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Events for August – December 2023

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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 (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 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 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 …	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 11:15am (MATH …	Algebraic Geometry Jeff Achter (Colorado State University) TBA
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
Oct. 03, 2023 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
Oct. 03, 2023 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
Oct. 03, 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 second part of a two part talk. Some combinatorial representations of affine Weyl groups (continued)
Oct. 03, 2023 3:30pm (MATH 3…	Topology Sarah Petersen X This talk will introduce Thom spectra, the spectrum of complex cobordism MU, and the formal group laws associated to complex oriented spectra. An Introduction to Spectra: Part III
Oct. 04, 2023 3:30pm (Math350)	Math Phys Connor McCranie (CU Boulder) View Online:https://cuboulder.zoom.us/j/94002553301 X For online participation: Zoom Meeting ID: 940 0255 3301 Passcode: 790356 Partial traces
Oct. 04, 2023 4:40pm (MATH 3…	Grad Student Seminar Sylvia Maher (University of Colorado) X By examining the classical homotopy theory of topological spaces and identifying the key features that make it work, we can abstract those features to arrive at the notion of a model category, which allows us to do homotopy theory in an abstract setting. In this talk we will see several examples of model categories and what we can do with them, from topological spaces to homological algebra. An Introduction to Model Categories
Oct. 05, 2023 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.
Oct. 05, 2023 11:15am (MATH …	Algebraic Geometry Charles Vial (Universität Bielefeld) TBA
Oct. 05, 2023 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
Oct. 10, 2023 1:25pm (online)	Algebra/Logic Tomasz Penza (Warsaw University of Technology, Poland) View Online:https://cuboulder.zoom.us/j/96896272260 X The Maltsev product $V ∘ W$ of varieties $𝒱$ and $𝒲$ of a type $Σ$ is a class that consists of all algebras $A$ of the type $Σ$ , which have a congruence $θ$ such that the quotient $A/θ$ belongs to $𝒲$ and each congruence class of $θ$ which is a subalgebra of $A$ belongs to $𝒱$ . The class $V ∘ W$ is closed under subalgebras, direct products, and isomorphic images. However it may not be closed under homomorphic images, and so it may not be a variety. I will present a new sufficient condition for $V ∘ W$ to be a variety and I will discuss various cases in which this sufficient condition is satisfied. Sufficient conditions for a Maltsev product of two varieties to be a variety
Oct. 10, 2023 2:30pm (MATH 3…	Lie Theory Paige Robertson (CU) X We will discuss a subfamily of the binary triangle family with weakly increasing columns. This subfamily has a natural poset structure through which we show that the maximal elements are counted by the Fibonacci numbers. We then examine those maximum elements which also have a maximum total sum, or weight, and show that this subset, and its corresponding maximum weights, are enumerated by known OEIS sequences. We conclude with a discussion of two methods of constructing these maximum elements with maximum weight, one recursive and one explicit. Enumerating Maximal Binary Triangles
Oct. 10, 2023 3:30pm (MATH 3…	Topology Jack Carlisle (University of Notre Dame) X In non-equivariant homotopy theory, formal groups play an essential role. These algebraic objects arise as the E cohomology of infinite complex projective space for E a complex oriented spectrum. If A is a finite abelian group, then there is a notion of an A - equivariant formal group law, and these objects arise naturally as the E cohomology of an A - equivariant analogue of infinite complex projective space. The purpose of this talk is to introduce the audience to these mysterious algebraic objects, and survey some important and recent results in this area. In particular, we will provide a direct proof (as opposed to the beautiful "global homotopy theoretic" proof of Hausmann) that the Hopf algebroid associated with the A - spectrum $M U_{A} \land M U_{A}$ represents isomorphisms of A-equivariant formal group laws. Formal groups in equivariant homotopy theory
Oct. 11, 2023 3:30pm (Math350)	Math Phys Daniel Spiegel (UC Davis) View Online:https://cuboulder.zoom.us/j/94002553301 X When a quantum system evolves in time or is subject to measurement, the state of the system changes according to the action of a unitary or a projection. But if we restrict our state to a subsystem, then more general types of evolutions appear to be possible, and are are mathematically described as completely positive maps. Completely positive maps are of interest to both operator algebraists and quantum information theorists, although the language used when working with them can vary between these communities. I will try to discuss them from both perspectives and will explain how Stinespring’s theorem characterizes these important objects. For online participation: Zoom Meeting ID: 940 0255 3301 Passcode: 790356 Completely positive maps and Stinespring’s theorem
Oct. 12, 2023 11:15am (MATH …	Algebraic Geometry Matt Watson (University of Colorado Boulder) X Du Val surface singularities are a type of mild singularity which appear on complex surfaces. In this talk, we will compute resolutions of Du Val singularities and give many equivalent definitions. If time allows, we will define canonical singularities for normal, quasiprojective varieties over C and show that the Du Val surface singularities are canonical. Du Val Surface Singularities
Oct. 12, 2023 2pm (Online)	Rep Theory Antoine Caradot (Henan University) View Online:https://cuboulder.zoom.us/j/97927772782 X Given a vertex operator algebra V, one can attach a graded Poisson algebra called the C2-algebra. The associated Poisson variety is an important invariant for V and is known as the associated variety of V. In this talk, we will introduce the cohomological variety of a vertex operator algebra, a notion cohomologically dual to that of the associated variety. First, we will motivate and define this variety, as well as give some of its structural properties. Then we will explain how to extract information on the Yoneda algebra defining this variety. Lastly, we will apply those results to the simple vertex operator algebras constructed from the Virasoro Lie algebra and finite dimensional simple Lie algebras. This is a joint work with Cuipo Jiang (Shanghai Jiao Tong University) and Zongzhu Lin (Kansas State University) The cohomological variety of a vertex operator algebra.
Oct. 12, 2023 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, part II
Oct. 17, 2023 1:25pm (online)	Algebra/Logic Erkko Lehtonen (Khalifa University, Abu Dhabi) View Online:https://cuboulder.zoom.us/j/96896272260 X Let $C1$ and $C2$ be clones on sets $A$ and $B$ , respectively. A set $K$ of functions of several arguments from $A$ to $B$ is called a $C1,C2)$ -clonoid if $K C1 ⊆ K$ and $C2 K ⊆ K$ . Sparks classified the clones $C$ on $0,1$ according to the cardinality of the lattice $L(JA, C)$ of $JA, C)$ -clonoids (here $J_{A}$ denotes the clone of projections on a finite set $A$ ): $L(JA, C)$ is finite if and only if $C$ contains a near-unanimity operation; $L(JA, C)$ is countably infinite if and only if $C$ contains a Mal'cev operation but no majority operation; $L(JA, C)$ has the cardinality of the continuum if and only if $C$ contains neither a near-unanimity operation nor a Mal'cev operation. We sharpen Sparks's result by completely describing the lattices $L(JA, C)$ and the $JA, C)$ -clonoids when $A = 0,1$ and $C$ contains a near-unanimity operation or a Mal'cev operation, i.e., when $L(JA, C)$ is finite or countably infinite. On clonoids of Boolean functions
Oct. 17, 2023 2:30pm (MATH 3…	Lie Theory Justine Fasquel (University of Melbourne) X W-algebras are a large family of vertex algebras associated to nilpotent elements of simple Lie algebras. They are usually constructed from affine Lie algebras by applying a quantum Hamiltonian reduction. In this talk, we will discuss the possibility to invert this process in order to reconstruct the representation theory of initial Lie algebras. We will illustrate the abstract theory with examples in small ranks. Inverse Hamiltonian Reduction : a tool for the representation theory of affine Lie algebras Sponsored by the Meyer Fund
Oct. 17, 2023 2:30pm (MATH 2…	Math Edu Leilani Arthurs (University of Colorado Boulder) X Conventionally, assessments are used to measure how much students have learned mid-semester or at the end of the semester; however, class assessments can also be used to promote student learning. In this 1-hour interactive workshop, participants will be introduced to alternative assessments that engage students in the learning process and can lead to improved outcomes compared to traditional instruction alone. By the end of this workshop, participants will be able to (i) differentiate three types of assessments, (ii) discuss the merits of alternative assessments, (iii) describe examples of alternative assessments, (iv) explain what it is like to experience such assessments from a student perspective, and (v) begin designing such assessments for use in their courses as early as this semester if they wish. Promote Student Learning Using Class Assessments
Oct. 17, 2023 3:30pm (MATH 3…	Topology Bridget Schreiner (University of Notre Dame) X A sequence of groups (or spaces) $G1 G2 G3…$ satisfies homological stability if $Hi(Gn)≅ Hi(Gn+1)$ for $n>>i$ . Regarding the sequence $G_{n}$ as the image of a functor from the category $N,+,0)$ and noting that stability occurs when $Hi(Gn+1,Gn)≅ 0$ , we can turn this into a statement about "differences" in the image of a functor $ℕ \to A b$ . This talk will introduce a generalization of the cross effects of Eilenberg and MacLane that is suited to answer questions about homological stability in this context. Cross effects and stability
Oct. 18, 2023 3:30pm (Math350)	Math Phys Howy Jordan View Online:https://cuboulder.zoom.us/j/94002553301 X For online participation: Zoom Meeting ID: 940 0255 3301 Passcode: 790356 Stratification of spaces of density matrices
Oct. 18, 2023 4:40pm (MATH 3…	Grad Student Seminar Sylvia Maher (University of Colorado Boulder) X By examining the classical homotopy theory of topological spaces and identifying the key features that make it work, we can abstract those features to arrive at the notion of a model category, which allows us to do homotopy theory in an abstract setting. In this talk we will see several examples of model categories and what we can do with them, from topological spaces to homological algebra. An Introduction to Model Categories (Part 2)
Oct. 19, 2023 11:15am (MATH …	Algebraic Geometry Dan Lyness (University of Colorado Boulder) TBA
Oct. 19, 2023 2pm (Online)	Rep Theory Harshit Yadav (University of Alberta) View Online:https://cuboulder.zoom.us/j/97927772782 X Unimodularity a classical notion shows up in various areas like linear algebra, lattices, Poisson algebras, etc. In this talk, we focus on unimodular Hopf algebras and unimodular tensor categories. We will introduce unimodular module categories and use them to construct Frobenius algebras and unimodular tensor categories. These ideas will be illustrated with examples drawn from Hopf algebras. On unimodularity in the theory of tensor categories
Oct. 24, 2023 12:10pm (MATH …	Kempner Andy Hone (University of Kent, Canterbury, UK) X We consider a family of nonlinear maps that are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g, as originally described by van der Poorten. Using the connection with the classical theory of J-fractions and orthogonal polynomials, we show that in the elliptic case g=1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos-4 sequence, which were found in particular cases by Chang, Hu, and Xin. We extend these formulae to the higher genus case, and find analogues of Chebyshev polynomials (which correspond to the case g=0). Moreover, for all g>0 we show that the iteration for the continued fraction expansion is equivalent to a discrete Lax pair with a natural Poisson structure, and the associated nonlinear map is a discrete integrable system, connected with solutions of the infinite Toda lattice. Continued fractions, hyperelliptic curves and higher genus Chebyshev polynomials
Oct. 24, 2023 1:25pm (Math 2…	Algebra/Logic Patrick Wynne (CU Boulder) View Online:https://cuboulder.zoom.us/j/96896272260 X A clonoid from an algebra A to an algebra B is a set of functions from finite powers of A into B that is closed under composition with term operations of A on the domain side and under composition with term operations of B on the codomain. We investigate clonoids from one module to another. When A and B are finite modules of coprime order (and the submodule lattice of A is distributive) we show that each clonoid from A to B is finitely generated and the lattice of clonoids from A to B is finite. We then extend these results to finite abelian Mal’cev algebras. We show how a 2-nilpotent Mal’cev algebra can be classified with the help of an associated clonoid between abelian Mal’cev algebras. This is joint work with Peter Mayr. Clonoids and nilpotent algebras
Oct. 24, 2023 2:30pm (Math 3…	Lie Theory Shashank Kanade (University of Denver) X I will give a gentle introduction to the combinatorial Rogers--Ramanujan identities. While these identities are over a century old, and have many proofs, the first representation-theoretic proof was given by Lepowsky and Wilson about four decades ago. Now-a-days, these identities are ubiquitous in several areas of mathematics and physics. I will mention how these identities arise from affine Lie algebras and quantum invariants of knots. The world of Rogers--Ramanujan identities Sponsored by the Meyer Fund
Oct. 24, 2023 3:30pm (MATH 3…	Topology David Chan (Michigan State University) X The algebraic K theory of a manifold M is a space whose homotopy groups record interesting geometric properties. Some of these invariants, like the Wall finiteness obstruction and Whitehead torsion, depend only on the fundamental group G of M and can also be defined in terms of the algebraic K theory of the group ring Z[G]. The relationship between the algebraic K theory of M and the algebraic K theory of Z[G] is made precise by a comparison map called the linearization. In this talk we will review these ideas and discuss some recent work on a refinement of the linearization map to the genuine equivariant algebraic K theory of spaces. This is joint work with Maxine Calle and Andres Mejia. A linearization map for equivariant algebraic K theory
Oct. 26, 2023 11:15am (MATH …	Algebraic Geometry Leo Herr (Universiteit Leiden) X Kato: Log structures are “magic powder” that makes mildly singular spaces appear smooth. Log schemes literally lie between ordinary schemes and tropical geometry, and are related to Berkovich Spaces. Problems in Gromov-Witten Theory demand intersection theoretic machinery for slightly singular spaces. Log structures have solved similar problems in Hodge Theory, D-modules, Connections and Riemann-Hilbert Correspondences, Abelian Varieties (esp. Elliptic Curves), etc. How can they be used to define a reasonable intersection theory for curve counting on singular spaces? We'll compare different answers to this question and prove basic formulas for log Gromov--Witten invariants. Log Gromov--Witten Theories
Oct. 26, 2023 2pm (Online)	Rep Theory Hongdi Huang (Rice University) View Online:https://cuboulder.zoom.us/j/97927772782 X A commutative algebra A together with a Lie bracket satisfying the Leibniz rule is called a Poisson algebra, which is named in honor of Siméon Denis Poisson. Poisson structures appear in many contexts, including string theory, classical (quantum) mechanics, and differential geometry. In this talk, we will talk about Poisson structure on weighted polynomial rings and introduce Poisson valuations. Furthermore, we will see that the Poisson valuations play an important role in characterizing the Poisson automorphism groups of certain Poisson algebras. Weighted Poisson polynomial rings and their Poisson valuations
Oct. 26, 2023 3:35pm (MATH 3…	Probability Lionel Levine (Cornell University) X The output of a language model like ChatGPT is a probability distribution for the next word (or fragment of a word). Mathematically, what is the model actually doing when it generates these probabilities? I’ll narrate how the model translates its input text into a sequence of vectors, which pass through a succession of alternating linear and nonlinear layers. Then we’ll examine this architecture with a view toward some big-picture open questions: Do language models “plan ahead” for future words, phrases, and sentences? If not, then how do they manage to generate coherent text? And if yes, then how does their training — they are trained myopically to maximize the predicted log probability for the next word — incentivize planning? I’ll illustrate the theory by fine-tuning GPT2 (an open-source precursor of ChatGPT) to do some funny and unusual things. I’ll also talk a bit about the experience of switching research fields, from pure math to AI safety. Language models for mathematicians
Oct. 27, 2023 1:20pm (MATH 3…	Geometry/Analysis Vincent R. Martinez (Hunter College & Graduate Center, CUNY) X We study dissipative perturbations of the 2D generalized surface quasi-geostrophic (gSQG) equations. This family contains the 2D Euler equations in vorticity form at one endpoint, the SQG equation at its midpoint, and the Okhitani equation at the most singular endpoint of the family. Recent work of Bourgain/Li, Elgindi/Masmoudi, Cordoba/Zoroa-Martinez, and Jeong/Kim have established ill-posedness in this family at critical regularity. This talk will consider “mild" perturbations of the gSQG equation which recover well-posedness in a setting of critical regularity, but additionally confers a mild degree of regularity instantaneously as well. These perturbations encompass an intermediate case in contradistinction with strongly dissipative perturbations, which instantaneously confer Gevrey regularity and recover well-posedness at critical regularity (Jolly/Kumar/M 2021), and inviscid regularization, which do not regularize solutions, but nevertheless recover local well-posedness at critical regularity (Chae/Wu 2010). We show that in this intermediate regime that one may recover local well-posedness at borderline Sobolev regularity, as well as a global existence theory at the 2D Euler endpoint. Moreover, we provide a general existence theory for an entire class of such perturbations that is effectively sharp in light of the recent ill-posedness results. This is joint work with A. Kumar (Florida State University). Well-posedness of mildly dissipative perturbations of a family of active scalar equations Sponsored by the Meyer Fund
Oct. 31, 2023 1:25pm (online)	Algebra/Logic Paolo Agliano (University of Siena, Italy) View Online:https://cuboulder.zoom.us/j/96896272260 X The concept of splitting of a lattice dates back to the 1940's and the general theory of splittings in the lattice of subvarieties of a variety has been used and abused for a long time. However one can also look at splitting in the lattice of subquasivarieties of a (quasi)variety since the theory is not much different (and in fact it is, in a certain sense, easier). We will talk briefly about the general theory and, as an application, we will prove a generalization of Baker's Finite Basis Theorem for quasivarieties. This is a joint work with Alex Citkin. Splittings and a (yet another!) generalization of Baker's Finite Basis Theorem
Oct. 31, 2023 2:30pm (MATH 2…	Math Edu Peter Rock (University of Colorado Boulder) View Online:https://cuboulder.zoom.us/j/91389610106 X For online participation: Zoom Meeting ID: 913 8961 0106 Passcode: math! I will be giving an overview of some of the materials that I have developed in my time as a course assistant and an instructor at CU. All of the materials that I will be showing during this talk are things that I currently use in my role as the course assistant for Calc III and which I have found to be helpful in fulfilling that role. I also believe that access to these materials would be a boon to the rest of the coordinated courses in our math program, so it is the goal of this talk to show off some of these tools and hopefully disseminate them amongst the various instructors, TAs, course coordinators, and course assistants of our department. The presentation itself will be composed of 3 parts: (1) A brief introduction to Canvas API tools with Python: how to not spend 4 hours editing all of your assignment due dates individually. (2) Diversifying homework assignments: how to build a problem portfolio and generate random assignments on the fly with LaTeX. (3) No more grading parties: how ideas from mastery reform grading can help make grading faster and more consistent in coordinated courses. Bonus: How to set up your shared overleaf file so that people can actually find things. All materials presented within this talk will be made available to those who are interested. I hope to see you there! Reorganizing Undergraduate Math Courses: Lessons from a Course Assistant
Oct. 31, 2023 3:30pm (MATH 3…	Topology Andrew Doumont X We often come across spaces that look nearly like manifolds but for some singularities. This talk will be an introduction to stratified spaces, which are a generalization of smooth manifolds that allow us to retain some important smooth manifold properties for spaces that aren't quite themselves smooth manifolds. Stratified Spaces
Nov. 01, 2023 4:40pm (MATH 3…	Grad Student Seminar Edouard Heitzmann (University of Colorado) X Hawking and Penrose’s Singularity Theorems are advanced results in General Relativity which describe Big Bang and Black Hole singularities given minimal topological information about the manifold they live in, using the language of Semi-Riemannian Geometry. This talk presents two proofs of the former Theorem: one stronger version which gives a picture of the Big Bang similar to the Robertson-Walker model, and the other which assumes less and proves less. Along the way, we will develop the theory of Focal Points and Index Forms, which play a pivotal role in these proofs, and provide some Physical intuition to justify the Timelike Convergence Condition assumed in Hawking’s Theorem. Hawking's Singularity Theorem
Nov. 02, 2023 11:15am (MATH …	Algebraic Geometry Sebastian Casalaina-Martin (University of Colorado Boulder) TBA
Nov. 02, 2023 2pm (Online)	Rep Theory Sergei Gukov (California Institute of Technology) View Online:https://cuboulder.zoom.us/j/97927772782 X At its core, this talk will be about a relation between characters of modules of logarithmic vertex algebras in the positive and negative Kazhdan-Lusztig (KL) zones. The main concrete result will be an easy-to-use step-by-step computational algorithm that produces a character of a log-VOA, say, in the positive KL zone from the expression in the negative KL zone, and vice versa. An equally (if not more!) valuable conceptual message of this talk will be an explanation that the same bijective relation plays an important role in very different areas of mathematics (and even physics) under other guises. Going to the other side
Nov. 06, 2023 3:30pm (Math350)	Math Phys Mert Okyay (CU Boulder) View Online:https://cuboulder.zoom.us/j/94002553301 X For online participation: Zoom Meeting ID: 940 0255 3301 Passcode: 790356 The geometric phase phase à la M. Berry
Nov. 07, 2023 1:25pm (MATH 2…	Algebra/Logic Nik Ruskuc (University of St Andrews, UK) View Online:https://cuboulder.zoom.us/j/96896272260 X Let S be a semigroup. A (right) S-act X is a representation of S by transformations on X. S-acts can be regarded as (unary) algebras in their own right. Thus we can talk about subacts, homomorphisms, quotients, generating sets and defining relations for S-acts. This in turn opens up a possibility of attempting to study/understand/classify semigroups according to properties of their S-acts. In this talk I will report on some recent work on some such properties: - Noetherian: S is Noetherian if every finitely generated S-act is finitely presented; - Coherent: S is coherent if every finitely generated subact of a finitely presented S-act is finitely presented; - The trivial S-act is finitely presented; - Pseudofiniteness: the trivial S-act is finitely presented and there is a uniform bound on the length of derivation sequences. I will also indicate some intriguing open problems that have arisen in each area. This work is currently being pursued under an EPSRC-funded project held jointly with Victoria Gould from York, and involves several collaborators, including Matthew Brookes, Craig Miller, James East and Thomas Quinn-Gregson. Semigroup actions and finiteness conditions Sponsored by the Meyer Fund
Nov. 07, 2023 2:30pm (MATH 3…	Lie Theory Chris Eblen (CU) X The classical case of Schur--Weyl duality links the representation theory of the general linear group to that of the symmetric group, enabling the study of both simultaneously. Generalizations of this kind of duality have yielded pathways to study the representation theory of some common groups in interesting ways. Such pathways are especially valuable when the group in question has an intractable representation theory, as in the case of the group of unipotent upper triangular matrices with entries in a finite field, $U_{n}$ . We discuss an analogue of Schur--Weyl duality for $U_{n}$ and a class of $U_{n}$ -modules called beach modules arising from the study of this duality within the framework of a particular supercharacter theory for $U_{n}$ . Unipotent Beach Modules and Schur--Weyl Duality
Nov. 07, 2023 3:30pm (MATH 3…	Topology Rachel Chaiser X I will give an introduction to the K-theory of Banach algebras, with a C-algebraic lean. I will define Banach algebras and C-algebras, so prior knowledge of these is helpful but not required. I will also discuss the relationship between K-theory for C*-algebras and K-theory for topological spaces. K-Theory for Banach Algebras
Nov. 08, 2023 3:30pm (Math350)	Math Phys Ezzedine El Sai (CU Boulder) X For online participation: Zoom Meeting ID: 940 0255 3301 Passcode: 790356 The Bures metric and Uhlman's connection
Nov. 09, 2023 11:15am (MATH …	Algebraic Geometry Bob Kuo (University of Colorado Boulder) X Smooth cubic surfaces (over the complex numbers) contain exactly 27 lines. Degenerating them, and following where the lines go, leads to a rich theory of combinatorics. Beniamino Segre, in his book "The non-singular cubic surfaces" (1942), uses the symmetries inherent in this combinatorial framework to classify the symmetries of cubic surfaces. In this talk, we'll do some early 20th-century math and follow how Segre classifies cubic surfaces defined over the real numbers. Classifying real cubic surfaces
Nov. 14, 2023 1:25pm (online)	Algebra/Logic Nóra Szakács (University of Manchester, UK) View Online:https://cuboulder.zoom.us/j/96896272260 X Inverse monoids are an important generalization of groups, and they form a variety of algebras. We focus on the class of inverse monoids called F-inverse: this is not closed under taking inverse subsemigroups or homomorphic images, but does form a variety in the signature enriched with an extra unary operation. Recently, Auinger, Kudryavtseva and M. Szendrei gave an elegant model for free F-inverse monoids in this extended signature. More generally, they describe the initial object in the category of X-generated F-inverse monoids with greatest group image G. We show that this category is equivalent to the category of finitary (i.e. algebraic) G-invariant closure operators on subgraphs of the Cayley graph of G, and adapt Stephen's approach to study the word problem in presentations of inverse monoids for F-inverse monoids in the enriched signature. Closure operators on group Cayley graphs, and presentations of F-inverse monoids
Nov. 14, 2023 2:30pm (MATH 3…	Lie Theory Karola Meszaros (Cornell) X The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as Fox’s conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial $ΔL(t)$ of an alternating link $L$ are unimodal. Fox’s conjecture remains open in general with special cases settled by Hartley (1979) for two-bridged knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsv\’ath and Szab\’o (2003) for the case of genus 2 alternating knots, among others. We settle Fox’s conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of $ΔL(t)$ , where $L$ is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal. This talk is based on joint work with Elena Hafner and Alexander Vidinas. Log-concavity of the Alexander polynomial Sponsored by the Meyer Fund
Nov. 14, 2023 2:30pm (MATH 2…	Math Edu Rebecca Machen (University of Colorado Boulder) View Online:https://cuboulder.zoom.us/my/rebeccamachen X In this training, participants will examine how their own mathematical backgrounds can impact their teaching choices in the classroom. The session will go beyond individual perspectives to promote a community mindset that fosters inclusivity and student belonging. Participants will explore the relationship between deficit language and its effects on students, learn about the pedagogical concepts of "warm demander and the hidden curriculum", and acquire practical strategies to transform the student experience in mathematics classrooms. This session welcomes anyone interested in learning more about how to promote student belonging in mathematical spaces. Promoting Student Belonging in Mathematics Classrooms through Equity-Driven Pedagogies
Nov. 14, 2023 3:30pm (MATH 3…	Topology Celia Hacker (Max-Planck-Institute Leipzig) View Online:https://cuboulder.zoom.us/j/94002553301 X At the intersection of Topological Data Analysis and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian and the resulting Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this talk, we introduce an approach to signal compression and reconstruction on complexes that leverages the tools of discrete Morse theory. The main goal is to reduce and reconstruct a cell complex together with a set of signals on its cells while preserving their global topological structure as much as possible. This is joint work with Stefania Ebli and Kelly Maggs. Signal processing on cell complexes using discrete Morse theory
Nov. 15, 2023 3:30pm (Math350)	Math Phys Shreya Arya (Duke University) X Brownian motion is the most basic diffusion process on Euclidean space. The goal of this talk is to understand diffusion on spaces beyond Euclidean space. While Brownian motion on manifolds has been extensively studied, expanding this construction to stratified spaces proves challenging. Sturm introduced an approach for constructing diffusion processes on metric measure spaces, relying on the theory of Dirichlet forms and their well-established connection to Markov processes. These forms generate diffusion only if the space satisfies a geometric condition known as the "measure contraction property," which can be viewed as a regularity condition for measures. We explore the geometric condition in the context of Whitney stratified spaces, equipped with the Riemannian distance and a volume measure. I will talk about the relationships between diffusion and Dirichlet forms and their generalization to stratified spaces. No prior knowledge of Dirichlet forms and Markov processes is assumed. This is ongoing work with Celia Hacker, Markus Pflaum and Sayan Mukherjee. Brownian Motion on Singular Spaces Sponsored by the Meyer Fund
Nov. 15, 2023 4:40pm (MATH 3…	Grad Student Seminar Jon Kim (University of Colorado) X When working with curves and surfaces, it's much nicer to work with them when they are smooth. Alas, we can't always have these nice assumptions, so we gotta ask when smoothness fails and whether if we can resolve it. Well, we know how to identify singularities, but if we can resolve them, we then gotta figure out how to do it in a manner where we can preserve information away from it. Thankfully, it turns out that we do have such a method called blowing-up. This talk will be an introduction to blow-ups in an illustrated manner to via worked examples with and (cartoon)-pictures. An Illustrated Introduction to Blow-ups
Nov. 16, 2023 11:15am (MATH …	Algebraic Geometry Adrian Neff (University of Colorado Boulder) X Stable maps spaces arise naturally when trying to answer questions in enumerative geometry, e.g., in the proof of Konstevich's formula for the number of rational plane curves passing through a fixed number of points. However, once the genus of the curve is greater than 0, the stable maps spaces become highly singular, having multiple irreducible components of varying dimensions. In order to have more refined enumerative counts, one tries to find desingularizations of these spaces. In this talk, I'll discuss the history of this desingularization problem for genus 1 stable maps to projective space, and conclude with a discussion of a new approach with the potential to work for higher genera (work in progress). Stable maps from genus 1 curves to projective space
Nov. 16, 2023 2pm (Online)	Rep Theory Agustina Czenky (University of Oregon) View Online:https://cuboulder.zoom.us/j/97927772782 X In this talk, we look at the classification problem for symmetric fusion categories in positive characteristic. We recall the second Adams operation on the Grothendieck ring and use its properties to obtain some classification results. In particular, we show that the Adams operation is not the identity for any non-trivial symmetric fusion category. We also give lower bounds for the rank of a (non-super-Tannakian) symmetric fusion category in terms of the characteristic of the field. As an application of these results, we classify all symmetric fusion categories of rank 3 and those of rank 4 with exactly two self-dual simple objects. Low rank symmetric fusion categories in positive characteristic.
Nov. 28, 2023 12:10pm (MATH …	Kempner Uri Shapira (Technion Israel, Israel) X The aim of this talk is to introduce the area of homogeneous dynamics to a general audience and to survey some key instances in which number theory interacts with it. I hope to touch upon the following: (1) Margulis' resolution of Oppenheim's conjecture regarding values of indefinite irrational quadratic forms at integral points. (2) Einsiedler-Lindenstrauss-Katok work towards Littlewood's conjecture in Diophantine approximation. (3) Linnik/Duke theorem that says (loosely) that integral points on large spheres are spread randomly on these spheres. (4) I will also try to mention a few results that I have been involved in proving. Homogeneous dynamics and number theory
Nov. 28, 2023 1:25pm (online)	Algebra/Logic Clemens Schindler (Technical University Vienna, Austria) View Online:https://cuboulder.zoom.us/j/96896272260 X The set of all increasing functions on the rational numbers carries an algebraic structure (the standard composition operation of functions, yielding a semigroup) and a topological structure (the topology of pointwise convergence in which a sequence of functions converges if and only if it is eventually constant at every argument). These structures are compatible in the sense that the operation is continuous with respect to the topology. Whenever one considers such a combined algebraic-topological structure, one can ask the following general question: Is the topology predetermined by the compatibility with the operations, i.e., does the algebra have a unique (Polish?) topology? In other words: Can the (Polish) topology be "reconstructed" from the algebra? I will present several examples of Polish groups and semigroups that do (not) have this property and introduce the known proof techniques. Afterwards, I will discuss why these successful techniques fail for the space of all increasing functions on the rational numbers and outline how this shortcoming can be mitigated to show that, indeed, the increasing functions on the rational numbers carry a unique Polish topology. This is joint work with Michael Pinsker. Unique Polish semigroup topology: novel techniques to crack the semigroup of increasing functions on the rational numbers
Nov. 28, 2023 3:30pm (MATH 3…	Topology Howy Jordan X Given a space, consider a decomposition into a pair of complementary open and closed subspaces. Artin gluing tells us which open sets of the subspaces can glue to an open set of the whole space and what data is needed to glue sheaves on the subspaces to obtain a sheaf on the whole space. This can be readily generalized to decompositions into locally closed subspaces. Artin gluing describes how to glue sheaves, hence etale bundles. In this talk we will consider how to also apply Artin gluing to fiber bundles. If the fibers are isomorphic we can obtain another fiber bundle, but in general we obtain stratified fiber bundles. In the case of tangent bundles, properties of the Artin gluing data are related to the Whitney regularity conditions. Artin Gluing of Fiber Bundles and the Whitney Conditions
Nov. 29, 2023 4:40pm (MATH 2…	Grad Student Seminar Daniel Lyness (University of Colorado) X Given a map between compact Riemann surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces and the degree of the map. In this talk we will discuss the various components of this formula towards giving a sketch of the proof, and seeing some simple consequences in Topology and Algebraic Curves. We will define CW complexes in order to compute the Euler characteristic of surfaces, and then analyse the rigid structure of maps between Riemann surfaces to define the degrees of such maps, before seeing how to relate these numbers. I intend this to be a very accessible talk which will introduce some interesting ideas from Topology and Algebraic Curves. There will be lots of pictures! The Riemann-Hurwitz formula
Nov. 30, 2023 11:15am (MATH …	Algebraic Geometry Jon Kim (University of Colorado Boulder) TBA
Nov. 30, 2023 2pm (Online)	Rep Theory Vladimir Kovalchuk (University of Denver) View Online:https://cuboulder.zoom.us/j/97927772782 X W-(super)algebras have generated great interest in recent years due to their numerous applications in mathematics and physics. The process of Hamiltonian reduction in stages suggests that W-(super)algebras often arise as extensions of tensor products basic building blocks. In type A, we expect that the building blocks are the Gaiotto-Rapcak Y-algebras which arise as 1-parameter quotients of the universal 2-parameter VOA of type W(2,3,…). For types B, C, and D, the quotients of the universal 2-parameter VOA of type W(2,4,…) provide some, but not all, of the necessary building blocks. In this talk we discuss a new universal 2-parameter VOA of type W(1^3,2,3^3,4,…), whose 1-parameter quotients are expected to account for the missing building blocks for W-(super)algebras of types B, C, and D. There are 8 infinite families of such quotients, which are analogues of the Gaiotto-Rapcak Y-algebras. We will explain the process behind the construction of this universal two-parameter VOA and discuss several applications. This is a join work with Thomas Creutzig and Andrew Linshaw. On the universal 2-parameter VOA of typeW(1^3,2,3^3,4,...).
CANCELED Nov. 30, 2023	Functional Analysis Andrew Toms (Purdue University) View Online:TBA Toward the rank problem for simple nuclear C*-algebra
Dec. 05, 2023 2:30pm (Zoom)	Math Edu Austin Mohr (Nebraska Wesleyan University) View Online:https://cuboulder.zoom.us/j/95717237725 X Mastery-based testing is an assessment style that challenges students to provide complete solutions to problems derived from clear course concepts. Students are allowed multiple attempts to demonstrate mastery with no penalty for failed attempts. I argue that mastery-based exams encourage sustained progress toward thorough understanding as compared with traditional points-based exams. Moreover, I believe they offer increased equity with respect to student experience in previous mathematics courses. We will conclude with brief guidelines on how you can easily implement mastery-based exams in your existing courses. Assessment That Promotes Persistence and Equity
Dec. 06, 2023 3:30pm (MATH 3…	Topology Bob Kuo (CU Boulder) X One can view closed symmetric monoidal categories as a linearization of category theory. It is a framework that allows one to talk about linear algebraic notions in categories without an obvious linear structure. Some examples include compactly generated pointed topological spaces, chain complexes, and spectra. In this talk, I'll introduce the definition of closed symmetric monoidal categories, and then discuss how the notions of duals and traces generalized to these examples. Traces and duality in closed symmetric monoidal categories
Dec. 06, 2023 4:40pm (MATH 2…	Grad Student Seminar Sylvia Maher (University of Colorado) X In this talk I will discuss derived functors, which are universal homotopy-theoretic approximations to functors between categories equipped with a notion of homotopy theory. I will show how they generalize derived functors from homological algebra, and then talk about homotopy (co)limits, an important class of derived functors. Model Categories (part 3): Derived Functors
Dec. 12, 2023 12:10pm (MATH …	Kempner Jun Ho Lee (Mokpo Nat'l Univ, Korea) X Arithmetic of the units of number fields gives a lot of information about number fields. According to Dirichlet's unit theorem, the structure of units of the orders in number fields. In this talk, we are interested in exact expression of the units. Moreover, we will share the conjectures for units. We will talk about our results in this direction if time permits. Note on the units of algebraic number fields
Dec. 12, 2023 1:25pm (online)	Algebra/Logic H. Peter Gumm (Philipps University Marburg, Germany) View Online:https://cuboulder.zoom.us/j/96896272260 X Universal Coalgebra is a general theory of state based systems, encompassing all sorts of automata, transition systems, Kripke structures, etc., be they deterministic, non-deterministic, probabilistic, second order, or else. This diversity of examples is covered by choosing as signature F of the mentioned coalgebras an appropriate Set-Functor, rather than just a sequence of arities, like in Universal Algebra. Preservation properties of that signature functor F determine much of the structure of the corresponding classes of coalgebras. Of particular interest, it turns out, is the question, whether F preserves preimages or (weak) pullbacks. This provides only the backdrop and the motivation for us to study a class of functors arising from classical Universal Algebra as "free-algebra-functors" where FV(X) is taken as the base set of the free algebra over X in a variety V. We show amongst other things, that FV(-) preserves preimages if and only if V can be defined by essentially balanced (also called regular) equations. Regarding preservation of kernel pairs, the picture is still fragmented. For instance, if V is a Malcev variety then FV weakly preserves kernel pairs, and if V is n-permutable and FV weakly preserves kernel pairs, then V is Malcev. In joint work with R. Freese we study the case when V is a variety L of lattices, with further order preserving operations permitted. It turns out that FL(-) always weakly preserves kernel pairs. In purely lattice theoretical terms, this shows that in any lattice variety L any balanced equation p(u1,...,um) = q(v1,...,vn), where {u1,...,um} = {v1,...,vn} must have an obvious reason, namely that both p and q are substitution instances of a common "ancestor"-term s, from which the said equation arises by further identification of variables, resulting in syntactically identical terms. For arbitrary nontrivial idempotent varieties V of algebras we additionally show, using Olshak's theorem, that (strong) preservation of pullbacks is impossible. Free algebra functors are special instances of copower functors which we can define in any concrete and cocomplete category. For instance, if M is a (commutative) monoid, the functor M[X] arising as X-fold sum of M, preserves preimages iff M is positive, and weakly preserves pullbacks iff M is (refinable) equidivisible. Lucid reasons behind balanced identities
Dec. 12, 2023 3:30pm (MATH 3…	Topology Justin Barhite X Traces arise in many different places in math: traces of matrices, characters of group representations, and even the Euler characteristic of a CW complex! There are very general notions of trace, expressed in the language of category theory, that capture these examples of traces and whose properties imply familiar results like the Lefschetz fixed point theorem and the induction formula for characters. The formalism of traces doesn't tell the whole story though; there are some constructions that feel trace-like in certain ways but also have a distinct flavor, and what's really needed to explain them from this category-theoretic perspective is a dual notion of "cotrace." I will talk about some of these things that I have been working to understand by developing a theory of bicategorical cotraces. Traces and Cotraces in Bicategories

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