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Events for April – August 2024

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Apr. 02, 2024 1:25pm (online)	Algebra/Logic Will Brian (University of North Carolina at Charlotte) View Online:https://cuboulder.zoom.us/j/96896272260 X I'm going to talk about an old question of van Douwen: Are the shift map and its inverse conjugate in the automorphism group of P()/fin? By the late 1980's, van Douwen and Shelah proved that it is consistent they are not conjugate. Specifically, any automorphism witnessing their conjugacy would need to be nontrivial (van Douwen), but it is consistent that all automorphisms are trivial (Shelah). In this talk I'm going to discuss the recently-proven complementary result: it is consistent that the shift map and its inverse are conjugate and, in fact, it follows from CH. Does P()/fin know its right hand from its left?
Apr. 02, 2024 2:30pm (MATH 3…	Lie Theory Carl Mautner (UC Riverside) X The Schur algebra is a finite-dimensional algebra that relates the representation theory of symmetric and general linear groups. In joint work with Tom Braden we introduce a new algebra we call the Hilbert Schur algebra, which is also finite-dimensional and has the Schur algebra as a quotient. I will describe this algebra, our motivation for defining it, and some of its interesting algebraic and combinatorial properties. Hilbert Schur algebras Sponsored by the Meyer Fund
Apr. 02, 2024 3:30pm (MATH 3…	Topology Emily Montelius View Online:https://cuboulder.zoom.us/j/93245421661 X Together we will watch and discuss some introductory and overview materials covering knot theory from a graduate topology level An introduction to knot theory
Apr. 02, 2024 10:10pm (MATH …	Functional Analysis Rufus Willett (University of Hawaii) X The HK conjecture predicts an isomorphism between the homology of certain groupoids, and the K-theory of their C-algebras. On the other hand, the Baum-Connes conjecture predicts a topological computation of the K-theory of groupoid C-algebras. Both conjectures are wrong: there are counterexamples to the HK conjecture due to Scarparo, Deeley, Ortega-Scarparo, and Chaiser; and to the Baum-Connes conjecture due to Higson-Lafforgue-Skandalis. I will explain some interactions between these two conjectures, in particular focusing on the precise sense in which Baum-Connes predicts that HK is wrong, and how one might try to fix these issues. I will not assume any prior knowledge of the HK or Baum-Connes conjecture, and will focus on the case of group actions rather than general groupoids (for simplicity). Some of this is based on joint work with Christian Bönicke, Clément Dell'Aiera, and Jamie Gabe; some on joint work with Robin Deeley; and some on work of Valerio Proietti and Makoto Yamashita (that I had nothing to do with). The HK and Baum-Connes conjectures Sponsored by the National Science Foundation
Apr. 03, 2024 12:20pm (Zoom)	Math Edu Ally Graf & Patrick Newberry (University of Colorado Boulder) View Online:https://cuboulder.zoom.us/j/91244797970 X Although we assess student knowledge through various methods, one common evaluation technique we use in our courses is a formal exam. In this talk we will share strategies to create an exam that showcases students' understanding of the course material, as well as how to reflect on what went well and what did not following the exam. Creating Exam Questions
Apr. 03, 2024 3:30pm (MATH 3…	Math Phys Rebecah Storms (CU Boulder, Department of Mathematics) View Online:https://cuboulder.zoom.us/j/94002553301 (Password: 790356) X I will give a brief refresh of jet bundles for those who are unfamiliar, and then I will formulate the Einstein Field Equation in this language. General Relativity
Apr. 04, 2024 9:15am (via zo…	Math Phys Albachiara Cogo (University of Tübingen) View Online:https://cuboulder.zoom.us/j/94002553301 X In this joint work with Cederbaum, Leandro and Dos Santos, we generalize to any dimension n+1 Robinson’s divergence formula used to prove the uniqueness of (3+1)-dimensional static black holes. To this end, we use a tensor first introduced by Cao and Chen for the analysis and classification of Ricci solitons. We thereby prove the uniqueness of black holes and of equipotential photon surfaces in the class of asymptotically flat (n+1)-dimensional static vacuum space-times, provided the total scalar curvature of the horizon is properly bounded from above. In the black hole case, our results recover those of Agostiniani and Mazzieri and partially re-establish the results by Gibbons, Ida, and Shiromizu, and Hwang and finally by Raulot in the case of a spin manifold; in the photon surface case, the results by Cederbaum and Galloway can also be proven. Our proof is not based on the positive mass theorem and avoids the spin assumption. Black Hole and Equipotential Photon Surface Uniqueness in (n+1)-dimensional Static Vacuum Spacetimes via Robinson's Method
Apr. 04, 2024 2pm (Online)	Rep Theory Christopher Raymond (Université Laval) View Online:https://cuboulder.zoom.us/j/97927772782 X In this talk, I want to give a pedagogical overview of Zhu's algebras and then describe an alternative approach to understanding their algebraic structure that has resurfaced in 3D-2D correspondences. First, I will introduce the approach to Zhu's algebras in physics and the resulting calculational framework. Following that, I'll describe the link between Zhu's algebras and Yangians and briefly discuss how these links are realised in physics if time permits. Zhu's algebras and related structures.
Apr. 09, 2024 3:30pm (MATH 3…	Topology Emily Montelius X In this talk I will construct the map between Algebraic and Topological K-Theory for Banach Algebras. We will complete this construction by following our Banach algebra through its general linear group construction, taking Milnor's classifying space, and then applying the plus construction. We will show that this construction agrees with the traditional definition of these K-Theories, while also returning maps between them. The comparison map between Algebraic and Topological K-Theory for Banach Algebras
Apr. 10, 2024 3:30pm (MATH 3…	Math Phys Connor McCranie (CU Boulder, Department of Mathematics) Hyperfunctions and Quantum Field Theory
Apr. 12, 2024 9:05am (Zoom)	Math Edu Joseph Timmer (University of Colorado Boulder) View Online:https://cuboulder.zoom.us/j/91244797970 X Using a rubric can make grading go smoothly and make consistency easier to attain. Creating a good rubric typically involves more than just writing out a solution to the problem. In this talk, we discuss what makes for good rubrics that ease the burden of grading and address the pros/cons of various schemes. If time permits, we will create a rubric and see these ideas in action! Creating and Adhering to Rubrics
Apr. 16, 2024 1:25pm (MATH 2…	Algebra/Logic Piotr Kawalek (TU Wien, Austria) View Online:https://cuboulder.zoom.us/j/96896272260 X In 1999, Goldmann and Russell initiated a systematic study of algorithmic complexity of equation satisfiability problems in finite groups. In this talk we present some of the most important historical results on the topic as well as some of the very recent progress. We also consider closely connected problems of satisfiability and equivalence of circuits in finite algebras from congruence modular varieties. Some of the most general results in this area were obtained through relating functions computable by short polynomials in these algebras with classes of bounded-depth modular circuits. We discuss these types of connections, which turn out to be especially useful in tractability proofs. Complexity of Satisfiability and Equivalence Problems in Finite Algebras Sponsored by the Meyer Fund
Apr. 16, 2024 3:30pm (MATH 3…	Topology Ralph Kaufmann (Purdue-University) View Online:https://cuboulder.zoom.us/j/94002553301 X We review the setup of topological invariants as they appear in physics in the interpretation of K-theory. The basic Z-valued invariant can be identified with the first Chern-class. This goes back to the TKNN integers appearing in the quantum Hall effect. More subtle invariants appear in the setting of symmetries. These can be identified as coming from Real K-theory (KR) theory and its shifted cousin, quaternionic K-theory (KQ). We will give a gentle introduction and present the mathematical and condensed matter classification. This is joint work with D. Li and E.B Kaufmann Topological Insulators and K-theory Sponsored by the Meyer Fund
Apr. 17, 2024 3:30pm (MATH 3…	Math Phys Birgit Kaufmann (Purdue-University) View Online:https://cuboulder.zoom.us/j/94002553301 X Starting from a periodic system, one can by means of Bloch theory or discretisation form an effective geometry. In the standard case, this is a bundle over the Brillouin torus, whose sections are the states. More generally, one can consider families of Hamiltonians parameterized over a base. Such geometries can be thought of as classical or C* geometries. In the latter form, they appear via quiver representations. We will present the background and tools for the analysis of these geometries. This is joint work with R. Kaufmann and S. Khlebnikov Effective geometries and their local and global invariants. Sponsored by the Meyer Fund
Apr. 18, 2024 2pm (Online)	Rep Theory Mamoru Ueda (University of Alberta) View Online:https://cuboulder.zoom.us/j/97927772782 X Creutzig-Diaconescu-Ma conjectured that there exists a homomorphism from the shifted affine Yangian to the universal enveloping algebra of an iterated W-algebra. They also conjectured that this homomorphism will induce a resolution of the generalized AGT conjecture. In this talk, I will explain how to construct a homomorphism from the affine Yangian of type A to the universal enveloping algebra of a W-algebra including the non-rectangular W-algebra. I expect that this homomorphism can be extended to the shifted affine Yangian. Affine Yangians and W-algebras
Apr. 18, 2024 3:35pm (MATH 3…	Probability Andrew Campbell (ISTA) X A universality of local eigenvalue statistics of Hermitian random matrices has been the subject of immense investigation over the past couple decades resulting in the now well established principle that local eigenvalue statistics are determined only by the symmetry class of the matrix (i.e. real symmetric vs complex Hermitian) and the shape of the local eigenvalue density. For example, generically at the edge of the spectrum the eigenvalue density of random matrices exhibits square-root decay, leading to the local eigenvalue statistics falling into the famous Tracy--Widom universality class. For non-Hermitian matrices substantially less is known, with full universality in the bulk and at the edge for a random matrix X with centered independent identically distributed entries (or IID matrix) established only quite recently. While this is quite a substantial accomplishment, it is nowhere near the generality available for Hermitian matrices. In this talk we will consider the edge statistics of a deformed IID matrix A+X, where A is some deterministic deformation of the IID matrix X. By using matrices with Gaussian entries as a middle step for comparison we will see that this model has implications for many important non-Hermitian random matrix models. This is based on joint work with Giorgio Cipolloni, Laszlo Erdos, and Hong Chang Ji. At the edge of deformed non-Hermitian matrices.
Apr. 19, 2024 2:30pm (MATH 3…	Geometry/Analysis Leandro Lichtenfelz (Wake Forest University) X The Euler equations of hydrodynamics, which model ideal fluid flows on some region , admit a beautiful description as geodesic equations on the group of volume-preserving diffeomorphisms of , as first shown by V. Arnold. A major difficulty in studying the geometry of this group is the fact that it is infinite-dimensional, so a natural idea is to try to approximate it in some sense by a finite-dimensional model. One such model was constructed by Zeitlin, based on a quantization scheme developed earlier by Hoppe. In this talk, I will give an overview of Zeitlin's model, and then describe some aspects related to its geometry, in particular conjugate points. This is joint work with Stephen Preston and Alice Le Brigant. Conjugate points on Lie groups and Zeitlin's model of hydrodynamics Sponsored by the Meyer Fund
Apr. 23, 2024 12pm (MATH 350)	Topology Topology Day : Kyle Ormsby (Reed College) X Model structures underpin the modern enterprise of abstract homotopy theory and provide a convenient framework in which we can pass from one notion of sameness (isomorphism) to a weaker one (weak equivalence). Despite their fundamental nature, model structures have historically been studied en masse or applied in specific cases, and very little is known about the totality of model structures on a given category. Homotopical combinatorics is an emerging field that remedies this situation by studying the enumerative combinatorics and structural properties of model structures on finite lattices. Specialized to a finite chain, we find rich connections with Catalan combinatorics, including (intervals in) the Tamari and Kreweras lattices. I will introduce the audience to model structures and sketch homotopical combinatorics as it currently stands, including the surprising assist its development received from equivariant infinite loop space theory. Counting model structures Sponsored by the Meyer Fund
Apr. 23, 2024 1pm (MATH 220)	Probability David Renfrew (Binghamton University) X We consider the density of states of structured Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up. Singularities in the spectrum of random block matrices
CANCELED Apr. 23, 2024	Algebra/Logic Paolo Marimon (TU Wien, Austria) View Online:https://cuboulder.zoom.us/j/96896272260 X We classify the operations which are minimal above the clone generated by a permutation group G acting on a set. Rosenberg (1986) showed that when G is trivial there are only five types of minimal operations. We show that for most permutation groups there are only three types of minimal operations. More generally, we carry such classification for any permutation group and specify the behaviour of the minimal operations on tuples with two or more elements in the same orbit. Our results are especially helpful for the study of constraint satisfaction problems on omega-categorical structures, where we improve and generalise results of Bodirsky and Chen (2007), and answer some questions of Bodirsky (2021). This is joint work with Michael Pinsker. Minimal operations over permutation groups
Apr. 23, 2024 2:30pm (MATH 3…	Lie Theory Thomas Creutzig (University of Alberta) X An affine Lie algebra is a central extension of the loop algebra of a finite dimensional Lie algebra. Certain modules, the vacuum modules at any complex level k, of the affine Lie algebra carry themselves an interesting algebraic structure, namely that of a vertex operator algebra (VOA). The representation theory of affine VOAs is rather rich, e.g. suitable categories of modules form ribbon categories and there are exciting connections to geometry, quantum groups, physics and much more. I want to give an overview on the state of the art of this area. Representation Theory of Affine VOAs Sponsored by the Meyer Fund
Apr. 23, 2024 3:30pm (MATH 3…	Topology Topology Day: Ping Xu (Penn State) X Dg manifolds are a useful geometric notion which unifies many important structures such as homotopy Lie algebras, foliations and complex manifolds. In this talk, we describe a Duflo-Kontsevich type theorem for dg manifolds. The Duflo theorem of Lie theory and the Kontsevich theorem regarding the Hoschschild cohomology of complex manifolds can both be derived as special cases of this Duflo--Kontsevich type theorem for dg manifolds. This is a joint work with Hsuan-Yi Liao and Mathieu Stienon. Duflo-Kontsevich type theorem for dg manifolds
Apr. 25, 2024 11:15am (MATH …	Algebraic Geometry Daniel Bragg (Utah) X We show that every Deligne-Mumford gerbe over a field occurs as the residual gerbe of a point of the moduli stack of curves. Informally, this means that the moduli space of curves fails to be a fine moduli space in every possible way. We also show the same result for a list of other natural moduli problems. This is joint work with Max Lieblich. A Stacky Murphy’s Law for the Stack of Curves Sponsored by the Meyer Fund
Apr. 25, 2024 12:20pm (Math …	Noncomm Geometry Dmitry Kaledin (Steklov Math Institute) X It has become accepted wisdom by now that when you localize a category with respect to a class of morphisms, what you get is not just a category but a category "with a homotopical enhancement". Typically, the latter is made precise through the machinery of "infinity-categories", or "quasicategories", but this is quite heavy technically and not really optimal from the conceptual point of view. I am going to sketch an alternative technique based on Grothendieck's idea of a "derivator". How to enhance categories, and why Sponsored by the Meyer Fund
Apr. 25, 2024 12:20pm (MATH …	Algebraic Geometry Dmitry Kaledin (Steklov Math Institute) X It has become accepted wisdom by now that when you localize a category with respect to a class of morphisms, what you get is not just a category but a category "with a homotopical enhancement". Typically, the latter is made precise through the machinery of "infinity-categories", or "quasicategories", but this is quite heavy technically and not really optimal from the conceptual point of view. I am going to sketch an alternative technique based on Grothendieck's idea of a "derivator". How to enhance categories, and why (Cross listed with Non-commutative Geometry) Sponsored by the Meyer Fund
Apr. 30, 2024 1:25pm (online)	Algebra/Logic Guenter Pilz (JKU Linz, Austria) View Online:https://cuboulder.zoom.us/j/96896272260 X Algebraists like decompositions, for doing parallel computing, etc. Some decompositions are very "successful", some others are not. We try to find the "reasons of sucess" by looking at various instants, including decompositions of semisimple rings. We also get "Generalized Discrete Fourier Transforms" for group rings and related structures. Finally, we try to solve equations of degree >4. This is joint work with Manuel Kauers. Which (sub)direct decompositions are "useful"?
Apr. 30, 2024 3:30pm (MATH 3…	Topology Sarah Petersen X This talk will describe some of the history and background of one of the oldest (now solved) problems in algebraic topology. An Introduction to the Kervaire Invariant Problem
May. 01, 2024 3:30pm (MATH 3…	Math Phys Howy Jordan (CU Boulder, Department of Mathematics) Infinite Dimensional Control Theory
May. 02, 2024 11:15am (MATH …	Algebraic Geometry Luca Schaffler (Universita Roma Tre) X Given the moduli space of hyperplanes in projective space, V. Alexeev constructed a family of compactifications parametrizing stable hyperplane arrangements with respect to given weights. In particular, there is a toric compactification that generalizes the Losev–Manin compactification for the moduli of points on the line. We study the first natural wall crossing that modifies this compactification into a non-toric one by varying the weights. In particular, we prove that in dimensions two the wall crossing corresponds to blowing up at the identity of the dense torus of the generalized Losev–Manin space. As an application, we show that any Q-factorialization of this blow-up is not a Mori dream space for a sufficiently high number of lines. This is joint work in progress with Patricio Gallardo. An explicit wall crossing for the moduli space of hyperplane arrangements Sponsored by the Meyer Fund
May. 02, 2024 12:15pm (Math …	Algebraic Geometry Farbod Shokrieh (University of Washington) X I will describe some connections between arithmetic of abelian varieties, non-archimedean/tropical geometry, and combinatorics. For example, we give formulas for (non-archimedean) canonical local heights in terms of tropical geometry. Our formulas extend classical computations of local height functions due to Néron (involving Néron models) and, in the case of elliptic curves, due to Tate (involving Bernoulli polynomials). Based on joint work with Robin de Jong. Canonical local heights and tropical theta functions Sponsored by the Simons Foundation
May. 02, 2024 3:35pm (MATH 3…	Probability Vishesh Jain (University of Illinois at Chicago) X Markov chains provide a natural approach to sample from various distributions on the independent sets of a graph. For the uniform distribution on independent sets of a given size in a graph, perhaps the most natural Markov chain is the so-called ``down-up walk''. The down-up walk, which essentially goes back to the foundational work of Metropolis, Rosenbluth, Rosenbluth, Teller and Teller on the Markov Chain Monte Carlo method, starts at an arbitrary independent set of size , and in every step, removes an element uniformly at random and adds a uniformly random legal choice. Davies and Perkins showed that there is a critical such that it is hard to (approximately) sample from the uniform distribution on independent sets for the class of graphs with vertices and maximum degree at most . They conjectured that for below this critical value, the down-up walk mixes in polynomial time. I will discuss a resolution of this conjecture, which additionally shows that the down-up walk mixes in (optimal) time . Based on joint work with Marcus Michelen, Huy Tuan Pham, and Thuy-Duong Vuong. Optimal mixing of the down-up walk on fixed-size independent sets
May. 07, 2024 10am (MATH 350)	Lie Theory Lucas Gagnon (York University) X A long-standing conjecture of Stanley and Stembridge asks whether certain symmetric function invariants of graphs are “e-positive,” i.e. expand positively in the basis of elementary symmetric functions. Efforts to resolve Stanley—Stembridge have raised the wider question of whether other symmetric functions are e-positive, particularly for symmetric invariants of other interesting objects. This talk will describe an instance of e-positivity arising in the study of modules of the unipotent upper triangular groups over a finite field. I use a collection of novel module homomorphisms to prove e-positivity, and these maps implicate a well-known algorithm from the study of Iwahori—Hecke algebras. Throughout, I will connect these results to previously known connections between the unipotent upper triangular group and symmetric functions in non commuting variables. Positivity in unipotent symmetric function invariants and Hecke traces Sponsored by the Meyer Fund
May. 07, 2024 12:10pm (MATH …	Kempner Luis David Garcia Puente (Colorado College) X Let G be a finite, connected graph. An arithmetical structure on G is an assignment of positive integers to the vertices such that, at each vertex, its label divides the sum of the labels at adjacent vertices (counted w/ multiplicity). Arithmetical structures naturally generalize the notion of the Laplacian matrix of a graph. They also appear in the context of arithmetical geometry. In this talk, I will present a general introduction to the topic, we will focus on the combinatorics of arithmetical structures on certain graph families including some counting results that relate arithmetical structures on a path graph and Catalan numbers. We will also discuss some results about the associated finite Abelian group of an arithmetical structure. What is an arithmetical structure? Sponsored by the Meyer Fund
Jun. 27, 2024 5pm (MATH 220)	Probability Stanislav Volkov (Lund University) X I will present some generalizations of the one-dimensional forest fire model with ignition occurring only at zero, which was analyzed in particular by Volkov (2009). In more recent work, we allow the rates at which the trees grow to depend on their location, and the fire can spread using long-range connections. We establish that the expected time required for the fire to reach a site at a distance from the origin is at most of the order . The talk is based on a recent joint paper with Mikhail Menshikov (Durham) and the late Francis Comets (Paris). Generalized one-dimensional forest fire model
Aug. 20, 2024 3pm (MATH 350)	Topology Sabrina Pauli (TU Darmstadt) X Enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions. A famous problem in this area is counting plane rational curves of degree d passing through 3d-1 given points. Tropical geometry has been a powerful tool in tackling enumerative geometry problems, notably marked by Mikhalkin's correspondence theorem. This theorem translates the count of plane curves into the count of their tropical counterparts, for which there are elegant solutions using only combinatorics. Results from A1-homotopy theory allow to ask questions in enumerative geometry over arbitrary base fields, not just algebraically closed ones, and still get meaningful answers, leading to a new area called A1-enumerative geometry. In the talk I will give an introduction to A1-enumerative geometry and explain how tropical geometry can be used to solve problems in this area. Tropical methods in A1-enumerative geometry Sponsored by the Meyer Fund
Aug. 20, 2024 3pm (MATH 350)	Algebraic Geometry Sabrina Pauli (TU Darmstadt) X Enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions. A famous problem in this area is counting plane rational curves of degree d passing through 3d-1 given points. Tropical geometry has been a powerful tool in tackling enumerative geometry problems, notably marked by Mikhalkin's correspondence theorem. This theorem translates the count of plane curves into the count of their tropical counterparts, for which there are elegant solutions using only combinatorics. Results from A1-homotopy theory allow to ask questions in enumerative geometry over arbitrary base fields, not just algebraically closed ones, and still get meaningful answers, leading to a new area called A1-enumerative geometry. In the talk I will give an introduction to A1-enumerative geometry and explain how tropical geometry can be used to solve problems in this area. Tropical methods in A1-enumerative geometry
Aug. 22, 2024 2:30pm (MATH 2…	Functional Analysis Tomasz Maszczyk (University of Warsaw) X Under the technical condition of separability of some topological space of cohomology, we show that the Godbillon-Vey number of a foliation of codimension one on a compact orientable 3-fold is topologically rigid if and only if the foliation admits a projective transversal structure. Here by the rigidity of the Godbillon-Vey number we mean that it is constant under the infinitesimal singular deformations of the foliation, and a foliation admits a transversal projective structure whenever it can be glued from the level sets of locally defined functions related by fractional linear transition maps. Title: Foliations with rigid Godbillon-Vey class
Aug. 29, 2024 2:30pm (Math 3…	Functional Analysis PIOTR M. HAJAC (IMPAN, Warsaw, Poland) X In 2006, Katsura discovered optimal conditions for morphisms of topological graphs to contravariantly induce morphisms between their C-algebras. We use these morphisms in the special case of discrete topology (usual directed graphs) to extend the well studied theory of unions of graphs to one-injective pushouts of graphs. Our main result is a pushout-to-pullback theorem that yields a one-surjective gauge equivariant pullback of graphs C-algebras, which suits the Mayer-Vietoris six-term exact sequence in K-theory and applications to noncommutative topology. To exemplify our theorem, first we vastly generalize the old concept of a derived graph to obtain locally derived graphs. Then we use locally derived graphs, which provide us with a very rich class of non-injective graph homomorphisms satisfying Katsura's conditions, to construct pushouts satisfying the assumptions of our pushout-to-pullback theorem. (Based on joint work with Mariusz Tobolski.) LOCALLY DERIVED GRAPHS AND GAUGE EQUIVARIANT PULLBACKS OF GRAPH C*-ALGEBRAS

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