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Events for the weeks of September 24th – October 6th

Mon, Sep. 25 3:35pm (MATH 3…	Grad Student Seminar Erik Knutsen (University of Colorado) X A homotopy representation for a compact Lie group G is a type of G-CW complex. Fausk, Lewis, and May showed that there is an isomorphism between the Grothendieck group of homotopy representations and the group of invertible G-Spectra, Pic(G). This enabled them to describe Pic(G) using a left exact sequence. In order to refine the description of Pic(G), they conjectured that there is an isomorphism between the Grothendieck group of homotopy representations and that of generalized homotopy representations for G. If G is a finite group or a torus it is known that there is such an isomorphism. I will discuss my work toward proving that conjecture for any compact Lie group G. Homotopy Representations (and Spectra)
Tue, Sep. 26 2:30pm (Zoom)	Math Edu David Grant (University of Colorado Boulder) View Online:https://cuboulder.zoom.us/j/95557714315 X The CU Math Department’s undergraduate program, especially its service courses, looks vastly different than it did 30 years ago. This is due to the efforts of a number of faculty, but all the changes show the indelible mark of Rob Tubbs, who for three decades worked to transform our program into the national model for education it is today. We will describe this evolution, and the logic behind its interrelated parts. The evolution of our Undergraduate Program: An homage to Robert Tubbs
Tue, Sep. 26 2:30pm (MATH 3…	Lie Theory Richard Green (CU) X The Weyl groups of type affine D are a family of infinite groups. I will show how these groups arise naturally as invertible operators acting on a specific type of discrete structure. This gives rise to combinatorially interesting one-parameter families of finite dimensional representations of the groups. This is the first part of a two part talk. Some combinatorial representations of affine Weyl groups
Tue, Sep. 26 3:30pm (MATH 3…	Topology Alex LaJeunesse X Last week we saw how cohomology theories correspond to objects called omega-spectra. We will use the language of model categories to study the homotopy theory of these objects and to define the Stable Homotopy Category (of spectra), which plays a role in topology similar to that played by the derived category in homological algebra. After investigating some properties of this category, we will define homotopy groups of general spectra and look at some tools that help us compute them. An Introduction to Spectra: Part II
Wed, Sep. 27 3:30pm (Math350)	Math Phys Sheagan John (CU Boulder) View Online:https://cuboulder.zoom.us/j/94002553301 X For online participation: Zoom Meeting ID: 940 0255 3301 Passcode: 790356 Hilbert-Schmidt operators and density matrices
Thu, Sep. 28 11:15am (MATH …	Algebraic Geometry Jeff Achter (Colorado State University) TBA
Thu, Sep. 28 11:15am (Math …	Noncomm Geometry John Lott (University of California, Berkeley) X I will describe some topological obstructions to the existence of complete Riemannian metrics with (nonuniformly) positive scalar curvature. The motivation comes from conjectures about the simplicial volume of closed manifolds. Scalar curvature on noncompact manifolds Sponsored by the Meyer Fund
Thu, Sep. 28 2pm (Online)	Rep Theory Fei Qi (University of Denver) View Online:https://cuboulder.zoom.us/j/97927772782 X Given two left modules $W1, W2$ for a meromorphic open-string vertex algebra $V$ , we will first use Huang's cohomology to describe the equivalence classes of modules $U$ fitting in the exact sequence $0 → W2 → U → W1 → 0$ while satisfying a technical convergence condition. Then, we will explain that the technical convergence condition automatically holds if $V$ contains a nice subalgebra $V0$ , such that $W1$ and $W2$ are semisimple $V0$ -modules, and the products of intertwining operators converge. Here $V0$ is not required to be conformally embedded into $V$ . Nor will we need $V$ -modules to form a tensor category. The result leads to a new method for computing $Ext1(W1, W2)$ . On extensions of left modules for a meromorphic open-string vertex algebra.
Thu, Sep. 28 2:30pm (MATH 3…	Functional Analysis Emily King (Colorado State University) X In areas a diverse as extremal combinatorics, signal processing, and quantum information, one is interested in finding subspaces with certain mathematical properties, called frames or fusion frames with additional adjectives depending on the desired outcomes. Often such optimal configurations have remarkable symmetry and are sometimes even orbits under group actions. Finally, when one constructs such a configuration, one might ask whether it is indeed "new." H* algebras, which may be thought loosely of as the Hilbert space analog of C* algebras, may be leveraged to answer this question. No prior knowledge of frames will be assumed. Subspace Configurations, Group Representations, and H* Algebras
Thu, Sep. 28 3:35pm (MATH 3…	Probability Shih Yu Chang (San Jose State University) X A concentration tail bound serves as a probabilistic inequality that establishes an upper limit on the extent to which a random variable is expected to deviate from its mean. These inequalities find valuable applications in probability theory, statistics, and machine learning, aiding in the comprehension of random variable behaviors and the management of risks associated with extreme outcomes. Among the frequently utilized concentration tail bounds are the Chernoff bound and the Hoeffding bound. Conversely, a tensor is a mathematical concept employed for the representation of multi-dimensional arrays of data. Tensors have a presence across various domains within mathematics and physics, and they find extensive utility in disciplines like physics, engineering, computer science, and machine learning. In this discussion, we will explore tail bounds pertaining to random tensors from five distinct viewpoints: (1) Tail bounds for the summation of random tensors, approached through Laplace transforms and Lieb's concavity principles; (2) General tail bounds derived through a majorization approach; (3) The consideration of T-product random tensors; (4) Examination of random tensor means; (5) Exploration of random double tensor integrals. In conclusion, we will deliberate on potential applications for tail bounds concerning random tensors, elucidating their relevance in diverse problem-solving scenarios. Random Tensors Tail Bounds
Fri, Sep. 29 1:20pm (MATH 3…	Geometry/Analysis Georgios Daskalopoulos (Brown University) X I will describe several open problems arising in Thurston's theory of stretch maps and geodesic laminations. This is joint work with Karen Uhlenbeck. Analytic questions arising in the theory of best Lipschitz maps Sponsored by the Meyer Fund
Fri, Sep. 29 2:30pm (MATH 3…	Algebra/Logic Charlotte Aten (University of Denver) View Online:https://cuboulder.zoom.us/j/96896272260 X Classical neural network learning techniques have primarily been focused on optimization in a continuous setting. Early results in the area showed that many activation functions could be used to build neural nets that represent any function, but of course this also allows for overfitting. In an effort to ameliorate this deficiency, one seeks to reduce the search space of possible functions to a special class which preserves some relevant structure. I will propose a solution to this problem of a quite general nature, which is to use polymorphisms of a relevant discrete relational structure as activation functions. I will give some concrete examples of this, then hint that this specific case is actually of broader applicability than one might guess. Discrete neural nets and polymorphic learning Sponsored by the Meyer Fund
Tue, Oct. 3 1:25pm (MATH 2…	Algebra/Logic Petr Vojtechovksy (University of Denver) View Online:https://cuboulder.zoom.us/j/96896272260 X In some varieties close to groups, there are two competing notions of solvability, one coming from the commutator theory of congruence modular varieties and the other from the classical solvability of group theory. In this talk we will explore the two notions of solvability. Among other results, we will show that they agree in finite Moufang loops. If time permits, I will also mention some recent results on supernilpotence of groups and loops. This is joint work with Ales Drapal and David Stanovsky. Solvability and supernilpotence in varieties close to groups Sponsored by the Meyer Fund
Tue, Oct. 3 2:30pm (MATH 2…	Math Edu Nathaniel Miller (University of Northern Colorado) X Inquiry-based learning (IBL) is a framework for teaching in which students engage actively with meaningful problems, collaborate with peers, and communicate their results. This talk will give an introduction to IBL teaching methods. We’ll start with an activity in which attendees will play the role of students in an IBL activity. Then we will discuss what IBL is and how you might consider including it (or more of it) in your classes. No prior experience with IBL is necessary! A Taste of IBL (Inquiry-Based Learning) Sponsored by the Meyer Fund
Tue, Oct. 3 2:30pm (MATH 3…	Lie Theory Richard Green (CU) X The Weyl groups of type affine D are a family of infinite groups. I will show how these groups arise naturally as invertible operators acting on a specific type of discrete structure. This gives rise to combinatorially interesting one-parameter families of finite dimensional representations of the groups. This is the second part of a two part talk. Some combinatorial representations of affine Weyl groups (continued)
Tue, Oct. 3 3:30pm (MATH 3…	Topology Sarah Petersen An Introduction to Spectra: Part III
Wed, Oct. 4 3:30pm (Math350)	Math Phys Connor McCranie (CU Boulder) Partial traces
Thu, Oct. 5 10am (Online)	Rep Theory Andoni de Arriba de la Hera (Instituto de Ciencias Matemáticas) View Online:https://cuboulder.zoom.us/j/97927772782 X The goal of the talk is to construct embeddings of the N=2 superconformal vertex algebra, motivated by mirror symmetry, into the chiral de Rham complex, provided that we have solutions to the Killing spinor equations. Our approach to the chiral de Rham complex is based on the universal construction by Bressler and Heluani, which applies to any Courant algebroid over a smooth manifold. The Killing spinor equations that are considered come from the approach to special holonomy based on Courant algebroids in generalized geometry and are inspired by the physics of heterotic supergravity and string theory. The embeddings are given in two different set-ups. Firstly, for equivariant Courant algebroids over homogeneous manifolds, where the construction reduces to embeddings into the superaffinization of a quadratic Lie algebra, and the Killing spinor equations become purely algebraic conditions that can be checked on explicit examples. As an application, we present the first examples of (0,2) mirror symmetry on compact non-Kähler complex manifolds. These results are included in axiv:2012.01851, recently published in International Mathematical Research Notices. Secondly, for transitive Courant algebroids over complex manifolds, where these equations are equivalent to the Hull-Strominger system, with origins in the heterotic sigma-model studied by physicists. Several examples have been studied where the obtained results are applied. These results are included in arxiv:2305.06836. This talk is based on my PhD thesis, and is a joint work with Luis Álvarez-Cónsul and Mario Garcia-Fernandez. Supersymmetric Vertex Algebras and Killing Spinors.
Thu, Oct. 5 11:15am (MATH …	Algebraic Geometry Charles Vial (Universität Bielefeld) TBA
Thu, Oct. 5 3:35pm (MATH 3…	Probability Nhan Nguyen (CU Boulder) X A random polynomial is a polynomial whose coefficients are random variables. A major task in the theory of random polynomials is to examine how the real roots are distributed and correlated in situations where the degree of the polynomial is large. In this talk, we examine two classes of random polynomials that have captured the attention of researchers in the fields of probability theory and mathematical physics: elliptic polynomials and generalized Kac polynomials. Correlations between the real roots of random polynomials

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Events for the weeks of September 24th – October 6th

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