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Events for October 2020 – February 2021

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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
Jan. 14, 2021 9:10am (Zoom 9…	Ulam Alexandr Kazda X The Constraint Satisfaction Problem is a problem where one is tasked with deciding the truth of the sentence "there exist values for variables x,y,... such that this conjunction of predicates on x,y,... holds." While CSP is a computational problem, it leads to many purely mathematical questions in logic, algebra, category theory, and combinatorics. CSP (and its variants) will be the main theme of this semester's Ulam seminar. The complexity of CSP depends on the expressive power of the predicates that we allow. In the talk, we will look at the project of classifying the complexity of CSP on finite structures where there turned out to be a clear dividing line between "easy" and "hard" languages. We will also briefly introduce some common variants of CSP. A brief history of the Constraint Satisfaction Problem
Jan. 14, 2021 1pm (Zoom (vir…	Rep Theory Shashank Kanade (University of Denver) X Principal characters of standard (i.e., highest weight, integrable) modules for affine Lie algebras have been a rich source of q-series and partition identities. The algebra A_1^{(1)} (or, sl_2^) was "understood" in this sense a few decades ago. On q-series side, this leads to identities of Gordon-Andrews and Andrews-Bressoud. In this talk, I'll present q-series identities related to the next "simplest" affine Lie algebra, namely, A_2^{(2)}. Here, we get six families of q-series identities confirming a conjecture of McLaughlin and Sills. The main machinery used is that of Bailey pairs and Bailey lattices. This is a joint work with Matthew C. Russell. (N.B.: These q-series include Vir(3,p) minimal model characters.) Principal characters of standard A_2^(2)-modules
Jan. 19, 2021 1pm (Zoom)	Charlotte Aten (University of Rochester) X We study a class of algebras we regard as generalized rock-paper-scissors games. We show how to construct an infinite family of such algebras via group actions and that certain subsets of this family generate the varieties generated by hypertournament algebras, an n-ary analogue of tournament algebras. The group action construction allows us to find automorphisms of these algebras and determine their congruence lattices. We'll discuss how these results may aid in finding a (necessarily infinite) basis for the variety generated by tournament algebras, which to this author's knowledge remains an open problem. Multiplayer rock-paper-scissors
Jan. 21, 2021 9:10am (Zoom 9…	Ulam Alexandr Kazda X Polymorphims play a major role in the dichotomy proof for the complexity of the Constraint Satisfaction Problem. However, polymorphisms also help us better understand relational structures. A popular type of relational structure is the directed graph (digraph). It turns out that polymorphisms of directed graphs are close to being as general as possible: Jakub Bulin, Dejan Delic, Marcel Jackson and Todd Niven have shown that for each relational structure A there is a digraph G such that A and G are, roughly speaking, the same with respect to having many types of polymorphisms. However, one notable exception are the so-called Maltsev polymorphisms; we will show that whenever a digraph has Maltsev polymorphism, it must also have a majority polymorphism. This is an implication not true in general relational structures. Polymorphisms of directed graphs
Jan. 21, 2021 5pm (Zoom (vir…	Rep Theory Cuipo Jiang (Shanghai JiaoTong University) X We prove that the universal affine vertex algebra associated with a simple Lie algebra g is simple if and only if the associated variety of its unique simple quotient is equal to g*. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra. This is a joint work with T. Arakawa and A. Moreau. Simplicity of vacuum modules and associated varieties.
Jan. 26, 2021 1pm (Zoom)	Jason Parker (Brandon University) X In [2], my PhD supervisors (Pieter Hofstra and Philip Scott) and I studied the new topos-theoretic phenomenon of isotropy (as introduced in [1]) in the context of single-sorted algebraic theories, and we gave a logical/syntactic characterization of the isotropy group of any such theory, thereby showing that it encodes a notion of inner automorphism or conjugation for the theory. In the present talk, I will summarize the results of my recent PhD thesis, in which I build on this earlier work by studying the isotropy groups of (multi-sorted) quasi-equational theories (also known as essentially algebraic, cartesian, or finite limit theories). In particular, I will show how to give a logical/syntactic characterization of the isotropy group of any such theory, and that it encodes a notion of inner automorphism or conjugation for the theory. I will also describe how I have used this characterization to exactly characterize the ‘inner automorphisms’ for several different examples of quasi-equational theories, most notably the theory of strict monoidal categories and the theory of presheaves valued in a category of models. In particular, the latter example provides a characterization of the (covariant) isotropy group of a category of set-valued presheaves, which had been an open question in the theory of categorical isotropy. [1] J. Funk, P. Hofstra, B. Steinberg. Isotropy and crossed toposes. Theory and Applications of Categories 26, 660-709, 2012. [2] P. Hofstra, J. Parker, P.J. Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Computer Science 341, 201-217, 2018. Isotropy groups of quasi-equational theories
Jan. 27, 2021 4pm (via zoom)	Math Phys Flor Orosz Hunziker (CU Boulder) X The minimal models are representations of the Virasoro algebra for particular central charges c_{p,q}. For these central charges the category of representations has finitely many irreducibles and it gives rise to a rational conformal field theory. In this talk, we focus on the non rational central charges where the number of irreducibles is infinite. We prove that there is a braided tensor category structure on a natural subcategory of representations of the Virasoro algebra for arbitrary central charge. This talk is based on joint work with Thomas Creutzig, Cuibo Jiang, David Ridout and Jinwei Yang. Meeting ID: 940 0255 3301 Passcode: 790356 Tensor categories for non rational Virasoro vertex algebras
Jan. 28, 2021 9am (Zoom (vir…	Rep Theory Ana Ros Camacho (Cardiff University) X Algebras in tensor categories appear in several interesting research areas, like e.g. VOA extensions or spin topological field theories, but they are usually tricky to find. In this talk, we will explain how to generalize a result by Ostrik and Natale on algebra objects in categories related to lattice VOAs to the case of so-called group-theoretical fusion categories. The algebra objects we find for these also have very good properties that we will describe in detail. We will assume little knowledge of categories. Joint work with the WINART2 team Y. Morales, M. Mueller, J. Plavnik, A. Tabiri and C. Walton Algebra objects in group-theoretical fusion categories.
Jan. 28, 2021 9:10am (Zoom 9…	Ulam Alexandr Kazda X The Promise Constraint Satisfaction Problem is a variant of the Constraint Satisfaction Problem (CSP). In PCSP, we are promised that either there exists a homomorphism from A to B, or there is even no homomrphism from A to a structure C; our goal is to decide between these two cases. An example of PCSP is PCSP(K_3,K_4), where we are deciding if the input graph is 3-colorable or not even 4-colorable. The natural generalization of polymorphisms to pairs of structure are minions of polymorphisms. Minons have very little structure; compared to algebras we lose the ability to compose operations. Surprisingly, what little structure is left still leads to nontrivial mathematics. In the talk, we will have a look at polymorphisms in general and polymorphisms from K_3 to K_4 in particular. Promise CSP: Arise my minions!
Feb. 02, 2021 10:30am (Zoom)	MathClub Kevin Manley (CU-Boulder) X In this talk, we’ll look at a few examples from Georg Cantor that clarified (mostly through frustration and confusion) several areas of mathematics. There’ll be a little set theory, a little calculus, a little geometry, and a bunch of head scratching. A Little Weirdness from Georg Cantor
Feb. 02, 2021 1pm (Zoom)	Matt Valeriote (McMaster University, Canada) X Given a subalgebra A of a direct product $Πi≤ nAi$ of algebras $A_{i}$ from some variety V, one can consider Proj $k(A)$ , the system of projections of A onto all k-element sets of coordinates. In general, A is not uniquely determined by Proj $k(A)$ , but if V happens to have a (k+1)-near unanimity term, then Kirby Baker and Alden Pixley show that this is the case. They also show that if a variety V satisfies this uniqueness property for all subalgebras of direct products of its members, then it must have a (k+1)-ary near unanimity term. In this talk I will consider the following existence question: Given a system of k-fold projections $Γ$ over some direct product $Πi≤ nAi$ of algebras $A_{i}$ from a variety V, under what circumstances will there exist a subalgebra A of $Πi≤ n Ai$ such that $Γ=$ Proj $k(A)$ ? An answer will be given that settles a question posed by George Bergman in a paper from 1977. This is joint work with Libor Barto, Marcin Kozik, and Johnson Tan. Near unanimity terms and the Baker-Pixley Theorem
Feb. 04, 2021 9:10am (Zoom 9…	Ulam Marcin Wrochna (University of Oxford) X Graph coloring is a classic hard algorithmic problem. In promise graph coloring, we ask if at least an approximation is feasible: given a graph promised to be 3-colorable, say, can we efficiently find a 100-coloring? To prove that even this relaxed problem is hard, we study polymorphisms (think: algebraic relations between solutions). It turns out that in some cases, basic topological tools give a very direct understanding of those polymorphisms, seeing them as continuous maps from toruses to spheres. Joint work with Andrei Krokhin, Jakub Opršal, Stanislav Živný. Topology in promise constraint satisfaction
Feb. 04, 2021 1pm (Zoom (vir…	Rep Theory Christoph Keller (University of Arizona) X Physicists are interested in holographic families of VOAs. These are families of VOAs that on the one hand have dim V_n `small' for `small' n, and on the other hand have some kind of large central charge limit. I will discuss the motivation behind these requirements and the connection to extremal VOAs. I will then discuss some attempts at constructing such families, namely permutation orbifold VOAs and lattice orbifold VOAs. This talk is based on joint work with Thomas Gemuenden. Holographic Families of VOAs
Feb. 04, 2021 4pm (Zoom)	Grad Student Seminar Adrian Neff (CU Boulder) X If I told you that there is an abundance of geometry lurking in the prime ideals of a ring, then you probably wouldn't believe me. In this talk, I'll (hopefully) convince you otherwise. I'll introduce the (underlying set of) affine schemes from a geometric point of view, starting with motivation from classical algebraic geometry, and after looking at the hidden geometry of rings, we'll take a look at why this perspective is "better" than the classical view. A geometric introduction to affine schemes
Feb. 09, 2021 2pm (MATH 350)	Lie Theory Nicholas Davidson (Reed College) X Webs are combinatorially defined diagrams which encode homomorphisms between tensor products of certain representations of Lie (super)algebras. I will describe some recent work which defines webs associated to the the type P Lie superalgebra, and then I will use these webs to deduce an analog of Howe duality for this Lie superalgebra. Type P Webs and Howe Duality
Feb. 10, 2021 10am (via zoom)	Math Phys Matthias Lesch (University of Bonn) X Abstract: The spectral flow is a priori an integer invariant of a path of unbounded self-adjoint Fredholm operators. Its origin lies in Atiyah-Patodi-Singers famous work on index theory for manifolds with boundary. In this overview talk I will first address at least two rigorous definitions. I will also explain the relation to the homotopy theory of the classifying spaces of the K-functor. Finally, I will hint at recent joint work with Bourne, Carey, and Rennie on a KO-valued version of spectral flow. The latter finds its explanation in the aforementioned relation between the spectral flow and the homotopy of the classifying spaces for the K-functor. The talk will be non-technical and address a general math/physics audience with a certain functional analytic interest. Meeting ID: 940 0255 3301 Passcode: 790356 Spectral Flow
Feb. 11, 2021 9:10am (Zoom 9…	Ulam Libor Barto (Charles University) X The Promise Constraint Satisfaction Problem (PCSP) is a generalization of the Constraint Satisfaction Problem (CSP) that includes approximation variants of satisfiability and graph coloring problems. In a [LICS '19] paper it was shown that a specific PCSP, the problem to find a valid Not-All-Equal solution to a 1-in-3-SAT instance, is not finitely tractable in that it can be solved by a trivial reduction to a tractable CSP, but such a CSP is necessarily over an infinite domain (unless P=NP). We further explore this phenomenon: we give a general necessary condition for finite tractability and characterize finite tractability within a class of templates - the "basic" tractable cases in the dichotomy theorem for symmetric Boolean PCSPs allowing negations by Brakensiek and Guruswami [SODA'18]. This is a joint work with Kristina Asimi. Finitely tractable PCSPs
Feb. 11, 2021 11am (Zoom)	Noncomm Geometry Manuel Rivera (Purdue University) X I will describe the sense in which the coalgebraic structure of the singular chains on a connected space determines its fundamental group in complete generality. An extension of Adams’ classical cobar theorem, from simply connected spaces to spaces with arbitrary fundamental group, lies at the bottom of this new observation. The key idea is to consider the singular chains as an E-infinity coalgebra under a notion of weak equivalence drawn from the Koszul duality theory of co/associative structures. This leads to an extension of rational homotopy theory of nilpotent spaces to spaces with arbitrary fundamental group, where the notion of weak equivalence between spaces is defined to be a continuous map inducing an isomorphism on fundamental groups and an isomorphism on higher rationalized homotopy groups. I will also describe how these constructions may be used to obtain a Hochschild type model for the free loop space of a non-simply connected space that is potentially useful for studying string topology. Algebraic models for non-simply connected spaces and their loop spaces
Feb. 11, 2021 4pm (Zoom (vir…	Rep Theory Tomoyuki Arakawa (Kyoto University) X The 4D/2D duality discovered by Beem et at in physics gives a remarkable connection between 4D N=2 SCFTs and VOAs. It gives not only many new interesting examples of VOAs but also new perspectives to known VOAs, such as Frenkel-Styrkas’s modified regular representation of the Virasoro algebra and Adamovic’s realization of N=4 small superconformal algebra. In this talk I will discuss the 4D/2D duality from the VOA perspective, starting from these examples. 4D/2D duality and VOA theory
Feb. 16, 2021 10:30am (zoom)	MathClub Keith Kearnes (CU-Boulder) X My goal will be to describe the relationship between algebra and geometry in the study of projective planes. The structure of projective planes
Feb. 16, 2021 1pm (Zoom)	Erhard Aichinger (JKU Linz, Austria) X The degree of a function $f$ between two abelian groups has been defined as the smallest natural number $d$ such that $f$ vanishes after $d+1$ applications of any of the difference operators $Δ_{a}$ defined by $Δa * f ,, (x) = f(x+a) - f(x)$ . Functions of finite degree have also been called generalized polynomials or solutions to Fr\'echet's functional equations. A pivotal result by A. Leibman (2002) is that $deg (f ∘ g) ≤ deg(f) ⋅deg (g)$ . We show how results on the degree can be used (i) to get lower bounds on the number of solutions of equations, and (ii) to connect nilpotency and supernilpotency. This leads to generalizations of the Chevalley-Warning Theorems to abelian groups, a group version of the Ax-Katz Theorem on the number of zeros of polynomial functions, and a computable $f$ such that all finite $k$ -nilpotent algebras of prime power order in congruence modular varieties are $f(k,.)$ -supernilpotent. The degree as a measure of complexity of functions on a universal algebra
Feb. 18, 2021 9:10am (Zoom 9…	Ulam Jakub Opršal (Durham University) X Since deciding whether a given graph is 3-colourable is NP-complete, and checking validity of a colouring is an easy problem, also the problem of finding a colouring of a 3-colourable graph with 3 colours is a well-known NP-complete. One might be interested in a relaxed version of this problem, e.g., finding a colouring of a 3-colourable graph that uses 6 colours. This falls into a more general scope of so-called promise constraint satisfaction problem (PCSP). We will describe the basics of a theory characterising when one promise constraint satisfaction problem can be reduced to another using a gadget reduction. A theory of gadget reductions for promise constraint satisfaction I
Feb. 18, 2021 1pm (Zoom (vir…	Rep Theory Bin Gui (Rutgers University) X Given a strongly rational unitary VOA $V$ , a Hermitian form on the space of its intertwining operators was introduced recently to understand the unitarity of the representation modular tensor category $Rep(V)$ . It was actually shown that, along with some natural assumptions, if this Hermitian form (which is necessarily non-degenerate) is positive, namely, if it is an inner product, then $Rep(V)$ is unitary. The crucial step of this story is to prove the positivity of the Hermitian form. In this talk, I give a geometric interpretation of this positivity problem using the idea (self)conjugate Riemann surfaces and (self)conjugate conformal blocks. Conjugation and positivity of conformal blocks
Feb. 23, 2021 1pm (Zoom)	Kristina Asimi (Charles University, Prague) X The Promise Constraint Satisfaction Problem (PCSP) is a generalization of the Constraint Satisfaction Problem (CSP). In a [LICS '19] paper it was shown that a specific PCSP, the problem to find a valid Not-All-Equal solution to a 1-in-3-SAT instance, is not finitely tractable in that it can be solved by a trivial reduction to a tractable CSP, but such a CSP is necessarily over an infinite domain (unless P=NP). We further explore this phenomenon: we give a general necessary condition for finite tractability and characterize finite tractability within a class of templates - the "basic" tractable cases in the dichotomy theorem for symmetric Boolean PCSPs allowing negations by Brakensiek and Guruswami [SODA'18]. This is a joint work with Libor Barto. Finitely tractable PCSPs
Feb. 23, 2021 2pm (MATH 350)	Lie Theory Monica Vazirani (UC Davis) X Building on the work of Calaque-Enriquez-Etingof, Lyubashenko-Majid, and Arakawa-Suzuki, Jordan constructed a functor from quantum $D$ -modules on general linear groups to representations of the double affine Hecke algebra (DAHA) in type $A$ . When we input quantum functions on GL(N) the output is $L(kN)$ , the irreducible DAHA representation indexed by an $N \times k$ rectangle. For the specified parameters $L(kN)$ is Y-semisimple, i.e. one can diagonalize the Dunkl operators. We give an explicit combinatorial description of this module via its Y-weight basis. This is joint work with David Jordan. The rectangular representation of the double affine Hecke algebra via elliptic Schur-Weyl duality
Feb. 24, 2021 4pm (via zoom)	Math Phys Howie Jordan (University of Colorado) X Contextuality is a fundamental feature of quantum mechanics whereby we realize that the measurements we make of a quantum system depend crucially on the context in which we make them. One might hope that there are preexisting parameters with definite values underlying quantum mechanics, which some more fundamental theory could use to explain the measurements we observe. Theorems of Bell, Kochen, and Specker established that this cannot be the case. Underlying such theorems is a common theme: information available locally (i.e., on restricted contexts) cannot be extended globally. This realization led to a reformulation of contextuality in terms of (pre)sheaves, a mathematical tool capturing the essence of such local-global problems. In this talk, we will overview this sheaf theoretic treatment of contextuality. Presheaves defined over the commutative subalgebras of a C*-Algebra play a central role. We reformulate the Kochen-Specker theorem, the original inspiration for this approach. We also view the broader idea of sheaf-theoretic contextuality from this setting, including the connection to sheaf cohomology and topos theory. Meeting ID: 940 0255 3301 Passcode: 790356 The Sheaf Theoretic Approach to Contextuality in Quantum Mechanics
Feb. 25, 2021 9:10am (Zoom 9…	Ulam Jakub Opršal (Durham University) X Since deciding whether a given graph is 3-colourable is NP-complete, and checking validity of a colouring is an easy problem, also the problem of finding a colouring of a 3-colourable graph with 3 colours is a well-known NP-complete. One might be interested in a relaxed version of this problem, e.g., finding a colouring of a 3-colourable graph that uses 6 colours. This falls into a more general scope of so-called promise constraint satisfaction problem (PCSP). We will describe the basics of a theory characterising when one promise constraint satisfaction problem can be reduced to another using a gadget reduction. A theory of gadget reductions for promise constraint satisfaction II
Feb. 25, 2021 4pm (Zoom)	Grad Student Seminar Chase Meadors (CU Boulder) X Many of our calculus 1-3 students know how to get their computers do their homework, so why can't we do the same for ours? We will explore the notion of type theories and how they can be used as a formal foundation of mathematics, in contrast to typical set-theoretic foundations. We will then explore how proof assistants like Lean actually apply type theory to formalize mathematics. If time permits, we will briefly discuss some of the main "points" of Homotopy Type Theory and current research into its implementation. A Tour of Type Theory
Feb. 25, 2021 5pm (Zoom (vir…	Rep Theory Robert McRae (Tshinghua University) X I will discuss work in progress related to proving semisimplicity of the module category for a suitable positive-energy, self-contragredient, C_2-cofinite vertex operator algebra V. The goal is to show that the category of V-modules is semisimple if the Zhu algebra of V is a semisimple algebra. The idea for proving this is to show that the braided tensor category of V-modules is rigid with a non-degenerate braiding, using tensor-categorical methods combined with the modular invariance methods used by Huang to prove the Verlinde conjecture for rational vertex operator algebras. On semisimplicity of modules for C_2-cofinite vertex operator algebras
Feb. 25, 2021 5:15pm (Zoom)	MathClub organized by Agnes Beaudry and Jonathan Quartin X What's undergraduate math research anyways? Who is it for? It might not be what you think it's like and it might be for you, even if you haven't realized it yet! So come to the Math Club on Thursday Feb 25 at 5:15 pm to learn more about undergraduate math research and the opportunities for math research experiences available to CU undergrads. And if you're wondering, does math research happen online? The answer is absolutely! Everyone is welcome! Math Research : Demystified

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