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Events for August – December 2020

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Sep. 10, 2020 1pm (Zoom (vir…	Rep Theory Andrew Linshaw (Denver University) Trialities of W-algebras
Sep. 17, 2020 2pm (Zoom)	Functional Analysis Andrew Stocker (CU Boulder) Hilbert C*-modules - Modules and Maps
Sep. 17, 2020 4pm (Zoom)	Grad Student Seminar Howie Jordan (CU Boulder) X We’ve all heard the hype: all you have to do is forget about “excluded middle” and then BOOM, your logic is constructive (in a certain sense). But like… why would you do that? And in what sense is it more constructive? If you aren’t aware, the Law (ha!) of Excluded Middle is the assertion that, for any particular sentence, either that sentence or its negation holds in your logic. In this talk we’ll look at what kind of logic you get when you drop excluded middle and discuss various motivations and interpretations of this, from the philosophical hooey of Brouwer to the structural interpretations arising from Gentzen’s methods. We will also look at a sample of some of the differences in practice and theory that arise when you don’t allow yourself the luxury of excluded middle in your proofs, in particular the interesting effects this has on the theory of real analysis! Noo don’t exclude the middle you’re so constructive aha
Sep. 24, 2020 1pm (Zoom (vir…	Rep Theory Jinwei Yang (University of Alberta) X Tensor categories of vertex operator algebras play an important role in the study of vertex operator algebras and conformal field theories. A central problem of tensor category theory of Huang-Lepowsky-Zhang is the existence of the vertex tensor category structure. We develop a few general methods to establish the existence of tensor structure on module categories for vertex operator algebras, especially for non-rational and non-C_2 cofinite vertex operator algebras. As applications, we obtain the tensor structure of affine Lie algebras at various levels, affine Lie superalgebra gl(1\|1), the Virasoro algebra at all central charges as well as the singlet algebras. We also study important properties, including constructions of projective covers, fusion rules and rigidities of these tensor categories. This talk is based on joint work with T. Creutzig, Y.-Z. Huang, F. Orosz Hunziker, C. Jiang, R. McRae and D. Ridout. Recent progress on tensor categories of vertex operator algebras
Sep. 24, 2020 2pm (Zoom)	Functional Analysis Shen Lu (CU Boulder) Hilbert C*-modules - Multipliers and morphisms
Sep. 24, 2020 4pm (Zoom)	Grad Student Seminar Chase Meadors (CU Boulder) X Coxeter groups are groups with a certain kind of presentation. Associated to each such group is an R-algebra called the Hecke algebra, and, in some cases, quotients of these algebras are isomorphic to a class of diagram algebras called Generalized Temperley-Lieb algebras. A diagram algebra is an algebra whose basis can be represented as pretty pictures, where multiplication is performed by "pasting pictures together" and following certain rules. Were this talk in person, I would undoubtedly bring construction paper, crayons, and Elmer's glue so we could all do calculations in these algebras together; alas, we will have to settle for looking at pretty pictures on my screen instead. At any rate, we will explore the connection between Coxeter groups and these diagram algebras and discuss some results and open conjectures. Coxeter groups and diagram algebras (pretty pictures included)
Sep. 29, 2020 10:30am (zoom)	MathClub Yano Casalaina-Martin (CU-Boulder) X I will discuss what it means for the zero set of a polynomial in several variables to be rational, as well as an open problem regarding cubic polynomials. Rationality of cubic hypersurfaces
Sep. 29, 2020 1pm (Zoom)	Peter Mayr (Boulder) X We review properties of varieties generated by finite simple algebras, their subvarieties and free objects. In particular we show that every congruence modular variety that is generated by a finite nonabelian simple algebra has a unique maximal subvariety. Varieties generated by finite simple algebras
Sep. 29, 2020 1pm (Zoom)	Peter Mayr (CU) X We review properties of varieties generated by finite simple algebras, their subvarieties and free objects. In particular we show that every congruence modular variety that is generated by a finite nonabelian simple algebra has a unique maximal subvariety. Varieties generated by finite simple algebras
Oct. 01, 2020 2pm (Zoom)	Functional Analysis Rachel Chaiser (CU Boulder) Hilbert C*-modules - Projections and Unitaries
Oct. 01, 2020 3pm (Zoom (vir…	Rep Theory Reimundo Heluani (IMPA, Brazil) X We prove a new Fermionic quasiparticle sum expression for the character of the Ising model vertex algebra, related to the Jackson-Slater q-series identity of Rogers-Ramanujan type. We find, as consequences, an explicit monomial basis for the Ising model, and a description of its singular support. We find that the ideal sheaf of the latter, defining it as a subscheme of the arc space of its associated scheme, is finitely generated as a differential ideal. We prove three new q-series identities of the Rogers-Ramanujan-Slater type associated with the three irreducible modules of the Virasoro Lie algebra of central charge 1/2. This is joint work with G. E. Andrews and J. van Ekeren and is based on arxiv.org:2005.10769 The singular support of the Ising model.
Oct. 01, 2020 4pm (Zoom)	Grad Student Seminar Andrew Campbell (CU Boulder) X Assume you attend one Fall 2020 Grad Student Seminar at random. Somehow you know there is at least a 1/20 chance the random talk is on the probabilistic method with examples in combinatorics and geometric functional analysis. This would imply there is at least one talk this semester with the above property, otherwise how could there possibly be any chance of such a thing happening? You now understand how the probabilistic method is used to prove the existence of mathematical objects. Since you are now an expert you know the probabilistic method is nonconstructive and the only way to find this talk is to show up every week. The Probabilistic Method
Oct. 06, 2020 1pm (Zoom)	Keith Kearnes (Boulder) X I will explain why any minimal strongly abelian variety with bounded essential arity is categorically equivalent to a minimal unary variety. Minimal strongly abelian varieties
Oct. 08, 2020 1pm (Zoom (vir…	Rep Theory Jethro Van Ekeren (UFF, Brazil) X (joint work with Lam, Moeller and Shimakura) If V is a holomorphic vertex algebra of central charge 24 then its weight one space V_1 is known to be a reductive Lie algebra which is either trivial, abelian of dimension 24 (in which case V is the Leech lattice vertex algebra) or else one of 69 semisimple Lie algebras first determined by Schellekens in 1993. Until now the only known proof of Schelekens result was a heavily computational one involving case analysis and difficult integer programming problems. Recently Moeller and Scheithauer have established a bound on the dimension of the weight one space of a holomorphic orbifold vertex algebra, using the Deligne bound on the growth of coefficients of weight 2 cusp forms. In this talk I will describe how the dimension bound together with Kac's very strange formula implies that all holomorphic vertex algebras of central charge 24 and nontrivial weight one space are orbifolds of the Leech lattice algebra. Since the automorphism group of the latter algebra is known one can, with a little more work, recover Schellekens result in this way. Schellekens list, the Leech lattice and the very strange Formula
Oct. 08, 2020 2pm (Zoom)	Functional Analysis Andrew Stocker (CU Boulder) Tensor products
Oct. 08, 2020 4pm (Zoom)	Grad Student Seminar Michael Wheeler (CU Boulder) X We all know the story of how Calculus started as a non-rigorous confusion of intuitive notions of the infinitely small and infinitely big and how eventually the field was put on rigorous footing with the invention of the limit. This story leaves out an interesting question: If the foundation using infinitesimal numbers was so treacherous, why is it that so many of the results obtained using this faulty foundation continued to be true when using the limit formulation, a formulation where all the proofs look different? In this talk, we will explore a modification of the Cauchy construction of the reals that results in an ordered field very closely resembling the real numbers except that it has infinitesimal and infinite elements (and so is not Archimedean). If time permits, we will talk about how some of the basics of Calculus can be formulated in this larger field. Reviving the Ghosts of Departed Quantities
Oct. 13, 2020 10:30am (zoom)	MathClub Kate Stange (CU-Boulder) X We'll take a little tour of some famous and less famous integer recurrence sequences, including the Fibonacci numbers and elliptic divisibility sequences, viewing them as shadows of geometric group laws on curves, which connects these sequences to current topics in research. Along the way, we'll use software to visually explore some of their properties. The integer shadows of curves
Oct. 13, 2020 1pm (Zoom)	Stefano Fiovaranti (JKU Linz, Austria) X We investigate finitary functions from Z_n to Z_n for a squarefree number n. We show that for squarefree n the lattice of all clones on Z_n which contain the addition of Z_n is finite and we provide an upper bound for the cardinality of this lattice. Furthermore, we prove that these clones can be generated by functions of arity at most the greatest prime factor of n. Expansions of squarefree abelian groups
Oct. 14, 2020 3pm (Zoom)	Math Phys Daniel Spiegel (CU Boulder Physics Department) X We aim to shed light on the description of topological charges and particle exchange statistics in the framework of algebraic quantum mechanics. The notions of observable, state and pure state, superselection sector, and locality are defined in terms of C*-algebras and form the basis of our discussion. We show how physically relevant superselection sectors may be obtained from localized, transportable endomorphisms of a quasi-local algebra. These endomorphisms act as charge creation operators and are endowed with a tensor product from which particle exchange statistics may be analyzed. We briefly discuss Kitaev’s toric code as an example of these ideas. Superselection Sectors and Braid Statistics in Algebraic Quantum Mechanics
Oct. 15, 2020 11am (Zoom)	Noncomm Geometry Zhengfang Wang (University of Stuttgart) X Analgous to Hochschild cohomology, we define the singular Hochschild cohomology of an algebra as the Yoneda algebra of the identity bimodule in the singularity category of bimodules. The singularity category was introduced by Buchweitz in 1986 and then rediscovered by Orlov in 2003. In this talk, we construct a natural complex to compute the singular Hochschild cohomology. We prove that this complex carries a natural B-infinity algebra structure. In particular, the singular Hochschild cohomology is a Gerstenhaber algebra. We will also talk about Keller’s recent theorem and conjecture for singular Hochschild cohomology. B-infinity algebra structure on singular Hochschild cohomology
Oct. 15, 2020 1pm (Zoom (vir…	Rep Theory Naoki Genra (University of Alberta) X Screening operators are useful tools to characterize free field realizations of vertex algebras, and give new perspectives in the structures of them. We explain screening operators of the beta-gamma system, affine vertex (super)algebras and W-(super)algebras. We also explain the applications to the coset constructions, representations and trialities of W-algebras. Screenings and applications
Oct. 15, 2020 2pm (Zoom)	Functional Analysis Shen Lu (CU Boulder) Induced representations
Oct. 20, 2020 10:30am (zoom)	Geometry/Analysis Braden Balentine (CU Boulder) X We discuss mild solutions to the Navier-Stokes equation on the $n$ -dimensional hyperbolic space $ℍ^{n}$ , $n ≥ 2$ . Substantial results have already been obtained by Pierfelice. In this talk, we discuss how to extend the Fujita-Kato theory of mild solutions from Euclidean space to the hyperbolic space and also examine the ways in which the theory can be improved in this setting. More precisely, due to the additional exponential time decay offered on $ℍ^{n}$ for dispersive and smoothing estimates of the heat kernel, we are able to significantly simplify the proofs of the $L^{n}$ and $L^{p}$ norm decay results as compared to the Euclidean setting. Additionally, we are able to show that the $L^{n}$ norm of a global solution decays to zero as $t$ goes to infinity on $ℍ^{n}$ , and does so at a much faster rate than is known in the Euclidean setting. The proof does not rely on dimension or the energy inequality in any way. As a necessary part of our work, we extend to $ℍ^{n}$ known facts in Euclidean space concerning the strong continuity and contractivity of the semigroup generated by the Laplacian. This work, together with Pierfelice's, contributes to providing a full Fujita-Kato theory on $ℍ^{n}$ . Global Time Behavior of $L^{p}$ -Mild Solutions to the Navier-Stokes Equation on the Hyperbolic Space
Oct. 20, 2020 1pm (Zoom)	Branimir Seselja (University of Novi Sad, Serbia) X By several postulates we introduce a new class of algebraic lattices, in which a main role play so called normal elements. A model of these lattices are weak-congruence lattices of groups, so that normal elements correspond to normal subgroups of subgroups. We prove that in this framework many basic structural properties of groups turn out to be lattice-theoretic. Consequently, we give necessary and sufficient conditions under which a group is Hamiltonian, Dedekind, Abelian, solvable, perfect, supersolvable, finite nilpotent. These conditions are given as lattice theoretic properties of a lattice with normal elements. Joint research with Andreja Tepavcevic and Jelena Jovanovic. Lattices with normal elements
Oct. 21, 2020 4pm (Zoom)	Topology Howie Jordan (CU Boulder) X This talk will be a gentle introduction to sheaves. Join Zoom Meeting https://cuboulder.zoom.us/j/98979344859 Meeting ID: 989 7934 4859 Passcode: 787231 Sheaves
Oct. 22, 2020 1pm (Zoom (vir…	Rep Theory Darlayne Addabbo (University of Arizona) X I will discuss the level two Zhu algebra for the Heisenberg vertex operator algebra and techniques used in determining its structure. I will also discuss more general results helpful in determining generators and relations for higher level Zhu algebras, and in particular, will provide an example to clarify the necessity of an extra condition required in the definition of higher level Zhu algebras. (Joint with Katrina Barron.) Higher level Zhu algebras for vertex operator algebras
Oct. 27, 2020 10:30am (zoom)	MathClub Judy Packer (CU-Boulder) X A brief history of the Heisenberg Uncertainty Principle will be given, with an eye towards introducing the (Weyl)-Heisenberg group. This group (which can be viewed as a subgroup of the group of invertible 3 x 3 real matrices, has many interesting applications, and a few will be discussed. The Heisenberg Uncertainty Principle and the Heisenberg group
Oct. 27, 2020 1pm (Zoom)	H. Peter Gumm (Philipps University Marburg, Germany) X Motivating our investigation, we begin by demonstrating how all sorts of transition systems studied in Computer Science - automata (deterministic, nondeterministic, probabilistic), Kripke structures, neighbourhood systems (including topological spaces) and many more - can be naturally modeled as instances of the concept of "universal coalgebra". Each of the mentioned cases requires an appropriate Set-functor replacing what we know in universal algebra as "signature". Various preservation properties of the signature functor F determine the structure theory of the class of all F-coalgebras. In our talk we concentrate on functors parameterized by some universal algebras, be it a complete lattice or a commutative monoid, and mainly on the functor $F_{Σ}$ given by constructing for a set X the free $Σ$ -Algebra $FΣ(X)$ . $F_{Σ}$ preserves preimages if and only if each $Σ$ -term which is weakly independent at a variable position x is (strongly) independent of x, (in short: $Σ$ implies its own 'derivative' $\Sigma'$). For every permutable variety V(\Sigma) the free algebra functor weakly preserves kernel pairs. For the converse we exhibit a syntactic condition stating that an equation p(x,x,y) = q(x,y,y) holds if and only both p(x,y,z) and q(x,y,z) arise as substitution instances of a common term s(x,y,z,u). We give a geometrical interpretation of this fact and use it to show that if $V(Σ)$ is n-permutable then $F_{Σ}$ weakly preserves kernel pairs iff $V(Σ)$ is Mal'cev. Free algebra functors as coalgebraic signatures
Oct. 28, 2020 4pm (zoom)	Topology Agnes Beaudry (CU Boulder) X This is the second talk in a sequence of introductory lectures whose goal is to discuss the notions of gerbes, bundle gerbes and their classification. Last week, we discussed sheaves of sets and sheaves of groups. In this talk, we will discuss sheaves of categories, also known as stacks. Topic: CU Topology Seminar Time: This is a recurring meeting Meet anytime Join Zoom Meeting https://cuboulder.zoom.us/j/98979344859 Meeting ID: 989 7934 4859 Passcode: 787231 Stacks & Gerbes - Part I
Oct. 29, 2020 1pm (Zoom (vir…	Rep Theory Ryo Sato (Academia Sinica,Taipei, Taiwan) X The Kazama-Suzuki coset vertex operator superalgebra associated with a simple Lie algebra g and its Cartan subalgebra h is a ``super-analog'' of the parafermion vertex operator algebra associated with g. At positive integer levels, the coset superalgebra turns out to be C_2-cofinite and rational by the general theory of orbifolds (Miyamoto) and Heisenberg cosets (Creutzig-Kanade-Linshaw-Ridout), respectively. On the other hand, at Kac-Wakimoto admissible levels, the coset superalgebra is not C_2-cofinite nor rational. In this talk we discuss a relationship between the category of weight modules for the admissible affine vertex algebra associated with g and that for the corresponding Kazama-Suzuki coset vertex superalgebra. In our discussion the inverse Kazama-Suzuki coset construction, which is originally due to Feigin-Semikhatov-Tipunin in the g=sl_2 case, plays an important role. As an application, for g=sl_2 at level -1/2, we determine all the fusion rules between simple weight modules of the Kazama-Suzuki coset vertex superalgebra and verify the conjectural Verlinde formula in this case (corresponding to Creutzig-Ridout's result in the affine side). The last part is based on the joint work with Shinji Koshida. Kazama-Suzuki coset vertex superalgebras at admissible levels.
Oct. 29, 2020 2pm (Zoom)	Functional Analysis Shen Lu (CU Boulder) Imprimitivity Bimodules
Nov. 03, 2020 1pm (Zoom)	Michael Kompatscher (Oxford University, UK) X Even though nilpotent algebras with a Maltsev term have many pleasant algebraic properties, they are not completely understood from a computational point of view. This concerns in particular the complexity of the circuit equivalence problem CEQV(A), i.e. the problem of deciding whether an equation of polynomials p(x_1,…,x_k) = q(x_1,…,x_k), encoded by circuits, holds for all values in the algebra A; and the closely related circuit satisfiability problem CSAT(A), which asks if there is a solution to such an equation. In a previous talk in Boulder I showed how, under the assumption of an open conjecture in circuit complexity, there are algorithms with quasipolynomial running time O(e^{log(n)^c}) for both CEQV(A) and CSAT(A), for all finite nilpotent Maltsev algebras A. Assuming the Exponential Time Hypothesis, Idziak, Kawalek and Krzaczkowski, recently showed proper quasipolynomial lower bounds (i.e. c>1) on the complexity for some nilpotent algebras. Thus, under the assumption of both conjectures, there are nilpotent algebras A such that CEQV(A) and CSAT(A) can be solved in quasipolynomial, but not polynomial time. In this talk I would like to discuss their proof and outline how to generalize it to all nilpotent algebras of Fitting length >= 3 (work in progress). CEQV and CSAT for nilpotent Maltsev algebras
Nov. 03, 2020 2pm (zoom)	Lie Theory Flor Orosz Hunziker (CU) X In this talk we will discuss the tensor structure associated with certain representations of the Virasoro algebra. In particular, we will show that there is a braided tensor category structure on the category of C1-cofinite modules for the Virasoro vertex operator algebras of arbitrary central charge. This talk is based on joint work with Thomas Creutzig, Cuibo Jiang, David Ridout and Jinwei Yang. Tensor Categories arising from the Virasoro algebra
Nov. 04, 2020 4pm (MATH 350)	Topology Marvin Qi (Cu Boulder) X This is the second talk in a sequence of introductory lectures whose goal is to discuss the notions of gerbes, bundle gerbes and their classification. In this talk, we will take a break from stacks and discuss the notion of bundle gerbes. Topic: CU Topology Seminar Time: This is a recurring meeting Meet anytime Join Zoom Meeting https://cuboulder.zoom.us/j/98979344859 Meeting ID: 989 7934 4859 Passcode: 787231 Bundle Gerbes
Nov. 05, 2020 1pm (Zoom (vir…	Rep Theory Yi-Zhi Huang (Rutgers University) X I will introduce an associative algebra A^{\infty}(V) constructed using infinite matrices with entries in a grading-restricted vertex algebra V. The Zhu algebra and its generalizations by Dong-Li-Mason are very special subalgebras of A^{\infty}(V). I will also introduce the new subalgebras A^{N}(V) of A^{\infty}(V), which can be viewed as obtained from finite matrices with entries in V. I will then discuss the relations between lower-bounded generalized V-modules and suitable modules for these associative algebras. This talk is based on the paper arXiv:2009.00262. Associative algebra and the representation theory of grading-restricted vertex algebras.
Nov. 05, 2020 2pm (Zoom)	Functional Analysis Rachel Chaiser (CU Boulder) Morita Equivalence
Nov. 10, 2020 10:30am (zoom)	MathClub Kyle Luh (CU-Boulder) X We'll discuss some mathematics that is motivated by a card trick. We'll then touch on some applications to cryptography. To avoid any disappointment, I should mention that this magic trick is not one you would pay to see. Markovian Kangaroos
Nov. 10, 2020 1pm (Zoom)	Zurab Janelidze (University Stellenbosch, South Africa) X A “form” is a category equipped with abstract subobject posets for its objects and “connections” between those posets induced by morphisms of the category. In this talk we present an application of this structure in developing a self-dual approach to homomorphism theorems for group-like universal algebras. We call a form satisfying the axioms necessary for obtaining the standard isomorphism theorems a “noetherian form”. Examples of noetherian forms abound (in some of the examples, though, the isomorphism theorems trivialize). For instance, any bounded lattice can be viewed as a noetherian form with a trivial base category. Furthermore, not only group-like universal algebras (e.g., modules, groups, rings with or without unit, loops, etc.), but even any variety of universal algebras can be viewed as a noetherian form, thanks to the fact that the category of sets has the structure of a noetherian form (a non-trivial fact, since the two natural forms, that of subsets and that of equivalence relations, are both non-neotherian). If time allows, we will also discuss some combinatorial problems suggested by the notion of a noetherian form. Noetherian forms
Nov. 10, 2020 2pm (zoom)	Lie Theory Hugh Thomas (Université du Québec à Montréal) X Full heaps are an interesting collection of posets defined by Richard Green, associated to an extended Dynkin diagram and a choice of two extending nodes. They were introduced for Lie-theoretic applications. I will describe a different way to construct them (in the simply-laced cases) based on the representation theory of quivers. I will not assume that the audience is familiar with the theory of quiver representations, so I will try to give a bit of an introduction to them. This is based on joint work with Al Garver and Becky Patrias, and is motivated by our study of affine type generalizations of the RSK correspondence. I won't have time to say much about that, but I will at the very least attempt to explain how we use full heaps to define an affine version of reverse plane partitions. (The talk will be given on Zoom at https://cuboulder.zoom.us/j/94126918788.) Simply-laced heaps via quiver representations
Nov. 10, 2020 4pm (Zoom)	Topology Morgan Opie (Harvard) X Given the ubiquity of vector bundles, it is perhaps surprising that there are so many open questions about them -- even on projective spaces. In this talk, I will outline some results about vector bundles on projective spaces, including my ongoing work on complex rank 3 topological vector bundles on CP^5. In particular, I will describe a classification of such bundles which involves a surprising connection to topological modular forms; a concrete, rank-preserving additive structure which allows for the construction of new rank 3 bundles on CP^5 from "simple" ones; and future directions related to this project. Topic: CU Topology Seminar Time: This is a recurring meeting Meet anytime Join Zoom Meeting https://cuboulder.zoom.us/j/98979344859 Meeting ID: 989 7934 4859 Passcode: 787231 Vector bundles on projective spaces
Nov. 12, 2020 1pm (Zoom (vir…	Rep Theory Antun Milas (SUNY-Albany) X We prove several families of q-series identities that are motivated by the correspondence between 4d N = 2 superconformal field theories (SCFTs) and vertex operator superalgebras. We also discuss identities coming from certain non-commutative q-series and quivers representation theory Some q-series identities related to characters of vertex algebras
Nov. 12, 2020 2pm (Zoom)	Functional Analysis Andrew Stocker (CU Boulder) The Rieffel Correspondence
Nov. 16, 2020 2pm (Zoom)	DeLong Barry Simon (Cal Tech) X This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. Tales of Our Forefathers
Nov. 17, 2020 1pm (Zoom)	Erkko Lehtonen (Universidade NOVA de Lisboa, Portugal) X The associative spectrum was introduced by Csákány and Waldhauser in 2000 as a method of quantifying the non-associativity of binary operations or the corresponding groupoids. The associative spectrum of a groupoid G is an integer sequence, the n-th member of which equals the number of distinct term operations induced on G by the bracketings of n variables. Graph algebras were introduced by Shallon in 1979 and provide a useful representation of directed graphs as algebras with a binary operation. In this talk, we report our work on associative spectra of graph algebras. We classify undirected graphs according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. For arbitrary digraphs, the situation is considerably more complicated. We provide a necessary and sufficient condition for a graph algebra to satisfy a given bracketing identity, expressed in terms of several numerical structural parameters associated, on the one hand, with the graph and, on the other hand, with a pair of bracketings. Based on this, we establish bounds on the possible associative spectra of graph algebras; such a spectrum is either a constant sequence bounded above by 2 or it grows exponentially. This stands in stark contrast with associative spectra of arbitrary groupoids, for which other constant and subexponential spectra are also possible. This is joint work with Tamás Waldhauser (University of Szeged). Associative spectra of graph algebras
Nov. 18, 2020 2pm (via zoom)	Math Phys Barry Simon (Caltech) X In recognition of the 100th anniversary of the Ising model, I will describe the use of reflection positivity in the theory of phase transitions. I will especially focus on presenting the method of Infrared Bounds. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Meeting ID: 940 0255 3301 Passcode: 790356 Reflection Positivity as a Tool in the Theory of Phase Transition Sponsored by the Meyer Fund
Nov. 19, 2020 9am (Zoom (vir…	Rep Theory Anne Moreau (Paris-Saclay) X To any vertex algebra one can attach in a canonical way a certain Poisson variety, called the associated variety. Nilpotent Slodowy slices appear as associated varieties of admissible (simple) W-algebras. They also appear as Higgs branches of the Argyres-Douglas theories in 4d N=2 SCFT’s. These two facts are linked by the so-called Higgs branch conjecture. In this talk I will explain how to exploit the geometry of nilpotent Slodowy slices to study some properties of W-algebras whose motivation stems from physics. In particular I will be interested in collapsing levels for W-algebras. This is a joint work (still in preparation) with Tomoyuki Arakawa and Jethro van Ekeren. Singularities of nilpotent Slodowy slices and collapsing levels for W-algebras
Nov. 19, 2020 2pm (Zoom)	DeLong Barry Simon (Cal Tech) X During the last 20 years there has been a considerable literature on a collection of related mathematical topics: higher degree versions of Poncelet’s Theorem, certain measures associated to some finite Blaschke products and the numerical range of finite dimensional completely non-unitary contractions with defect index 1. I will explain that without realizing it, the authors of these works were discussing Orthogonal Polynomials on the Unit Circle (OPUC). This will allow us to use OPUC methods to provide illuminating proofs of some of their results and in turn to allow the insights from this literature to tell us something about OPUC. This is joint work with Andrei Martínez-Finkelshtein and Brian Simanek. Poncelet’s Theorem, Paraorthogonal Polynomials and the Numerical Range of Truncated GGT matrices
Nov. 19, 2020 4pm (Zoom)	Grad Student Seminar Rachel Chaiser (CU Boulder) X C-algebras have an incredible amount of both algebraic and analytic structure. We give a flavor for how the two worlds collide via examples and building up some of the basic structure, with an eye toward classifying commutative C-algebras and stating everyone's favorite equivalence of categories. I'll also talk about some of my favorite C-algebra facts. An introduction to C-algebras
Nov. 24, 2020 10:30am (zoom)	MathClub Marty Walter (CU-Boulder) X We will give a snap shot of the second edition of my book, "Mathematics for the Environment: Teaching Math As If Survival Matters." How is the Pandemic a warmup for other crises like global warming, collapse of industrial agriculture, universal pollution, the sixth extinction and more. A mathematical perspective implemented by elementary mathematics is useful in redesigning our civilization so that it (and we) shall survive. Doing Math As If Survival Matters
Nov. 24, 2020 1pm (Zoom)	David Broodryk (University of Cape Town, South Africa) X Many categorical conditions can be translated into the language of universal algebra as Mal'cev conditions. For example, much of the theory of Mal'cev varieties can be reproduced in the purely categorical context of Mal'cev categories. Then, the famous result that a variety is congruence-permutable if and only if it has a Mal'cev term t(x,y,y)=x and t(x,x,y)=y can be seen as a syntactical characterisation of this categorical condition. In this talk, we present such a characterisation for the categorical condition of co-extensivity, where a category C is said to be co-extensive if for each pair of objects X,Y in C, the canonical functor ×:X/C×Y/C?(X×Y)/Cis an equivalence. As a motivating example, we begin by considering the property of commutative semirings that to present a commutative semiring S as a product is exactly to find two elements of S which sum to 1 and whose product is 0. This very naturally implies that the variety of commutative semirings is co-extensive, and it then becomes interesting to ask exactly which other varieties are co-extensive. In order to answer this question, we first characterise the weaker condition of left co-extensivity. We then show that any co-extensive variety must have what we call a 'diagonalising term'. Lastly, we complete the characterisation by finding a sufficient and necessary set of identities this term must satisfy in order for the variety to be co-extensive. Characterisation of co-extensive varieties of universal algebras
Nov. 24, 2020 2pm (MATH 350)	Lie Theory Tianyuan Xu (CU Boulder) X Lusztig's $a$ -function on a Coxeter group $W$ is a function that takes a constant nonnegative integer value on each Kazhdan--Lusztig cell of $W$ . The function plays a key role in the study of the cells of $W$ and the cell representations of the Hecke algebra of $W$ , but it is also very difficult to compute. For an element $w$ in $W$ , we have $a(w)=0$ if and only if $w$ is the identity and $a(w)=1$ if and only if $w$ has a unique nonempty reduced word; moreover, the elements of $a$ -values 0 and 1 each form a single two-sided cell. We report on some extensions of these results about the set $W2$ of elements with $a$ -value 2. In particular, we classify Coxeter groups $W$ where $W2$ is finite, count $W2$ , and describe the partition of $W2$ into cells when $W2$ is finite. If time allows, we will also discuss a connection between cell representations associated to $W2$ and the reflection representation of $W$ in types $A D E$ . (Joint work with Richard Green.) On Kazhdan--Lusztig cells of $a$ -value 2
Nov. 25, 2020 5pm (Zoom (vir…	Rep Theory Shoma Sugimoto (Kyoto University) X For a finite dimensional simply-laced simple Lie algebra $g$ and an integer $p≥ 2$ , we can attach the logarithmic $W$ -algebra $W(p)Q$ . When $g=sl2$ , $W(p)Q$ is called the triplet $W$ -algebra, and studied by many people as one of the most famous examples of $C2$ -cofinite but irrational vertex operator algebra. However, apart from the triplet $W$ -algebra, not much is known about the log $W$ -algebras $W(p)Q$ . In this talk, after we construct $W(p)Q$ and their modules $W(p,λ)Q$ geometrically along the preprint of Feigin-Tipunin, first we show the simplicity, $Wk(g)$ -module structure, and character formula of $W(p,λ)Q$ when $\sqrt{p} \bar{λ}$ is in the closure of the fundamental alcove. In particular, for $p≥ h-1$ , $W(p)Q$ is simple and decomposed into simple $Wk(g)$ -modules. Second we give a purely $W$ -algebraic algorithm to calculate nilpotent elements in the Zhu's $C2$ -algebra of $W(p)Q$ much easier than straightforward way. Using this algorithm to the cases $g=sl3$ and $p=2,3$ , we show that $W(p)Q$ is $C2$ -cofinite in these cases. On the log W-algebra.
Dec. 01, 2020 1pm (Zoom)	Tamas Waldhauser (University of Szeged, Hungary) X Let A be a finite algebra and let C be the clone of term operations of A. A set of tuples is called and algebraic set, if it is the set of all solutions of a system of equations over A. These algebraic sets are the main objects of study in the so-called universal algebraic geometry initiated by Boris Plotkin. It is straightforward to verify that algebraic sets are always closed under C, the centralizer of the clone C. We are interested in algebras for which the converse is also true, i.e., where solution sets of systems of equations can be characterized as C-closed sets of tuples. We prove that this holds if and only if the algebra is polymorphism-homogeneous, and we determine such algebras in several classes of algebras, namely two-element algebras, finite lattices, finite semilattices, finite abelian groups, finite monounary algebras. This is joint work with Endre Tóth (University of Szeged). Solution sets and polymorphism-homogeneity
Dec. 02, 2020 3pm (via zoom)	Math Phys Andrew Hamilton (CU Boulder Astrophysics Department) X The spinors of the group Spin(N) of rotations in N spacetime dimensions are indexed by a bitcode with [N/2] bits. A well-known promising grand unified group that contains the standard-model group is Spin(10). Fermions in the standard model are described by five bits durgb, consisting of two weak bits d and u, and three color bits r, g, b. If a sixth bit T is added, necessary to accommodate a time dimension, then the enlarged Spin(11,1) algebra contains the standard-model and Dirac algebras as commuting subalgebras, unifying all four forces. The largest subgroup of Spin(11,1) that commutes with the Poincare group is Spin(5)xSpin(6), suggesting that the latter is a partial unification on the way to complete unification in Spin(11,1). The Spin(5)xSpin(6) algebra contains a subalgebra with precisely the properties of the electroweak Higgs field. The Spin(5)xSpin(6) symmetry contains, and is spontaneously broken by, a U(1) symmetry related to the U_{B-L}(1) symmetry. Grand unification is associated with a change in the dimensionality of spacetime. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Meeting ID: 940 0255 3301 Passcode: 790356 Unification of the four forces in the Spin(11,1) geometric algebra
Dec. 03, 2020 1pm (Zoom (vir…	Rep Theory Chiara Damiolini (Rutgers University) X In this talk I will discuss certain properties of sheaves of covacua and conformal blocks attached to modules over vertex operator algebras. After briefly recalling how these objects are constructed from a geometric point of view, I will focus on the conditions required to construct Cohomological Field Theories from these sheaves. If time permits I will also discuss open problems which naturally arise. This is based on joint works with A. Gibney and N. Tarasca. Cohomological Field Theories from vertex operator algebras
Dec. 03, 2020 4pm (Zoom)	Grad Student Seminar Andrew Stocker (CU Boulder) X Symbolic dynamics is the study of dynamical systems represented as infinite sequences of symbols --- these are called shift spaces. We will talk about several different types of shift spaces with varying complexity, and about how to represent a general dynamical system as a shift space. Symbolic Dynamics
Dec. 08, 2020 1pm (Zoom)	Michael Hoefnagel (University Stellenbosch, South Africa) X Consider the Mal’tsev condition on a variety which asserts the existence of an n-ary term p satisfying some equations of the form p(x_1 , . . . , x_n ) = y. Such a Mal’tsev condition can be reformulated within the internal language of an abstract category, using the so-called ‘matrix method’ of Z. Janelidze. The corresponding categorical condition is captured by a matrix of positive integers, and is formally called a closedness property of internal relations, or a ‘matrix property’ for short. Some of these properties follow from (conjunctions of) others, for example, the matrix property corresponding to the equations defining a majority term follows from the matrix property corresponding to a Pixley term. This then is the categorical theorem corresponding the fact that if p(x, y, z) is a Pixley term, then m(x, y, z) = p(x, p(x, y, z), z) is a majority term. The main aim of this talk is to present some recent work together with Z. Janelidze and P.-A. Jacqmin which has produced an algorithm for deciding implications of (conjunctions of) matrix properties in the light context of left-exact categories. Such implications of matrix properties are context sensitive, i.e., they depend on what further conditions the base category satisfies. In particular, we will see that implications of matrix properties in the context of varieties of algebras can be different from the context of left exact categories. A classification of left exact categories motivated from universal algebra
Dec. 08, 2020 2pm (MATH 350)	Lie Theory Nat Thiem (CU Boulder) X There are many instances throughout mathematics where the representation theory of algebraic structures becomes prohibitively challenging. In the theory of finite groups, there is a reasonable work-around that is known as a supercharacter theory. The basic idea is to limit attention to modules that are better behaved than irreducible modules, but that still retain many of the nice properties we expect from irreducibles. This talk gives an introduction to this approach with a focus on techniques that might have a chance of extending to other types of algebraic structures. Many such techniques allow the user to tailor a representation theory to specific applications, a process I will illustrate with some examples in combinatorics. An introduction to approximating representation theories
Dec. 10, 2020 1pm (Zoom (vir…	Rep Theory Shigenori Nagatsuka (University of Tokyo) X Recently, dualities among W-superalgebras and their affine cosets conjectured by Gaiotto-Rapcak have been established in many cases by Creutzig-Linshaw and Creutzig-Linshaw-Kanade by using universal objects of such algebras. Independently, Creutzig-Genra and I proved the duality in the case of subregular W-algebras and principal W-superalgebras of type A by using free field realizations of those algebras. This point of view upgrades the duality to a "reconstruction theorem" of one of the algebra from the other one. The simplest example is the Kazama-Suzuki coset construction of N=2 superconformal algebra from the affine sl2 vertex algebra and its inverse by Feigin-Semikhatov-Tipunin. In this talk, I will explain this reconstruction theorem and then its application to the representation theory of principal W-superalgebra side in the rational cases. This talk is based on on-going project with Thomas Creutzig, Naoki Genra and Ryo Sato Duality of subregular W-algebras and principal W-superalgebras of type A and their representations in rational cases.
Dec. 10, 2020 4pm (Zoom)	Grad Student Seminar Pat Wynne (CU Boulder) Turing Computability & Post’s Correspondence Problem
Dec. 17, 2020 1pm (Zoom (vir…	Rep Theory Drazen Adamovic (University of Zagreb) X We will review recent results appearing in the last five years including the representation theory of affine vertex algebras beyond the category O, semi-simplicity of representations at collapsing levels and some applications to logarithmic vertex algebras. Affine Vertex Algebras, collapsing levels and representation theory

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