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Events for June – October 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 …	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
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 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 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 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 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 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 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

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