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Events for May – September

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Thu, Sep. 17 5pm (MATH 350)	Algebraic Geometry Jon Kim (University of Colorado Boulder) X The moduli space of cubic surfaces with a marked line lies naturally between the moduli space of fully marked and unmarked cubic surfaces and comes equipped with a W(D_5)-action corresponding to the stabilizer of the marked line. Hodge theoretic compactifications of these moduli spaces have been extensively studied by many, including Allcock—Carlson—Toledo, Dolgachev—van Geemen—Kondo, and Casalaina-Martin—Grushevsky—Hulek. In previous work, by using KSBA wall crossing, we described many different birational models of this moduli space via KSBA stable pairs. In this talk, we will discuss recent results regarding the W(D_5)-invariant birational geometry of these KSBA moduli spaces and ongoing work on the log minimal model program for these KSBA moduli spaces with respect to their log canonical divisor. The log MMP for the moduli space of cubic surfaces with a marked line
Mon, May. 11 2pm (MATH 350)	Algebra/Logic Michael Pinsker (TU Wien) X For a permutation group G on a set X, a mixed identity is an expression of the form w(x_1,...,x_n,g_1,...,g_m)=1, where w is a word in the language of groups with constants g_1,...,g_m from G and free variables x_1,...,x_n. We say that the mixed identity holds in G if substitution of arbitrary elements of G for the free variables yields a true statement. The mixed identity is singular if deletion of all constants yields a true statement in the free group: for example, xgx^{-1}=1 (where x is a variable and g a constant) is singular. We consider the question which non-singular mixed identities hold in a given group G. The group G is oligomorphic if it acts with finitely many orbits on X^n, for all n. Based on several examples, Bodirsky, Schneider, and Thom have conjectured that in oligomorphic groups only singular identities can hold. We confirm this conjecture for all oligomorphic G without algebraicity, i.e. with the property that for any tuple t of elements of X, the stabilizer of t in G does not stabilize any element outside t. In fact, we obtain this result under even more general conditions satisfied e.g. by the general linear group of any infinite vector space over a finite field. This is joint work with Paolo Marimon (Oxford). Mixed identities in oligomorphic groups Sponsored by the Meyer Fund
Mon, May. 11 3pm (MATH 350)	Algebra/Logic Andrei Bulatov (Simon Fraser University) X We study constraint satisfaction problems (CSPs) where the constraint languages are defined by finite automata, giving rise to automata-based CSPs. The key notion is the concept of Automatic Constraint Satisfaction Problem ($\AutCSP$) where constraint languages and instances are specified by finite automata. The $\AutCSP$ captures infinite yet finitely describable sets of relations, enabling concise representations of complex constraints. Studying the complexity of the $\AutCSP$s illustrates the interplay between classical CSPs, automata, and logic, sharpening the boundary between tractable and intractable constraints. We show that checking whether an operation is a polymorphism of such a language can be done in polynomial time. Building on this, we establish several complexity classification results for the $\AutCSP$. In particular, we prove that Schaefer’s Dichotomy Theorem extends to the $\AutCSP$ over the Boolean domain, and we provide algorithms that decide tractability of some classes of $\AutCSP$s over arbitrary finite domains via automatic polymorphisms. An important part of our work is that our polynomial-time algorithms run on $\AutCSP$ instances that can be exponentially more succinct than their standard CSP counterparts. Automatic constraint satisfaction problem Sponsored by the Meyer Fund
Tue, Aug. 4 2:30pm (Math 3…	Lie Theory Robert McRae (Tsinghua University) View Online:https://cuboulder.zoom.us/j/6741562716 X Let $𝔤$ be a finite-dimensional simple Lie algebra and let $Lk(g)$ be the simple affine vertex operator algebra associated to $𝔤$ at a level $k$ such that $k+hinQ>0$ , where $h^{\lor}$ is the dual Coxeter number of $𝔤$ . We explain how to use recent results on braided tensor categories to show that the category of $C1$ -cofinite $Lk(g)$ -modules is a semisimple rigid braided tensor category. For certain levels, $Lk(g)$ is known to be the same as the universal affine vertex operator algebra of $𝔤$ and level $k$ , so in these cases, it follows that all generalized Verma modules induced from finite-dimensional simple $𝔤$ -modules are simple, and the category of $C1$ -cofinite $Lk(g)$ is equivalent to a minor modification of the tensor category of finite-dimensional $𝔤$ -modules. This makes it possible to construct many new vertex operator algebras by extending tensor products of affine vertex operator algebras associated to $𝔤$ at different levels. Semisimplicity of ordinary modules for simple affine vertex operator algebras at positive rational shifted levels Sponsored by the Meyer Fund
Fri, Aug. 21 1:25pm (Math 3…	A. I. Agnès Beaudry, Katherine E. Stange, and Jonathan Wise X AI has arrived. We will survey the current landscape, give an overview of the upcoming course & seminar series, and invite everyone to share their experiences. The AI seminar is aimed at a general mathematical audience. AI and Math
Thu, Aug. 27 11:15pm (MATH …	Algebraic Geometry Sebastian Casalaina-Martin and Jonathan Wise X Our algebraic geometry research faculty, Sebastian (Yano) Casalaina-Martin and Jonathan Wise will each give a quick informal talk about their current research. This is a great opportunity for new graduate students who are interested in working in algebraic geometry! AG Open House
Fri, Aug. 28 1:25pm (Math 3…	A. I. Katherine Stange (CU Boulder) X An overview of the architecture of an LLM. The Math Behind LLMs
Tue, Sep. 1 2:30pm (MATH 3…	Lie Theory Richard Green (CU) X A poset congruence is the analogue for partially ordered sets of a lattice congruence. If a group G acts on the underlying set of a poset P, then we call a poset congruence "G-invariant" if the G-action respects the partition of P into congruence classes. This talk will discuss specific examples where G is a Weyl group of type A or D. This is partly based on joint work with Tianyuan Xu (University of Richmond). Invariant poset congruences
Wed, Sep. 2 4:40pm (MATH 3…	Grad Student Seminar Edouard Pierre Francois Heitzmann (CU Boulder) X As part of a recent research problem, I needed a cheap and sharp way to compute anytime-valid refineable lower bounds for directional derivatives of a high-dimensional function over a coordinate diamond. The Interval Branch and Bound (B&B) algorithm turned out to be a good fit for this task. In this talk I will tell you about the Mathematics involved in this algorithm – including the so-called “Fundamental Theorem of Interval Arithmetic” – as well as its history and the context I had to apply it in. A Conservative Optimization Algorithm Using the Fundamental Theorem of Interval Arithmetic
Fri, Sep. 4 1:25pm (Math 3…	A. I. Edgar Costa (MIT) X Large language models become more reliable when surrounded by well-designed instructions, reusable skills, persistent records, and independent checks. I will explain how to build and use these supporting structures for rigorous technical work. How to Tame Your LLM: Harnesses for Rigorous Work
Tue, Sep. 8 11am (MATH 350)	Number Theory Taku Ishii (Seikei University) X Explicit formulas for spherical Whittaker functions on GL(n,R) were obtained by Stade around 1990. As an application, Stade computed archimedean zeta integrals for automorphic L-functions on GL(n)×GL(n) and GL(n)×GL(n-1). In this talk, I will discuss joint work with Stade on extending these results to classical groups. Archimedean Whittaker functions and archimedean zeta integrals Sponsored by the Meyer Fund
Tue, Sep. 8 5pm (MATH 350)	Math Club Rebekah Jones X Come to our first meeting on Tuesday, Sept 8th, 5-6pm to learn about what we do at Math Club and what we have planned for the semester. We’ll also have some fun math games/activities and, as always, free pizza! If you RSVP at the link below, you can vote on your pizza preferences. RSVP link: https://forms.gle/HFgNJZg8RhNfVWJg6 Welcome Day
Wed, Sep. 9 3:30pm (Math350)	Math Phys Andrew Doumont (CU Boulder) X Many techniques for studying symplectic manifolds (e.g. Gromov-Witten invariants) rely on the ability to count certain pseudoholomorphic curves. In order to do this counting one must have some notion of fundamental class on the moduli space of pseudoholomorphic curves. However, such moduli spaces are generally very poorly behaved. In this talk, we will discuss Kuranishi spaces which are a geometric structure one can equip these moduli spaces with in order to define a virtual fundamental class. Kuranishi Space
Fri, Sep. 11 10:10am (Math …	Math Edu Rebekah Jones X Join us to discuss potential topics and speakers for the Seminar on Undergraduate Math Education (SUME) this semester. We encourage all those interested in attending SUME this semester to attend! SUME Open House
Fri, Sep. 11 1:25pm (Zoom (…	A. I. Ken Ono (Axiom Math) X Mathematics is entering a period of both promise and unease. AI systems now discover counterexamples and conjectures, produce research-level arguments, and generate formally verified proofs in systems like Lean. This talk offers a working map of the landscape, organized around three distinct modes: discovery, reasoning, and verification, and why confusing them is the fastest way to overclaim. After surveying recent landmarks, I will present case studies from my own work at Axiom Math. I will discuss what is real, what is hype, and how rigor survives. The speaker is on Zoom. We suggest you attend on zoom (better experience for the speaker). Math 350 will be available for those who wish to share in-person viewing. Mathematics in the Age of AI: Discovery, Verification, and the Future of Proof
Tue, Sep. 15 1:25pm (MATH 2…	Algebra/Logic Keith Kearnes & Peter Mayr (CU) X The foundations group will talk about its research. Foundations Open House
Tue, Sep. 15 11pm (350)	Number Theory Jamie Juul (Colorado State University) X Iterating a polynomial or rational function produces a tree of preimages of a point, and the Galois group of the resulting tower of fields acts on this tree by symmetries — a dynamical analogue of the Galois representations attached to elliptic curves, and a natural place to ask how large this group of symmetries can be. I will describe a general criterion, based only on the combinatorics of a map's critical orbits, that determines whether this group is as large as possible, and if not, exactly how it falls short at every level. Along the way I'll work out an explicit family of examples where all the critical orbits eventually collide, and describe how recent work of König, Neftin, and Rosenberg makes the criterion apply automatically to a broad class of polynomials. Rational Functions with Large Iterated Monodromy Groups
Wed, Sep. 16 3:30pm (Math350)	Math Phys Markus J. Pflaum (CU Boulder) X This is an introductory talk to the semester topic of Mathematical Foundations of Quantum Theory. In the last 20 minutes potential topics for talks will be discussed and assigned. Students participation is strongly encouraged. State spaces in quantum mechanics Part 1 and seminar organization
Fri, Sep. 18 1:25pm (Math 3…	A. I. Gregory Berkolaiko (Texas A&M) X Claude Code is a command-line AI assistant that reads and edits files directly in a project directory. This talk reports on using it for three tasks: (a) reading a paper in an unfamiliar area and writing a report on it, a task often faced by a student; (b) revising a paper draft; and (c) course preparation, specifically the design of student projects. For (a) we work through "Bakry-Émery curvature on graphs as an eigenvalue problem" by Cushing, Kamtue, Liu and Peyerimhoff (Calc. Var. PDE, 2022). The second half concerns guardrails: writing a rules file the assistant is expected to follow, keeping the human as the decision-maker on every edit, and tracking changes through git and Overleaf. We will see cases where the assistant caught errors a careful reader would miss, cases where it introduced errors of its own, and cases where it simply ignored an explicit instruction. The talk is an introduction, aimed at those who have not yet used Claude Code or a comparable tool (Cursor, Aider). Math editing with Claude Code: uses and guardrails
Wed, Sep. 23 4:40pm (MATH 3…	Grad Student Seminar Taylor Rogers (CU Boulder) X "Homotopy coherent algebra provides a family of fields interpolating between characteristic 0 and p for each prime p. Finite group representation theory extends to these higher categorical contexts, as do generalizations of finite groups called $π$ -finite spaces. In this talk we will translate representation theory to these homotopical contexts. Further, time permitting, we will see how some properties of representation theory change over these higher categorical fields and why one might care... and I may tell a joke or two. " Representation Theory of Finite Spaces
Fri, Sep. 25 1:25pm (Math 3…	A. I. Sachi Hashimoto X New technologies often change our tools before they change the way we work. Large language models can now produce convincing mathematical code very rapidly through mathematical "vibe coding" (prompting AI and then accepting code if it seems to work). But when this code is intended for use in a publication or inclusion in a database, the stakes are higher: errors in the code become errors in the mathematical literature. Is the alternative to "vibe coding" reading the code line-by-line? I will talk about how we can build software development workflows for the AI age. Starting from the basics of version control and Git, and assuming no prior experience, I will discuss iterative AI-assisted development, subagents, code review, testing, and continuous integration. The talk will be aimed at mathematicians interested in developing large-scale mathematical software and databases without giving up oversight and trust. AI-Assisted Coding for Mathematicians: How to Build Software Without Vibe Coding
Tue, Sep. 29 2:30pm (MATH 3…	Lie Theory Katrina Barron (Notre Dame University) X We recall some aspects of the representation theory for Lie algebras and how some number theory makes an appearance through characters for modules. We motivate why this occurs by putting this in the setting of vertex operator algebras which are the algebras that naturally arise from the geometry of Conformal Field Theory (or string theory). We discuss how analytic properties can be proved in certain settings by the construction of a modular linear differential equation, including recently in a new setting through joint work with Florencia Orosz Hunziker and Gaywalee Yamskulna. Why should we expect number theoretic properties to arise in Lie theory and group theory? Sponsored by the Meyer Fund

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