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Events for the weeks of October 11th – 23rd

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Tue, Oct. 13 11am (350)	Number Theory Faculty & Postdoc Number Theory Open House
Tue, Oct. 13 1:25pm (online)	Algebra/Logic Charlotte Aten View Online:https://cuboulder.zoom.us/j/96896272260 X In order to study a certain 3-dimensional analogue of the Mandelbrot set, I needed a notion of holomorphy for the corresponding 3-dimensional real algebra. While this motivation (to generalize the basic elements of complex dynamics) is analytic, the focus of this talk will be the connection between holomorphic functions over a real algebra and first-order properties of the algebra. It turns out that, under the assumption of a single (left or right) divisible element, the holomorphic functions over an arbitrary real algebra form a vector space. Properties such as the Product Rule and Chain Rule are equivalent to purely algebraic properties of the underlying real algebra. When our real algebra has a multiplicative identity element, the Product Rule is precisely equivalent to being both commutative and associative. I will present proofs of some of these statements, which are quite succinct once the proper groundwork has been done. Holomorphy over real algebras
Tue, Oct. 13 1:30pm (Math 3…	Homotopy Theo Johnson-Freyd (Perimeter Institute for Theoretical Physics and Dalhousie University) X Abstract: Kitaev has argued that the spaces of n-dimensional invertible phases of matter, as n varies, fit together into the structure of an Omega-spectrum. It follows that each n-dimensional phase determines a (possibly trivial) character on the nth stable homotopy group of spheres. I will present an enforced-gaplessness theorem (conditional on some reasonable semisimplicity assumptions about the category of topological phases of matter): if this character is nontrivial, and not one of the six nontrivial Kervaire invariants, then the invertible phase is a topological insulator: a material all of whose boundary conditions have gapless conducting modes. This is joint work in progress with David Reutter. With six exceptions, nontrivial characters of spheres do not admit gappable boundary conditions Sponsored by the Meyer Fund
Tue, Oct. 13 1:30pm (Math350)	Math Phys Theo Johnson-Freyd (Perimeter Institute for Theoretical Physics and Dalhousie University) X Abstract: Kitaev has argued that the spaces of n-dimensional invertible phases of matter, as n varies, fit together into the structure of an Omega-spectrum. It follows that each n-dimensional phase determines a (possibly trivial) character on the nth stable homotopy group of spheres. I will present an enforced-gaplessness theorem (conditional on some reasonable semisimplicity assumptions about the category of topological phases of matter): if this character is nontrivial, and not one of the six nontrivial Kervaire invariants, then the invertible phase is a topological insulator: a material all of whose boundary conditions have gapless conducting modes. This is joint work in progress with David Reutter. With six exceptions, nontrivial characters of spheres do not admit gappable boundary conditions
Tue, Oct. 13 2:30pm (MATH 3…	Lie Theory Kelsey Brown (Colorado State University) X In the 1990's, Proctor classified the d-complete posets, which correspond to λ-minuscule elements of Weyl groups of simply laced Kac-Moody algebras. These have the Jeu de Taquin property for the natural generalization of standard Young tableaux theory to posets. We define a notion of highest weight tableaux for several infinite families of d-complete posets. These constructions allow us to develop local algorithms for coswitching pairs of tableaux on these families, a coplactic version of tableau switching that, in the known setting of types A and B, connects naturally to Schubert calculus, crystals, and promotion and evacuation. Semistandard tableaux and (co)switching on d-complete posets
Wed, Oct. 14 3:30pm (Math350)	Math Phys Markus Pflaum (University of Colorado Boulder) State spaces in quantum mechanics Part 4: Uhlmann connection
Wed, Oct. 14 4:40pm (MATH 3…	Grad Student Seminar Alex Lajeunesse (CU Boulder) X The rational numbers admit an infinite family of distinct absolute value functions, apart from the normal one. Completing with respect to these absolute values leads to the fields of p-adic numbers for primes p. This talk will introduce these fields, focusing on their nature as bridges between characteristic 0 and p. We will discuss how Galois theory can be transported along this bridge, culminating in the definition of perfectoid fields and the tilting correspondence of Fontaine-Wintenberger and Scholze. This is intended to be a warm-up for the DeLong lectures being given by Prof. Bhargav Bhatt the following week. The tilting correspondence in mixed characteristic
Thu, Oct. 15 11:15am (MATH …	Algebraic Geometry Jacob Cleveland (Colorado State University) X We show that birational toric Landau–Ginzburg models which are related by a crepant variation of GIT have isomorphic state spaces under mild assumptions, generalizing the state space LG/CY correspondence of Chiodo–Ruan. We prove that this isomorphism is canonical, and is induced by a Fourier–Mukai transform. In the special case that a toric LG model is a so-called geometric phase, we identify the LG state space with the cohomology of the associated hypersurface. The State Space Correspondence for Toric Landau-Ginzburg Models
Fri, Oct. 16 10:10am (Math …	Math Edu Emily Montelius X Undergraduate research can take on many forms. In this talk I will share my strategies for finding appropriate topics and problems, how I scaffold introductory material, and methods to get students asking and solving interesting questions. I will discuss options for intended research outcomes. I will also discuss how to create smaller math projects with undergraduates in mind. Strategies for Guiding Undergraduate Research
Tue, Oct. 20 1:25pm (online)	Algebra/Logic Michael Kompatscher (Charles University Prague) View Online:https://cuboulder.zoom.us/j/96896272260 X An (abstract) minion M is a functor from the category of finite sets to the category of sets. So, for every finite set X, there is a set M(X) (the 'X-ary' part of M), and for every function f: X -> Y there is a function M(f) : M(X) -> M(Y) such that M preserves the identity and composition. Every variety V naturally gives rise to a minion F_V, where F_V(X) is the free algebra generated by X and F_V(f) : F_V(X) -> F_V(Y) is the unique homomorphism extending f: X -> Y. In this talk I would like to discuss a nice (multilinear) description of this minion for the variety of n-step nilpotent groups. This description is based on concepts from combinatorics on words (subword counting) and a result by Fox, Lyndon and Chen from 1958, and motivated by the study of uniform algorithms for constraint satisfaction problems (CSPs). The minion of free n-step nilpotent groups
Tue, Oct. 20 2:30pm (MATH 3…	Lie Theory Spencer Daugherty (CU Boulder) TBD
Tue, Oct. 20 3:30pm (MATH 3…	Topology Coby Glass X Topological Data Analysis (TDA) is very promising in the field of cybersecurity anomaly detection, but is traditionally too computationally bottlenecked for use in real-time environments. This talk explores the pipeline from industrial application to hardware-accelerated mathematics, beginning with the challenges of embedding non-geometric cybersecurity data into valid metric spaces for Vietoris-Rips filtrations. After demonstrating a functional TDA-to-machine-learning pipeline, we will briefly examine the algebraic optimizations that make this possible on modern computing architectures. Finally, the presentation will discuss the potential for hardware-accelerated TDA, specifically designing systolic arrays to compute matrix rank over GF(2) via chunked Gaussian elimination. Topological Anomaly Detection on Real-Time Data
Wed, Oct. 21 3pm (KCEN MPR …	DeLong Bhargav Bhatt (Institute for Advanced Study & Princeton University) View Online:https://cuboulder.zoom.us/j/97834671785 X What if two numbers were considered close not when their difference is small, but when it is divisible by a large power of a fixed prime $p$ ? Following this seemingly strange idea leads to the $p$ -adic numbers, a concept introduced in the late nineteenth century that has transformed number theory and found applications throughout mathematics. This lecture will explain where the $p$ -adic numbers came from, how they differ from the familiar real numbers, and why they are useful. The p-adic world Sponsored by the Meyer Fund
Thu, Oct. 22 11am (KCEN MPR…	DeLong Bhargav Bhatt (Institute for Advanced Study & Princeton University) View Online:https://cuboulder.zoom.us/j/97834671785 X Over the past fifteen years, the discovery of perfectoid geometry and related ideas has provided new lenses through which to study $p$ -adic spaces at an extraordinarily fine scale. The resulting methods have resolved longstanding questions across algebraic geometry, number theory, homotopy theory, and beyond. In this lecture, I will recount some of this story, emphasizing the applications that motivated the conceptual developments. Recent developments in p-adic geometry Sponsored by the Meyer Fund
Fri, Oct. 23 1:25pm (Zoom)	A. I. Max Weinreich (CUNY) The Crisis of AI-Generated Mathematics

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