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Events for May – September 2023

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May. 02, 2023 11am (MATH 350)	Topology Hiro Lee Tanaka (Texas State University) X If you pour water on a statue, the water will descend in predictable ways. Amazingly, if you know exactly how streams of water on a statue can break, you can reconstruct the statue's shape. This fact is a toy example of Morse theory, a field of topology that tries to understand shapes of things in terms of streams of least resistance (these streams are sometimes called gradient flows). I want to introduce this theory in some fun, concrete examples, and as time allows, talk about a mysterious, recent discovery that connects two disparate branches of math: The geometry of Morse theory encodes exactly the algebra of associativity (yes, like associativity of multiplication). Topology Day: Streams of water, the shapes of things, and the geometry of associativity Sponsored by the Meyer Fund
May. 02, 2023 1:25pm (online)	Algebra/Logic Will Boney (Texas State University) View Online:https://cuboulder.zoom.us/j/96896272260 X Generalized indiscernibles can be built in first-order theories by generalizing the combinatorial Ramsey’s Theorem to classes with more structure, which is an active area of study. Trying to do the same for infinitely theories (in the guise of Abstract Elementary Classes) requires generalizing the Erdos-Rado Theorem instead. We discuss various results about generalizations of the Erdos-Rado Theorem and techniques (including large cardinals and forcing) to build generalized indiscernible. Building generalized indiscernibles in AECs with set theory
May. 02, 2023 2:30pm (MATH 3…	Topology Teena Gerhardt (Michigan State University) X The field of algebraic topology has exposed deep connections between topology and algebra. One example of such a connection comes from algebraic K-theory. Algebraic K-theory is an invariant of rings, defined using tools from topology, that has important applications to algebraic geometry, number theory, and geometric topology. Algebraic K-groups are very difficult to compute, but advances in algebraic topology have led to many recent computations which were previously intractable. Equivariant homotopy theory, a branch of algebraic topology which studies topological objects with a group action, has been particularly important in the study of algebraic K-theory. In this talk I will give a gentle introduction to algebraic K-theory and its applications, and explain the interesting role of equivariant homotopy theory in this story. Topology Day: Algebra in topology and topology in algebra: an introduction to algebraic K-theory Sponsored by the Meyer Fund
May. 02, 2023 3:30pm (MATH 3…	Topology Yuriy Mileyko (University of Hawaii) X The last couple of decades saw a considerable increase in applications of topological techniques within other disciplines, particularly in data analysis, leading to a creation of a nascent field of topological data analysis. The premise of the latter is that data have shape, and that shape matters. Mathematically, this typically translates into an assumption that the data under consideration represent a sample from some topological space whose homotopy type is of interest. For computational reasons, the focus shifts to recovering homology groups of the underlying space, and several important results have been obtained to address this important task. In this talk I will cover some of the key results which allow one to compute homology groups of a subspace of a metric space given only a finite (possibly noisy) sample from it. Not surprisingly, such results rely on certain assumptions about the global geometry/topology of the subspace. I will then show that these assumptions (and results) become more subtle if the goal is to recover local rather than global homology groups of the subspace. Topology: Day Recovering homology from samples: from global to local Sponsored by the Meyer Fund
May. 03, 2023 4:40pm (MATH 3…	Grad Student Seminar Maya Ornstein (CU Boulder) TBA
May. 04, 2023 9am (MATH 350)	Algebraic Geometry Jonathan and Yano X Jonathan and Yano will talk a bit about what it's like to do research in algebraic geometry. Younger grad students are especially encouraged to come. Algebraic geometry open house
May. 04, 2023 3:35pm (MATH 3…	Probability Oanh Nguyen (Brown University) X The Kac polynomial is one of the most studied models of random polynomials. It has the form $fn(x) = ∑i=0n xii xi.$ It is known that the empirical measure of the roots converges to the uniform measure on the unit disk. On the other hand, at any point on the unit disk, there is a hole in which there are no roots, with high probability. Disproving a natural conjecture for these holes, we will show that all the hole radii are of the same order. We will also discuss open problems and recent fascinating conjectures concerning the flow of roots. Joint work with Hoi Nguyen. Hole radii for the Kac polynomials
May. 04, 2023 4pm (MATH 220)	Init Conditions Howy Jordan X A category of sets is foundational to mathematics, and a category of complete inner product spaces is foundational to modern physics. The former has been captured by categorical language through the work of topos theory. As for the latter, the culmination of reconstruction programs led to a categorical characterization in 2022 by Heunen and Kornell. This could be summarized by saying that the category of Hilbert spaces and continuous linear maps is the dagger-abelian dagger monoidal category, with a simple monoidal separating unit and with directed colimits of dagger monics. In the first part of this talk, we'll consider the axioms of elementary topoi, which capture Set-like categories, and see how to distinguish Set among these. We'll introduce the idea of subobjects in a category, and see that their classifying objects correspond to truth values giving us classifying functions of subsets. As time permits, we'll encounter related objects, like power objects and exponentials. In the second part, we'll consider axioms for the category of Hilbert spaces. In particular, we'll introduce dagger and monoidal structures on categories, overview how a monoidal unit can obtain the structure of a field, and see how the axioms impose an orthomodular structure on the subobjects. The Category of Sets, The Category of Hilbert Spaces
May. 08, 2023 10am (Math 350)	Thesis Defenses Connor Meredith (CU) View Online:https://cuboulder.zoom.us/j/96126023746 X We determine the possible sizes of algebras of finite type that are nilpotent, nonabelian, dualizable, and generate a congruence modular variety. We also show that localization preserves dualizability, then use this result to construct a finite nilpotent algebra A that is nondualizable, yet such that no algebra in ISP(A) has a supernilpotent nonabelian congruence. Finally, we show that if A is a Mal’cev algebra, then higher commutators in any localization of A are closely controlled by higher commutators in subalgebras of A. Nilpotence and Dualizability of Algebras of Finite Type
Aug. 14, 2023 2pm (Math350)	Math Phys Alexei Kitaev (CalTech) Short Range Entangled States
Aug. 31, 2023 11:15am (MATH …	Algebraic Geometry Jonathan, Yano, Charles, and Jeff X Jonathan, Yano, Charles Vial, and Jeff Achter will talk a bit about what it's like to do research in algebraic geometry. Younger grad students are especially encouraged to come. Algebraic Geometry Open House
Sep. 05, 2023 3:30pm (MATH 3…	Topology Peter Patzt (University of Oklahoma) X I will talk about some of the many applications of the Steinberg module in algebraic topology. The Steinberg module was first introduced in representation theory by Steinberg but later Solomon and Tits discovered a geometric description as the top homology of the Tits building. Borel and Serre found it to be the virtual dualizing module of arithmetic groups, which gives access to their high dimensional cohomology groups. It comes up in spectral sequences introduced by Quillen and by Rognes, respectively, that can be used to compute algebraic K-theory. If time permits, we will also see how the split Steinberg module can be used to improve the homological stability range of GL_n(Z) (concerning low dimensional homology of GL_n(Z)). Some applications of the Steinberg module in algebraic topology Sponsored by the Meyer Fund
Sep. 07, 2023 3:35pm (MATH 3…	Probability Janos, Kyle & Sean (CU Boulder) X Janos, Kyle & Sean will talk about their current research and what it's like to work in the probability group. The talks will be informal and there will be plenty of time for questions. Probability Open House
Sep. 12, 2023 3:30pm (MATH 3…	Topology Organizational Meeting N/A
Sep. 13, 2023 3:30pm (Math35…	Math Phys M.J. Pflaum (CU Boulder) View Online:https://cuboulder.zoom.us/j/94002553301 X The semester theme will be "mathematics of quantum entanglement" Organizational Meeting
Sep. 13, 2023 4:40pm (Math 3…	Grad Student Seminar Chase Meadors (University of Colorado) X Infinitely large natural numbers, infinitesimally small real numbers, transcendental elements of a field: these are all examples of some sort of "limit" object. In this talk, I'll explain how these concepts are literally limits in the topological sense. We'll introduce lattices, argue that they are the perfect kind of structure for organizing logical information, and then motivate the concept of a filter. Filters show up in both lattice theory and topology, hinting at the connection we will make precise. Limits in Logic
Sep. 14, 2023 11:15am (MATH …	Algebraic Geometry Jonathan Wise (University of Colorado Boulder) TBA
Sep. 14, 2023 2pm (Online)	Rep Theory Shashank Kanade (University of Denver) View Online:https://cuboulder.zoom.us/j/97927772782 X I will explain how the characters of various rational and non-rational VOAs of type A are obtained from sl_r invariants of torus links. Specifically, we will consider principal W algebras, (1,p) singlet and (1,p) triplet VOAs. Invariants of torus links and characters of VOAs.
Sep. 14, 2023 2:30pm (MATH 3…	Functional Analysis Alonso Delfín (CU Boulder) X In this talk, I will discuss some of the main results in https://arxiv.org/abs/2208.14605. For a pair of C-algebras (A,B), representing an (A, B) C-correspondence on a pair of Hilbert spaces (H_1, H_2) roughly consists in naturally realizing the correspondence as a closed subspace of L(H_1, H_2). This concept is a generalization of R. Exel theory for Hilbert A-B bimodules, originally introduced in 1993. Exel's methods were used as a tool to prove, in its full generality, that any two Morita equivalent C-algebras have isomorphic K-theory. Extending this theory to C-correspondences yields necessary and sufficient conditions for an (A,B) C-correspondence to be a Hilbert A-B bimodule. Another consequence is that, if a right Hilbert A-module X is represented on (H_1, H_2), we then get faithful representations of L_A(X) and K_A(X), the algebras of adjointable and compact-adjointable maps, on the Hilbert space H_1. This will play a crucial role in my talk next week, where I will talk about the objects we get when the Hilbert spaces are replaced by general L^p spaces for p in (1, infinity). Representations of C-correspondences on pairs of Hilbert spaces.
Sep. 19, 2023 1:25pm (online)	Algebra/Logic Keith Kearnes (CU Boulder) View Online:https://cuboulder.zoom.us/j/96896272260 X A variety V is a Schreier variety if every if every subalgebra of a V-free algebra is free. I will explain a link between the concept of a Schreier variety and tame congruence theory. Namely, I will explain why a locally finite variety is a Schreier variety if and only if it consists of minimal algebras. Locally finite Schreier varieties
Sep. 19, 2023 2:30pm (Math 3…	Lie Theory ALT organizers X The organizers of the ALT seminar (Flor Orosz Hunziker, Richard Green, and Nat Thiem) will give introductions to their research and their respective styles as advisors. All are welcome, but newer students exploring the various research groups in the department are especially encouraged to attend. ALT Open House
Sep. 19, 2023 3:30pm (MATH 3…	Topology Juan Moreno X At its core, algebraic topology is about associating algebraic invariants to topological spaces. These are invariant in the sense that if you change your space by a homotopy equivalence you should get an isomorphic algebraic object. We will begin with the observation that often these invariants do not only respect homotopy, but also suspension, an operation that increases the dimension of a space. We call such invariants stable and will focus on the most important examples of them: generalized cohomology theories. The Brown representability theorem will then tell us that if we are interested in stable invariants then we should really be working with somethings called spectra instead of spaces. An Introduction to Spectra: Part I
Sep. 20, 2023 3:30pm (Math35…	Math Phys Markus Pflaum (CU Boulder) View Online:https://cuboulder.zoom.us/j/94002553301 X For online participation: Zoom Meeting ID: 940 0255 3301 Passcode: 790356 Introduction to C*-algebras and their state spaces
Sep. 21, 2023 11:15am (MATH …	Algebraic Geometry Jonathan Wise (University of Colorado Boulder) X I'll describe an approach to constructing compact moduli of linear series for variable curves. Limit linear series
Sep. 21, 2023 2pm (Online)	Rep Theory Marijana Butorac (University of Rijeka) View Online:https://cuboulder.zoom.us/j/97927772782 X In this talk I will present the construction of combinatorial bases of standard modules with rectangular highest weights for affine Lie algebras, which relies on the construction of quasi--particle bases of the Feigin--Stoyanovsky principal subspaces. This talk is based on a joint project with S. Kozic and M. Primc. Combinatorial construction of standard modules for affine Lie algebras.
Sep. 21, 2023 2:30pm (MATH 3…	Functional Analysis Alonso Delfín (CU Boulder) X In this talk, I will introduce the concept of a concrete L^p-correspondence over a pair of L^p-operator algebras. Definitions will be motivated by the C* results covered in my previous talk. Several examples of L^p-correspondences will be given and I will also show how, in some cases, we can get an L^p version of the Cuntz-Pimsner algebras. Time permitting, I will focus on L^p-modules, a particular type of L^p-correspondence that generalizes the notion of a Hilbert module, by presenting the current known theory and also discussing some open problems. L^p-correspondences and their L^p Cuntz-Pimsner algebras.
CANCELED Sep. 21, 2023	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
Sep. 25, 2023 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)
Sep. 26, 2023 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
Sep. 26, 2023 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
Sep. 26, 2023 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
Sep. 27, 2023 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
Sep. 28, 2023 11:15am (MATH …	Algebraic Geometry Jeff Achter (Colorado State University) TBA
Sep. 28, 2023 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
Sep. 28, 2023 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.
Sep. 28, 2023 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
Sep. 28, 2023 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
Sep. 29, 2023 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
Sep. 29, 2023 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

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