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Events for the weeks of September 20th – October 2nd

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Tue, Sep. 22 1:25pm (MATH 2…	Algebra/Logic Keith Kearnes (CU) View Online:https://cuboulder.zoom.us/j/96896272260 X I will discuss the classification of those finite algebras $A$ with the following property: there is no identity $s≈ t$ that is satisfied by $A$ with probability $p$ for any number $p$ satisfying 1/\|A\| < p < 1. Finite algebras whose equational probability spectrum contains a big gap
Wed, Sep. 23 3:30pm (Math350)	Math Phys Howy Jordan (CU Boulder) X This is a joint talk with the topology seminar. The space of mixed states of a quantum system can be decomposed by ranks. For purposes of differential topology and geometry, one wants subspaces in a decomposition to fit together in a natural way, in which case it is called a stratification and the pieces called strata. Moreover, in order to have a useful notion of stratified tangent bundle, certain regularity conditions such as Whitney (a)- and (b)-regularity are considered. In this talk, we discuss the necessary background in stratification theory and discuss the regularity conditions satisfied by the decomposition of mixed states by rank. State spaces in quantum mechanics Part 2: Stratifications
Wed, Sep. 23 3:30pm (MATH 3…	Topology Howy Jordan (CU Boulder) X This is a joint talk with the mathematical physics seminar. The space of mixed states of a quantum system can be decomposed by ranks. For purposes of differential topology and geometry, one wants subspaces in a decomposition to fit together in a natural way, in which case it is called a stratification and the pieces called strata. Moreover, in order to have a useful notion of stratified tangent bundle, certain regularity conditions such as Whitney (a)- and (b)-regularity are considered. In this talk, we discuss the necessary background in stratification theory and discuss the regularity conditions satisfied by the decomposition of mixed states by rank. State spaces of quantum systems Part 2: Stratifications
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 10:10am (Math …	Math Edu Ally Graf X A flipped classroom is a method of instruction where the main ``lecture" occurs outside the classroom without the instructor, and time in class focuses on higher-order cognitive activities such as collaborative learning. In this presentation, I will use personal experience both as a student and facilitator in flipped classrooms to facilitate a discussion on current research and what should be considered when implementing this teaching method. A Reflection on Flipped Classrooms
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
Tue, Sep. 29 3:30pm (MATH 3…	Topology Emily Montelius (CU Boulder) Introduction to Topological Data Analysis and seminar organization
Wed, Sep. 30 2:30pm (Math350)	Math Phys Markus J. Pflaum (CU Boulder) State spaces in quantum mechanics Part 3: Invariant Theory
Wed, Sep. 30 4:40pm (MATH 3…	Grad Student Seminar Colin Jackson (CU Boulder) X Some numbers are rational, a few more are algebraic, but most are transcendental. We will briefly cover some definitions, investigate some special cases of numbers, and finally look at some positive results in transcendental number theory. Rational, Algebraic, or Transcendental?
Thu, Oct. 1 11:15pm (MATH …	Algebraic Geometry Sachi Hashimoto X Let X be a curve of genus at least 2. By Faltings’ theorem, we know that X has finitely many rational points, but how can we actually determine this set in practice? Around 40 years before Faltings’ proof, Chabauty proved this finiteness theorem for certain curves by studying the embedding of X inside an abelian variety, the Jacobian variety of X. In the first part of my talk, I will explain Chabauty’s original method and the modern day extension “Geometric linear Chabauty” that makes this effective. In the second part of my talk, I will describe recent progress to extend Chabauty’s method to a broader class of curves by studying line bundles over the Jacobian (arising from pullbacks of the Poincare bundle). The math in these talks arises out of work of Edixhoven and Lido, and also reflects joint work with Spelier, Lido, Lombardo, Mascot, and Parent. Geometric methods for determining rational points on curves I and II
Fri, Oct. 2 1:25pm (Math 3…	A. I. TBA TBA

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