfied by the total ampl itude. The discussion of the splitting property has been significantly imp roved by an important remark by Daniel Friedan.\n LOCATION:https://stable.researchseminars.org/talk/MathematicalPhysics/69/ END:VEVENT BEGIN:VEVENT SUMMARY:Palle Jorgensen (Iowa) DTSTART:20221011T153000Z DTEND:20221011T170000Z DTSTAMP:20260404T110744Z UID:MathematicalPhysics/70 DESCRIPTION:Title: Reflection positivity\nby Palle Jorgensen (Iowa) a s part of Wales MPPM Mathematical Physics\n\n\nAbstract\nThe talk aims to extend tools from reflection positivity (RP) to non-commutative harmonic a nalysis\, spectral theory\, and new duality theory for unitary representat ions of Lie groups.\n\nIn its original form\, reflection positivity (RP)\, also referred to as Osterwalder-Schrader positivity\, came to serve as a crucial link between problems in quantum physics and in mathematics. More precisely\, the original variant of RP serves to link Euclidean field theo ry (math) to relativistic quantum field theory (physics). Thereby linking difficult questions for relativistic quantum fields\, on one side\, with\ , on the other side\, Euclidean fields. Tools and solutions available on o ne side have often yielded insights on the other side. The RP-axioms link the two. The basic symmetry groups are different\, i.e.\, the Poincare gro up vs the Euclidian group. Hence the associated harmonic analysis and spec tral theory are different of course. In recent work by the speaker and mul tiple co-authors have aimed to expand the framework of RP to the theory of representations of Lie group\, spectral theory\, and operator algebras. A mong other things\, this viewpoint yields insight into the role of the Mar kov property\, as opposed to Osterwalder-Schrader positivity.\n\nAs a gene ral principle\, the Reflection positivity (RF) correspondence has proved u seful in mathematics and in many neighboring areas. In its original form\, RP successfully combines powerful tools from analysis\, from geometry\, f rom representation theory to questions in quantum physics. For example\, d ue to work by many authors\, the RF-correspondence has served to link abel ian (commutative) properties of Gaussian processes/fields in the Euclidean setting\, to the context of non-commutativity in the study of quantum fie lds. And by now\, RP has further become a powerful tool in non-commutative harmonic analysis\, and in the theory of unitary representations of Lie g roups.\n LOCATION:https://stable.researchseminars.org/talk/MathematicalPhysics/70/ END:VEVENT BEGIN:VEVENT SUMMARY:Sahand Seifnashri (IAS Princeton) DTSTART:20221018T153000Z DTEND:20221018T170000Z DTSTAMP:20260404T110744Z UID:MathematicalPhysics/71 DESCRIPTION:Title: Non-Invertible Symmetries and their Representations\nby Sahand Seifnashri (IAS Princeton) as part of Wales MPPM Mathematical Physics\n\n\nAbstract\nI will discuss generalized symmetries in two-dimen sional QFTs that can be non-invertible. Such finite non-invertible symmetr ies are described by a fusion category rather than a finite group. Using t he results in the mathematical physics literature\, we find the appropriat e notion of representation for non-invertible symmetries needed to derive selection rules on correlation functions. We use this understanding to der ive a Cardy-like formula for 2d CFTs with a finite non-invertible symmetry . More specifically\, we derive a universal formula for the asymptotic den sity of states transforming in an irreducible representation of a fusion c ategory symmetry.\n\nBased on arXiv:2208.05495 and in collaboration with Y ing-Hsuan Lin\, Masaki Okada\, and Yuji Tachikawa.\n LOCATION:https://stable.researchseminars.org/talk/MathematicalPhysics/71/ END:VEVENT END:VCALENDAR