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Events for June – October 2022

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Jun. 08, 2022 11am (Math 220)	Number Theory Uri Shapira (Technion) X * HYBRID: Speaker will be on zoom. Join by zoom or join the watch party in Math 220 * https://cuboulder.zoom.us/j/91928134679 The passcode is the j-invariant of the elliptic curve with CM by the Gaussians (or email Kate). Abstract: In the last three and a half decades, starting with the pioneering solution of the Oppenheim conjecture on the values of quadratic forms by Margulis, the interplay between the ergodic theory of homogeneous spaces and number theory solidified. Moreover, exciting new connections between the fields keep arising even today. The aim of my talk will be to demonstrate this with a circle of problems which might be best described as problems in the "geometry of integral vectors". To each integral vector v in Z^d one can associate several natural geometric and arithmetic objects. For example: (1) The "direction" of v, namely its radial projection to the unit sphere. (2) The residue class of v modulo a fixed integer k. (3) The "orthogonal lattice" to v, defined by intersecting Z^d with the hyperplane orthogonal to v. Each of these objects resides in a natural probability space which allows one to ask statistical questions as the integral vector varies. I will survey classical and recent results describing the limit statistics of these objects as the integral vector varies in certain sets. For example: (1) A ball of radius T as T goes to infinity (Schmidt). (2) A sphere of radius T as T goes to infinity (Linnik/Duke/Aka - Einsiedler - S) (3) The integral points on certain non-compact quadratic surfaces (S - Sargent). (4) The integral points approximating an irrational line in R^d (S - Weiss). Focussing on one of the above examples, I will try to explain how the corresponding problem can be casted as a question in homogeneous dynamics. Applying ergodic theory to number theory
Jul. 11, 2022 5pm (CASE E-340)	Topology Inna Zakharevich (Cornell University) X Like many mathematicians, I enjoy analyzing how my favorite things work (often to extreme degrees). In this talk we'll play around with some of the mathematics of origami. How can we fold paper? More fundamentally, what does it even mean to fold it? And once it's folded, what questions can we ask? In this talk we'll mess around with solving polynomials, the difference between local and global properties, and maybe even get to (flat-)fold some paper. The Mathematics of Origami Sponsored by the National Science Foundation
Jul. 12, 2022 5pm (CASE E-340)	Topology John Lind (California State University Chico) X I will present a new type of generative music based on the "abelian sandpile model" - a discretized model for the flow of heat built using rich combinatorial structures. The music is created by encoding the mathematics into an algorithm which then plays synthesizers according to the rules laid out by the dynamical system. I will explain how this works, and we will listen to examples. As we will see, the mathematics behind the music is regulated by a discrete form of the Laplacian operator and an analogy between graphs and Riemann surfaces. Abelian Sandpile Music Sponsored by the National Science Foundation
Jul. 14, 2022 5pm (CASE E-340)	Topology Jonathan Campbell (Center for Communications Research, La Jolla) X In any given dimension, there are a finite number of possible crystal structures that can exist. In this talk I'll show how group cohomology can be used to classify such crystal structures (and I'll define group cohomology!) We will work out some of the classification for 2-dimensional crystal structures, the so-called "wall paper" groups. Group Cohomology and Crystals Sponsored by the National Science Foundation
Aug. 30, 2022 3:30pm (MATH 3…	Topology Howy Jordan (CU Boulder) X The open sets of a topology space have a nice algebraic structure, and it turns out that much of classical topology can be formulated in terms of this structure without reference to any underlying set of points. Like many such algebraic lattice structures, we can regard it as a system of propositions in logic with meet (intersection) and join (union) in the lattice corresponding to logical "and" and to logical "or". From this perspective, a topological space is a propositional theory with models of the theory given by the points of the space. With suitable separation properties, the models are entirely determined by the open sets. In many familiar cases, we can give the lattice of opens first and then derive the points from this. In general however, such a lattice might have no points at all. In logical terms, some propositional theories can be consistent but still fail to have any set-based models. Such generalized lattices of opens are often called locales, and the corresponding theories are those of geometric propositional logic. In this talk, we'll consider the fundamental relationship between locales and traditional topology, discuss why one might be interested in point-free topology, and consider examples of consistent theories which fail to have models in the usual sense. Locales and Geometric Propositional Logic
Aug. 31, 2022 3:20pm (Math35…	Math Phys Daniel Spiegel (CU Boulder) X The theme of this semester’s mathematical physics seminar is quantum information theory. To get us started, I’ll introduce density operators, which play a fundamental role in the theory and serve as a tool for understanding subsystems of a composite quantum system. After defining what a density operator is, I’ll cover some of their basic operations and results, including the partial trace, Schmidt decomposition, and purification. Zoom access: https://cuboulder.zoom.us/j/8063594807 Introduction to Density Operators
Aug. 31, 2022 4:30pm (MATH 3…	Grad Student Seminar Chase Meadors (CU Boulder) X A "Galois Connection" is a formalism that abstracts the relationship between certain algebraic structures I won't mention and automorphism groups studied in classical Galois theory. Galois connections are ubiquitous in Logic, and most notably are used to define the relationship between the "Syntax" and "Semantics" of a system of logic. We will take a look at Galois connections and see several interesting examples of the kind just mentioned (with pictures!) Galois Theory (but I never mention fields)
Sep. 01, 2022 9am (Math 220)	Algebraic Geometry Isabel Vogt (recorded) Brill Noether theory over the Hurwitz space
Sep. 06, 2022 11am (Math 350)	Number Theory Eli Orvis and Kate Stange (University of Colorado Boulder) X As preparation for a few seminar slots spent learning about the newest bombshell attacks (https://www.quantamagazine.org/post-quantum-cryptography-scheme-is-cracked-on-a-laptop-20220824/) on the SIDH/SIKE protocol in supersingular elliptic curve isogeny-based cryptography, Eli will spend twenty minutes describing how the cryptographic system works, and then Kate will spend twenty minutes on past attacks on the protocol. Next week David will discuss some overview of genus 2 tools and in the third week we will discuss the new attacks. (Plan subject to change!) All are welcome, as we will start from the basics (assuming the audience knows some elliptic curves, but not the cryptography). Isogeny-based cryptography: SIDH and past attacks
Sep. 07, 2022 3:20pm (Math350)	Math Phys Marissa McMaster (CU Boulder) X As the study of quantum mechanics advanced, mathematicians noticed that experimental quantum mechanical results were not consistent with classical propositional logic. In 1936, Garrett Birkhoff and John von Neumann sought to reconcile these inconsistencies with their lattice formulation of a quantum logic. In this talk, we will explore why propositional logic fails when applied to quantum systems, detail the Birkhoff-von Neumann proposal for a quantum logic, and explore the utility and criticisms of the study of quantum logic. An introduction to the Birkhoff-von Neumann lattice and quantum logic
Sep. 08, 2022 2:30pm (MATH 3…	Functional Analysis Robin Deeley (CU Boulder) X Smale spaces form an important class of hyperbolic dynamical systems. Putnam and Ruelle have constructed C-algebras associated with a Smale space. I will discuss this construction and the structure of these algebras. The talk will be accessible to graduate students; in particular, no knowledge of Smale spaces is required. Smale space C-algebras Part I
Sep. 13, 2022 11am (Math 350)	Number Theory David Grant (University of Colorado Boulder) X This is to get us started on our mini study group on the role of genus 2 curves and abelian surfaces in breaking SIDH/SIKE, the isogeny-based cryptography scheme (that was spectacularly broken a month ago). David has kindly offered to give us an overview on genus 2. All are welcome. Genus 2 curves and isogenies of their Jacobians
Sep. 13, 2022 2:30pm (Math350)	Lie Theory Nat Thiem (CU) X This is the first talk in a two part series, where the ultimate goal is to discuss some positivity conjectures in the ring of symmetric functions. Symmetric functions and their variations are fundamental examples of combinatorial Hopf algebras. The first talk gives a short introduction to these algebras with the underlying motivation of preparing for combinatorial positivity questions. For important context, we will also discuss some related Hopf algebras: quasisymmetric functions and noncommutative symmetric functions. (+)symmetric functions as combinatorial Hopf algebras
Sep. 13, 2022 3:30pm (Zoom)	Topology Connor Meredith (CU Boulder) X A Stone space is a compact, totally disconnected, Hausdorff space. In model theory, Stone spaces naturally arise as spaces of complete types of a structure. Various model theoretic properties will be controlled by the topological properties of these spaces. In this talk, we will define model theoretic "types" and their associated Stone space structure. For the purpose of better understanding Stone spaces, we will also present Stone's Duality Theorem, as well as a more general duality theorem due to Hu. https://cuboulder.zoom.us/j/97617189995 Stone Spaces
Sep. 14, 2022 5pm (MATH 350)	Grad Student Seminar Erik Knutsen (CU Boulder) X What are spectra? Wouldn’t you like to know? You may have heard of them as somehow determining reduced cohomology theories. You may not have heard of them as a homotopy theorist’s “abelian groups.” In either case, be prepared to be amazed. In this talk, you will be given a definition of what they are. In this talk, you may learn another definition of what they are. In this talk, there is hope that you might be told why they matter. Exclusively in Math 350 this Wednesday at 5pm, come see the spectacle of… Spectra! A Friendly Introduction to Spectra
Sep. 15, 2022 9am (Math 220)	Algebraic Geometry Roberto Villaflor (recorded) Serre's GAGA correspondence
Sep. 15, 2022 2:30pm (MATH 3…	Functional Analysis Robin Deeley (CU Boulder) X Smale spaces form an important class of hyperbolic dynamical systems. Putnam and Ruelle have constructed C-algebras associated with a Smale space. I will discuss this construction and the structure of these algebras. Smale space C-algebras Part II
Sep. 20, 2022 11am (Math 350)	Number Theory David Grant X We will continue David's exposition on genus 2 curves and isogenies of their Jacobians. Genus 2 curves and isogenies of their Jacobians
Sep. 20, 2022 2:30pm (MATH 3…	Lie Theory Nat Thiem (CU) X Stanley associated a symmetric function to each finite graph in such a way that the function encodes proper vertex colorings on that graph. The s-positivity and e-positivity of these functions has been of great interest in recent years, especially since neither positivity is even true for all graphs. For unit interval graphs we know s-positivity and e-positivity is conjectured. This talk translates these conjectures into representation theoretic questions in the framework of combinatorial Hopf algebras. Stanley chromatic symmetric functions and positivity conjectures
Sep. 20, 2022 3:30pm (MATH 3…	Topology Howy Jordan (CU Boulder) X By interpreting theories in categories other than set, we recover completeness theorems from finitary logic in an infinitary setting. Topoi, categories of sheaves, have enough structure to interpret large classes of theories, and we obtain notions of local truth. A sheaf of rings on a sober space X gives for instance a ring object in Sh(X) and this ring is a local ring iff each germ is a local ring. In this talk, we set up the notion of interpreting a theory's axioms in a topos. We also consider consequences for the reals and some algebraic structures. https://cuboulder.zoom.us/j/97617189995 Internal Logic of Topoi
Sep. 21, 2022 3:30pm (Math350)	Math Phys Connor Mccranie (CU Boulder) X In this talk, I will introduce a model of quantum mechanics based on the theory of groupoids. I will show how one can construct a 2-groupid from Schwinger's algebra of so-called "selective measurements". We will look at some examples and see how one can recover the standard Hilbert space picture of quantum mechanics. Groupoids and Schwinger's picture of quantum mechanics
Sep. 21, 2022 4:40pm (MATH 3…	Grad Student Seminar Nick Jamesson (CU Boulder) X What does the formal study of logic look like? This is not an uncommon question for mathematicians who work outside of logic to have. We will shed some light on this question by looking at model theory, a branch of logic that deals with models, which can be thought of as "possible worlds where a set of sentences are true." For instance, the integers mod 2 are a model in which the sentence "1+1=0" is true. Since logic is known for theorems that sound cool (think Gödel's incompleteness theorems) we will prove a few basic statements that sound cool using a powerful tool, the compactness theorem for first order logic. We'll also apply the compactness theorem to regular old math to give a model theoretic proof that every field is contained in an algebraically closed field. Demystifying Logic: A Crash Course in Model Theory
Sep. 22, 2022 2:30pm (MATH 3…	Functional Analysis Laura Scull (Fort Lewis College) X We start by considering spaces with singularities: in these spaces, the local structure is not homogeneous, but varies throughout the space, with certain points associated with certain singularity types. Lie groupoids provide a way of encoding these spaces. However, this encoding is not unique: different Lie groupoids may represent the same underlying space. This leads to the idea of 'Morita equivalence' of Lie groupoids: two groupoids are Morita equivalent if they encode the same underlying geometric structure. To make this formal, we turn to the notation of 'localization', where we take a class of arrows and create a 'bicategory of fractions' in which these are formally inverted, becoming isomorphisms. This is the framework we use to build a category that represents our spaces: we will wind up with a bicategory of fractions of Lie groupoids in which all Morita equivalences become isomorphisms. The goal of this talk is to explain this categorical construction and convince you that it is geometrically motivated and meaningful. I will start with the geometric intuition of how Lie groupoids are used to encode singular spaces, and how Morita equivalence arises as a natural consequence, with examples and pictures. I will then explain the construction, starting from the familiar idea of localizing a prime ideal in a commutative ring and extending to more complicated contexts, culminating in the bicategory of fractions we are interested in. At the end I will mention applications and current work in this area. No background with Lie groupoids or bicategories is assumed and all scary commutative diagrams will be thoroughly explained and motivated. Morita Equivalence of Lie Groupoids Sponsored by the Meyer Fund
Sep. 27, 2022 11am (Math 350)	Number Theory Castryck/Decru (YouTube) X In this late-breaking use of an empty seminar slot :), we will watch some subset of Castryck & Decru's YouTube talks on the subject of their break of SIDH. We'll pause and discuss as needed to sort out the details together. These attacks use genus two isogenies to completely break the SIDH variant of supersingular isogeny based cryptography. Breaking SIDH
Sep. 27, 2022 1:25pm (Zoom)	Keith Kearnes (CU) X I will discuss some properties of varieties which are not Maltsev-definable relative to the class of all varieties, but which are Maltsev-definable relative to the class of varieties having a Taylor term. Relative Maltsev definability
Sep. 27, 2022 2:30pm (MATH 3…	Lie Theory Jay Taylor (University of Manchester) X Complex irreducible characters of finite groups have two main invariants that are used to measure their rationality: their character field, and their Schur indices. In this talk we’ll survey what is known about these invariants and the many questions that remain open. Rationality Questions for Finite Reductive Groups Sponsored by the Meyer Fund
Sep. 27, 2022 3:30pm (MATH 3…	Topology Howy Jordan (CU Boulder) X Fragments of logic can be internalized in appropriately structured categories, and topoi have enough structure to interpret full higher order theories. There are two caveats. First, the internal logic is constructive in general. Second, there are two natural notions of morphisms between topoi and the more common of these, geometric morphisms which act like continuous maps between spaces, only preserve a certain (quite large) fragment of logic. We'll consider in particular how this affects the real numbers. The classical constructions of the naturals, integers, and rationals go through as expected. The reals, however, split into multiple notions. Classically equivalent ways to construct the reals become distinct objects, and these correspond to interesting topologies on the real line. We consider the corresponding objects in a few categories and discuss the relevance on theories of normed algebras in topoi. https://cuboulder.zoom.us/j/97617189995 Internal Logic of Topoi, II
Sep. 28, 2022 4:40pm (MATH 3…	Grad Student Seminar Howy Jordan (CU Boulder) X Given a theory describing some class of mathematical objects (groups, rings, orders, ...) we can construct the sentences one might consider about these, model the symbols as being about an actual set with actual functions, relations, etc, and talk about how manipulating sentences to get proofs relates to truth in the models. Thanks to category theory, we can replace all of this with a lot of abstract nonsense! And who doesn't like abstract nonsense? In this talk we'll introduce some category theory and construct a category representing syntax and provability in a given theory. We'll discuss how this allows us to talk about models as a special case of structure preserving maps. As in the classical case, we get an adjunction (generalized Galois connection) between syntax and semantics. We'll also use this to engage in some shenanigans! Given a non-commutative algebraic structure, we'll pretend its commutative, even recovering classical results about commutative structures in the non-commutative case. We'll talk about how the real numbers become more subtle and interesting, with distinct characterizations giving different structures. We'll even model a theory that describes surjections of N onto R. (Don't worry, the surjections of N onto R aren't real and they can't hurt you.) Categorical Logical Shenanigans
Sep. 29, 2022 3:35pm (MATH 3…	Probability Marcus Michelen (University of Illinois Chicago) X Consider the random symmetric n x n matrix whose entries on and above the diagonal are independent and uniformly chosen from {-1,1}. How often is this matrix singular? A moment's thought reveals that if two rows or columns are equal then the matrix is singular, so the singularity probability is bounded below by 2^{-n(1 + o(1))}. Proving any sort of upper bound on the singularity probability turns out to be difficult, with results coming slowly over the past two decades. I'll discuss a recent work which shows the first exponential upper bound on this probability. Along the way, I'll discuss some tools---both old and new---which are powerful and (hopefully) interesting in their own right. This talk is based on joint work with Marcelo Campos, Matthew Jenssen, and Julian Sahasrabudhe. How often is a random symmetric matrix singular?
Oct. 04, 2022 11am (Math 350)	Number Theory YouTube (Castryck/Decru) X We will continue what we were up to last week, since we didn't even finish the first video. :) Attacks on Isogeny based cryptography continued
Oct. 04, 2022 2:30pm (MATH 3…	Lie Theory Justin Willson (CU) X Green and Xu defined the 2-root lattice of a simply laced Weyl group by looking at pairs of orthogonal roots in the root systems of simply laced Weyl groups with Y-shaped Dynkin diagrams. In this talk we will discuss the existence of elements of squared length 2 in the 2-root lattice in hyperbolic root systems as well as progress made on the conjecture that there are no such elements in the affine and finite cases. Lattice roots arising from 2-root lattices of type ADE
Oct. 05, 2022 4:40pm (MATH 3…	Grad Student Seminar Andrew Campbell (CU Boulder) X You may be surprised to learn that differentiation of polynomials is still an active and even growing area of research. We will look at some recent work of Steinerberger answering the title question for polynomials with real roots, and the surprising connection to eigenvalues of matrices. Time permitting we may discuss the challenges of the complex root case. While there are interesting connections to probability, combinatorics, and operator algebras we will try to keep the vast majority of the talk at the level of calculus, linear algebra, and undergraduate analysis. How do polynomial roots move under repeated differentiation?
Oct. 06, 2022 9am (Math 220)	Algebraic Geometry Bob Kuo (CU Boulder) Modular compactifications of the space of genus 1 curves (part 1)
Oct. 06, 2022 3:35pm (MATH 3…	Probability Leonid Petrov (University of Virginia) X Imagine two cars, slow (S) and fast (F), driving to the right on a discrete 1-dimensional lattice according to some random walk mechanism, and such that the cars cannot pass each other. We consider two systems, SF and FS, depending on which car is ahead. It is known for some time (through connections to symmetric functions and the RSK correspondence) that if at time 0 the cars are immediate neighbors, the trajectory of the car that is behind is the same (in distribution) in both systems. However, this fact fails when the initial locations of the cars are not immediate neighbors. I will explain how to recover the identity in distribution by suitably randomizing the initial condition in one of the systems. This result arises in our recent work on multiparameter stochastic systems (where the parameters are speeds attached to each car) in which the presence of parameters preserves the quantum integrability. This includes TASEP (totally asymmetric simple exclusion process), its deformations, and stochastic vertex models, which are all integrable through the Yang-Baxter equation (YBE). In the context of car dynamics, we interpret YBEs as Markov operators intertwining the transition semigroups of the dynamics of the processes differing by a parameter swap. We also construct Markov processes on trajectories which "rewrite the history" of the car dynamics, that is, produce an explicit monotone coupling between the trajectories of the systems differing by a parameter swap. Based on a joint work with Axel Saenz. Rewriting History in Integrable Stochastic Particle Systems
Oct. 08, 2022 9:30am (Math 3…	Number Theory Front Range Number Theory Day X We are excited to announce that we will be hosting Front Range Number Theory Day on October 8th, 2022, here at CU Boulder! The goal of the FRNTD is to provide a venue for faculty, graduate students, and undergraduates on the Front Range who are interested in number theory to meet, learn, and collaborate. For more information and to register, please visit our website: https://sites.google.com/colorado.edu/front-range-number-theory-day/home. If you would like to give a five-minute talk, please include your title on the registration form. We hope to see you there! Front Range Number Theory Day
Oct. 11, 2022 10am (MATH 350)	Number Theory Matthew de Courcy-Ireland (EPFL, Lausanne) X We discuss a family of cubic surfaces defined by x^2+y^2+z^2=xyz+k modulo prime numbers. The solutions form a graph, where each vertex (x,y,z) is joined to the other solution of the same quadratic in any of the three variables. These moves are related to a nonlinear action of the modular group PGL(2,Z) on the surface. We outline some ways these equations arise in geometry, and how we became interested in showing that the associated graphs cannot be embedded in the plane. We sketch a proof that 2, 3, and 7 are the only primes for which the graph is planar. For larger primes, we give an estimate of the Euler characteristic of a surface in which the graph can be embedded. Non-planarity of Markoff graphs mod p
Oct. 11, 2022 11am (MATH 350)	Number Theory Matthew Litman (University of California Davis) X Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p. In this talk we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in P^1 x P^1 x P^1 cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of B-G-S we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily W_k of such surfaces, we construct finite orbits in W_k(C) by studying small orbits that appear in W_k(F_p) for many values of p and k. This talk is based on joint work with E. Fuchs, J. Silverman, and A. Tran. Markoff-type K3 Surfaces: Local and Global Finite Orbits
Oct. 11, 2022 1:25pm (MATH 2…	Marcos Mazari-Armida (CU Boulder) X The classical model-theoretic notion of superstability was introduced by Shelah in the late sixties for first-order theories. In this talk we will show that superstability is a natural algebraic property by characterizing some classical classes of rings, such as noetherian rings and perfect rings, via superstability of certain classes of modules. Characterizing some classical rings via superstability
Oct. 11, 2022 2:30pm (MATH 3…	Lie Theory Richard Green (CU) X Kazhdan and Lusztig defined partitions of an arbitrary Coxeter group W into left, right, and two-sided cells. Lusztig defined an integer a(w) for each element w of W, and proved that the a-function takes constant values on cells. In general, the problem of determining these cells, and the function a, is difficult. Elements of a-value 0 or 1 are well understood, and this talk will describe the structure of cells of elements of a-value 2 in the cases where the number of such elements is finite. This is joint work with Tianyuan Xu. The structure of Kazhdan—Lusztig cells of a-value 2
Oct. 12, 2022 3:30pm (Math350)	Math Phys Markus J. Pflaum (CU Boulder) Whightman functions, the Borchers–Uhlmann algebra and GNS type reconstruction of quantum fields
Oct. 12, 2022 4:40pm (MATH 3…	Grad Student Seminar Juan Moreno (CU Boulder) X Homotopy groups are important invariants of both spaces and spectra, but they're not everything. In this talk, I'll introduce Postnikov invariants which contain more information about the space or spectrum. Then we'll determine some Postnikov invariants arising from the physics of topological phases. Postnikov Invariants & Topological Phases
Oct. 13, 2022 9am (Math 220)	Algebraic Geometry Adrian Neff (CU Boulder) Modular compactifications of the space of genus 1 curves (part 2)
Oct. 13, 2022 12:20pm (Math …	Diversity Office of Institutional Equity and Compliance X Attitudes and stereotypes are developed about people and groups of people that affect our understanding and actions without our awareness or intentional control. This workshop examines unconscious bias and identifies key areas where it affects our work, interactions, and decision-making processes. Participants explore personal and group practices that can help reduce bias and increase inclusion, engagement, and productivity. Recognizing and Mitigating Unconscious Bias
Oct. 13, 2022 3:35pm (MATH 3…	Probability Dan Stroock (MIT) Applications of the Gaussian Isoperimetric Inequality
Oct. 18, 2022 1:25pm (Zoom)	Samson Leung (Carnegie Mellon University) X Let K be an abstract elementary class and \lambda=LS(K). K can be axiomatized by a sentence in L_{(2^\lambda)^+,\lambda^+}, allowing game quantification. This is parallel to Shelah-Villaveces result which demands a much higher complexity of junctions but without game quantification. Shelah’s presentation theorem gives K = PC(T, \Gamma, L(K)) where T is a first-order theory in an expansion of L(K) and \Gamma is a set of T-types. We provide a better bound of \|\Gamma\| in terms of I_2(\lambda,K). We also give conditions under which the categoricity in two successive cardinals implies the existence of models in the next cardinal. This improves the result of Shelah. Axiomatizing AECs and applications
Oct. 19, 2022 3:30pm (Math350)	Math Phys Sheagan John (CU Boulder) X A pair of orthonormal bases of the d-dimensional complex space are mutually unbiased if the square of their standard hermitian inner product is equal to 1/d. We briefly motivate why this notion of mutually unbiased bases (MUBs) is of great importance in quantum information theory, then discuss the existing mathematical tools and difficulties in constructing MUBs, including the relationship to orthogonal pairs of special linear Lie algebras. Mutually Unbiased Bases and Lie Algebras
Oct. 19, 2022 4:40pm (MATH 3…	Grad Student Seminar Adrian Neff (CU Boulder) X Michael Artin supposedly said that you can understand most concepts in moduli through the moduli space of triangles (source: someone told me this once). With this advice in mind, this talk will do just that: we will go through multiple variations on the moduli space of triangles in order to see various behaviors of moduli problems, and hopefully by the end (unless Artin was a liar), we will all understand moduli spaces. An Introduction to Moduli Spaces
Oct. 20, 2022 11:15am (Math …	Ulam Boris Tsygan (Northwestern University) X We review the topic of noncommutative differential forms, following the works of Karoubi, Cuntz-Quillen, Cortinas, Ginzburg-Schedler, and Waikit Yeung. In particular we give a new proof of the theorem of Ginzburg and Schedler that compares extended noncommutative De Rham cohomology to cyclic homology. This theorem is a stronger version of a theorem of Karoubi. We also review the relation of noncommutative forms to crystalline cohomology (the classical one as well as the noncommutative version due to Cortinas), to operations on Hochschild and cyclic complexes, and with De Rham theory of representation schemes (after Ginsburg-Schedler and Waikato Yeung). A survey of noncommutative differential forms.
Oct. 20, 2022 11:15am (Math …	Noncomm Geometry Boris Tsygan (Northwestern University) X We review the topic of noncommutative differential forms, following the works of Karoubi, Cuntz-Quillen, Cortinas, Ginzburg-Schedler, and Waikit Yeung. In particular we give a new proof of the theorem of Ginzburg and Schedler that compares extended noncommutative De Rham cohomology to cyclic homology. This theorem is a stronger version of a theorem of Karoubi. We also review the relation of noncommutative forms to crystalline cohomology (the classical one as well as the noncommutative version due to Cortinas), to operations on Hochschild and cyclic complexes, and with De Rham theory of representation schemes (after Ginsburg-Schedler and Waikato Yeung). A survey of noncommutative differential forms.
Oct. 20, 2022 2:30pm (MATH 3…	Functional Analysis Frederic Latremoliere (University of Denver) X We introduced a distance between spectral triples, inspired from our earlier work on the quantum Gromov-Hausdorff distance, and we have established a few first examples of convergence for this distance. In this talk, after reviewing our distance, we will discuss properties which are continuous, in an appropriate sense, for our distance, which includes the spectra of the Dirac operators in spectral triples. Spectral properties of the convergence of spectral triples Sponsored by the Simons Foundation
Oct. 25, 2022 1:25pm (MATH 2…	Michael Wheeler (CU Boulder) X A non-standard framework over a base set A is an extension of a fragment of set theory containing A that realizes many types with parameters from A and preserves bounded quantifier formulas. In particular, if A contains an indexing set I and U is an ultrafilter over I, then a saturated enough non-standard framework over A will contain elements that appear to be principal generators for U. The properties of the ultrafilter are reflected in properties of their hyperprincipal generators in a non-standard framework. In this talk we will go over some background information about non-standard frameworks and then discuss how the properties of being Regular and Good (key properties related to Keisler's Order for countable first-order theories) are reflected in their hyperprincipal generators. Hyperprincipal generators for regular and good ultrafilters
Oct. 25, 2022 2:30pm (MATH 3…	Lie Theory Richard Green (CU) X After reviewing the character theory of Coxeter groups of types A, B, and D, we will study a particular family of representations of Coxeter groups of type D. Each representation comes equipped with two very different bases, one of which can be described in terms of orthogonal roots. Orthogonal roots and representations of Coxeter groups
Oct. 25, 2022 3:30pm (MATH 3…	Topology Vasileios Maroulas (University of Tennessee) X Topological data analysis (TDA) studies the shape patterns of data. Persistent homology (PH) is a widely used method in TDA that summarizes homological features of data at multiple scales and stores them in persistence diagrams (PDs). In this talk we will discuss a random persistence diagram generation (RPDG) method that generates a sequence of random PDs from the ones produced by the data. RPDG is underpinned by (i) a model based on pairwise interacting point processes for inference of persistence diagrams, and (ii) by a reversible jump Markov chain Monte Carlo (RJ-MCMC) algorithm for generating samples of PDs. An example on a materials science problem will demonstrate the applicability of the RPDG method. Random persistence diagram generator and an application to materials Sponsored by the Meyer Fund
Oct. 26, 2022 4:40pm (MATH 3…	Grad Student Seminar Levi Lorenzo (CU Boulder) X To the outsider, the idea of “noncommutative geometry” is admittedly confusing. Commutativity is an algebraic property, so what about geometry makes it commutative (or not)? No, it doesn’t concern whether Sphere x Torus = Torus x Sphere. What it does concern is the algebras of (continuous, smooth, L^\infty, etc.) functions, which commute under pointwise multiplication. These algebras (along with certain operators) determine the (topological, smooth, Riemannian, spin, measure theoretic, etc.) structure on the manifold. This allows us to recover the geometric structure in algebraic language. Remove the commutativity requirement from those algebraic objects and you arrive at noncommutative geometry. These tools shed light on and can prove results in the classic arena, but they also give a way to understand the structures of many noncommutative spaces that arise, for example, as quotients of ordinary spaces, leaf spaces of foliations, dynamical systems, quantum groups, groupoids, and many examples from physics. In this talk, we introduce the primary objects at play and tools at hand in Connes’ program for noncommutative geometry and equip ourselves to journey away from the familiar and into the land of that which does not commute. Step into a (Noncommutative) World
Oct. 27, 2022 9am (MATH 350)	Algebraic Geometry Jonathan Wise TBA
Oct. 27, 2022 2:30pm (MATH 3…	Functional Analysis Christopher Schafhauser (University of Nebraska) X In the early '90s, Elliott conjectured that separable simple nuclear C-algebras are classified up to isomorphism by their K-theory groups and traces, in analogy with the Connes--Haagerup classification of separably acting injective factors by their type and flow of weights. In the last few years, Elliott's classification program has been completed under the two additional hypotheses: Z-stability and the UCT. In the von Neumann algebraic setting, the classification of injective factors was followed by a delicate analysis of the symmetries of such factors. For example, Oceanu proved every amenable group admits a outer action on the hyperfinite II_1 factor R, which is unique up to cocycle conjugacy, and Popa and Takesaki proved the automorphism group of R is contractable. In the spirit of the latter result, I will discuss recent joint work with Jamie Gabe on the homotopy type of automorphism groups of ``classifiable'' C-algebras. Homotopy groups of automorphism groups of classifiable C*-algebras. Sponsored by the Meyer Fund
Oct. 27, 2022 3:35pm (MATH 3…	Probability Benedek Valko (University of Wisconsin Madison) X If one removes the first row and column of an n by n random Haar unitary matrix then the eigenvalues of the resulting matrix are inside the unit disk. Zyczkowski and Sommers (2000) derived the joint pdf of these eigenvalues, and showed that they form a determinantal process. Killip and Kozhan (2017) extended this result to circular beta ensembles (with beta=2 corresponding to the unitary case), describing the eigenvalues of the truncated model via a random recursion. (For general beta the resulting ensemble is not determinantal.) We derive the point process limit of the truncated ensemble together with the scaling limit of the normalized characteristic polynomials. The limiting objects are closely connected to the random analytic function appearing as the limit of the normalized characteristic polynomials of the (full) circular beta ensemble. Joint with Yun Li (Tsinghua University, Beijing) Scaling limits of the truncated circular beta ensemble

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