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Events for July – November 2022

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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
Nov. 01, 2022 11am (Math 350)	Number Theory James Rickards (CU Boulder) X An Apollonian circle packing is an arrangement of tangent circles in the plane. Packings where all curvatures are integral has been a rich topic of study in the last 20 years. Instead of fixing a packing and considering the possible curvatures in the packing, we will invert the problem: fix a curvature, and consider all packings that have a circle of this curvature. One natural way to study this set is to ask: "What is the distribution of the minimal curvatures of these packings?" This problem leads to a staircase-like distribution, which we name the "Apollonian Staircase." By exploring a connection to binary quadratic forms and hyperbolic geometry, we will describe and prove some properties of the staircase. The Apollonian Staircase
Nov. 01, 2022 1:25pm (Zoom)	Bernardo Rossi (University Linz, Austria) X Universal algebraic geometry, first introduced by B. Plotkin, extends some basic notions of classical algebraic geometry to universal algebra. For a clone $C$ on a set $A$ , a subset of $A^{n}$ is called algebraic if it is the solution set of a system of equations from the $n$ -ary part of $C$ . A basic fact in classical algebraic geometry is that the union of two algebraic sets is algebraic. This is no longer true in the present setting, and clones with this special property are called equationally additive. We prove that on a finite set with at least three elements there is a continuum number of equationally additive constantive clones. We characterize the Mal'cev algebras whose clone of polynomial functions is equationally additive in terms of properties of the binary term condition commutator, and equationally additive E-minimal algebras in terms of their TCT-type. Joint work with E. Aichinger and M. Behrisch. On when the union of two algebraic sets is algebraic
Nov. 01, 2022 2:30pm (MATH 3…	Lie Theory Manabu Hagiwara (Chiba University (Japan)) X This talk introduces the topics of coding theory and its nontrivial relationship with root systems and perfect codes, with examples. The talk will focus on introductory aspects. Mainly two topics will be discussed: one is the connection between the combinatorial structures in representation theory and deletion perfect codes. The other is the connection between the sphere-packing problem and bit-flipping perfect codes. Root Systems and Perfect Codes
Nov. 02, 2022 4:40pm (MATH 3…	Grad Student Seminar Matt Watson (CU Boulder) X In classic algebraic geometry, we want to study the zero sets of polynomials. In modern algebraic geometry, smart people develop algebraic tools and methods for studying the geometry of these zero sets. This talk is not about modern algebraic geometry. Instead, we will explore the world of complex donuts (compact Riemann surfaces) and their holomorphic and meromorphic functions. By the end of this talk, we will know a few basic results in this theory and see them in action on the Riemann sphere. Hopefully this helps to give some geometric intuition for anyone taking algebraic geometry next semester. If time permits, we may also state the Riemann-Roch Theorem in the complex curve setting and give a friendly introduction to sheaves. ???? The Geometric Intuition Behind Algebraic Geometry
Nov. 03, 2022 11:15am (Math …	Noncomm Geometry Emil Prodan (Yeshiva University) X I will show that each metamaterial has a groupoid behind it and that all physically sound Hamiltonians can be generated from the left-regular representation of the groupoid’s C-algebra. The K-theories of this groupoid algebra classify the various dynamical features that can be observed in the metamaterial. Furthermore, in the presence of defects, the space of units of the groupoid displays closed invariant subspaces leading to a decomposition of this unit space that prompts and exact sequence of C-algebras. The so called bulk-defect correspondence principle can be formulated in terms of this exact sequence and a specific Kasparov product. Furthermore, since exact sequences of C*-algebras are classified by certain Kasparov groups, we can use Kasparov’s theory to investigate and classify geometric defects of metamaterials. Examples will be provided. Metamaterials by groupoid methods and Kasparov’s K-theory
Nov. 03, 2022 3:35pm (Virtual)	Probability Jin Feng (University of Kansas) X The deterministic Carleman equation can be considered as an one dimensional two speed fictitious gas model. Its associated (2 scale) hydrodynamic limit gives a nonlinear heat equation. The first rigorous derivation of such limit was given by Kurtz in 1973. In this talk, starting from a more refined stochastic model giving the Carleman equation as mean field, we derive a macroscopic fluctuation structure associated with the hydrodynamic limit. The purpose of such derivation is to illustrate a new, Hamiltonian-based, functional analytic and variational approach for the understanding of hydrodynamics limit program in general. A large deviation result is established through an abstract Hamilton-Jacobi method applied to this specific setting. The principle idea is to identify a two scale averaging structure in the context of Hamiltonian convergence in the space of probability measures. This is achieved through 1). A change of coordinate to the density-flux description of the problem; 2). Extending the weak KAM theory (in Hamiltonian dynamical system) to infinite particle context for explicitly identifying effective Hamiltonian; 3). Developing results for a new class of PDEs: Hamilton-Jacobi equation in the space of measures. This is a joint work with Toshio Mikami and Johannes Zimmer. Hydrodynamic limit fluctuations from nonlinear heat equation given by stochastic Carleman particles
CANCELED Nov. 08, 2022	Number Theory Ian Whitehead (Swarthmore) TBA
Nov. 08, 2022 1:25pm (Zoom)	Joshua Wiscons (California State University, Sacramento) X We discuss a geometry of involutions arising in "quasi-Frobenius" groups of finite Morley rank: groups possessing a quasi-self-normalizing subgroup H whose conjugates intersect trivially. The resulting geometry is that of a generically defined projective 3-space, and our focus is on whether it is genuine or not: a distinction that separates SO3(R) and PGL2(C). We conjecture natural conditions on quasi-Frobenius groups of finite Morley rank to identify PGL2(K), discuss progress on the conjecture (by us and, more recently, by Zamour), and highlight applications. This is joint work with Adrien Deloro. Geometry of involutions in ranked groups
Nov. 09, 2022 3:30pm (MATH350)	Math Phys Howy Jordan (CU Boulder) X Classical information theory answers fundamental questions about the storage and transmission of data. Suppose the goal is to transmit events that occur according to some specified probability distribution. How much information is inherent in the result of each event? How can the events be encoded (into bits, say) for efficient transmission of this information? Intuitively, a more likely outcome is less surprising and less informative, and so requires less in the way of storage. In this talk, we'll introduce the fundamental quantity of entropy (and its variations) which provide a foundation for precise answers to the above and related questions. We illustrate its use in the analysis of data encoding and transmission. We also set the stage for a comparison of classical information sources with quantum sources. The seminar will be held in hybrid mode. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356 An Introduction to Classical Information Theory
Nov. 09, 2022 4:40pm (MATH 3…	Grad Student Seminar Henry Maher (CU Boulder) X Category theory studies things and relationships between those things. However, in many settings, there are also relationships between those relationships; for example, there are homotopies between continuous maps between spaces, or natural transformations between functors between categories. This data is naturally described through the language of higher categories. In this talk, I aim to give a motivation of higher category theory by explaining its relationship to homotopy theory, as well as provide some examples of results from higher category theory. An introduction to higher category theory
Nov. 10, 2022 9am (MATH 350)	Algebraic Geometry Hannah Larson (Harvard / Clay) X There are certain natural cohomology classes on the moduli space M_{g,n}-bar of n-pointed, genus g stable curves known as "tautological classes." I will define these classes and explain the significance of the special situation when all classes are tautological. In joint work with Samir Canning, we find many new small values (g, n) for which all classes on M_{g,n}-bar are tautological. Via an inductive argument, these results lead to applications for arbitrary (g, n). For example, in joint work with Samir Canning and Samuel Payne, we show that the point counts of M_{g,n}-bar over finite fields are surprisingly well approximated by polynomials. Moduli spaces of stable curves
Nov. 10, 2022 2:30pm (MATH 3…	Functional Analysis Alex Kumjian (University of Nevada) X When is the canonical map from a k-graph into its fundamental groupoid an injection? We also discuss a class of examples, associated to actions of triangle groups on buildings, that led Robertson and Steger to introduce higher rank Cuntz-Pimsner algebras and thereby inspired the notion of higher rank graph (or k-graph) C*-algebras. The fundamental groupoid of a k-graph was introduced by Pask, Raeburn and Quigg in '04 and an example was given that shows that a k-graph need not embed. This is an interim report on continuing work with my colleagues Nathan Brownlowe, David Pask and Aidan Sims. Embeddability of k-graphs and a class of examples associated to triangle group actions on buildings. Sponsored by the Meyer Fund
Nov. 10, 2022 3:35pm (MATH 3…	Probability Ping Zhong (University of Wyoming) X Random variables in Voiculescu’s free probability theory can model the limits of suitable random matrix models. The Brown measure of a free random variable is a replacement for the eigenvalue counting measure of square matrices. The R-diagonal operators are a large family of free random variables that are natural limit operators of various well-known non-normal random matrix models. I will report some recent progresses on the Brown measure of the sum of two free random variables, one of which is R-diagonal. We show that subordination functions in free additive convolution can detect information about the Brown measure. These Brown measures are related to the limit eigenvalue distributions of deformed i.i.d. and deformed single ring random matrix models. Part of this talk is based on joint work with Hari Bercovici. Brown measure and eigenvalue distributions of deformed random matrices
Nov. 11, 2022 10am (MATH 350)	Functional Analysis Christopher Seaton (Rhodes College) X In symplectic geometry, the quotient of a symplectic manifold $M$ by a Lie group $G$ acting as symplectomorphisms does not usually inherit a symplectic structure. Rather, one forms the \emph{symplectic quotient}, the quotient by $G$ of a subset of $M$ defined as a fiber of the \emph{moment map} $μ : M \to 𝔤^{}$ where $𝔤$ is the Lie algebra of $G$ and $𝔤^{}$ its dual; this subset of $M$ is called the \emph{shell}. At a regular value of the moment map, the shell is a submanifold of $M$ and its quotient is a symplectic manifold. Otherwise, the shell and symplectic quotient are singular varieties. To consider the kinds of singularities that arise, one can restrict to the case when $M = V⊕ V$ for a finite-dimensional linear $G$ -representation $V$ . In this case, there is a natural homogeneous quadratic moment map $μ V⊕ Vto g$ such that $0$ is a singular value, hence the shell $N = μ-1(0)$ and symplectic quotient $N/!/ G$ are singular. For many cases of $G$ and $V$ , it has been demonstrated by Bellamy--Schedler, Becker, Terpereau, and others that $N/!/ G$ has \emph{symplectic singularities}; equivalently, $N/!/ G$ is Gorenstein with rational singularities and its smooth locus admits a holomorphic symplectic form. In some of these cases, the variety $N$ already has rational singularities. We will present recent results demonstrating that for a given $G$ , the shell $N$ has rational singularities and the symplectic quotient $N/!/G$ has symplectic singularities in the case of ``most" $G$ -modules $V$ . This includes the important case where $G$ is semisimple and $V = pg$ with $p ≥ 2$ , optimizing results of Aizenbud--Avni and generalizing a theorem of Budur. We will also discuss applications indicating that the representation varieties of surface groups have rational singularities and the corresponding character varieties have symplectic singularities. Joint work with Hans-Christian Herbig and Gerald Schwarz Rational singularities of the zero fiber of the moment map Sponsored by the Meyer Fund
Nov. 12, 2022 9am (Math350)	Math Phys Groupoidfest 2022 speakers X https://math.colorado.edu/groupoidfest2022/ Groupoidfest Sponsored by the Meyer Fund
Nov. 13, 2022 9am (Math350)	Math Phys Groupoidfest 2022 speakers X https://math.colorado.edu/groupoidfest2022/ Groupoidfest 2022 Sponsored by the Meyer Fund
Nov. 15, 2022 1:25pm (Zoom)	Jakub Oprsal (ISTA Klosterneuburg, Austria) X We will revisit Taylor's 1977 seminal paper `Varieties obeying homotopy laws' through the lens of recent developments in universal algebra and constraint satisfaction problems. The paper studies Maltsev conditions that influence homotopical properties of topological algebras in a variety satisfying such a condition. We recall that a topological algebra is a topological space together with operations that are continuous. Among other insights, we will provide a characterisation of idempotent varieties whose congruence lattices satisfy the meet-semidistributive law in terms of properties of topological algebras in the variety. Topological characterisation of varieties with meet-semidistributive congruences
Nov. 16, 2022 3:30pm (Math350)	Math Phys Daniel Spiegel (CU Boulder) X Following up on Howy Jordan’s talk, I’ll define the quantum mechanical von Neumann entropy and cover some of its basic properties. This talk will touch on Sections 11.3 and 11.4 of Nielsen and Chuang’s quantum information theory book. The seminar will be held in hybrid mode. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356 Introduction to von Neumann entropy
Nov. 16, 2022 4:40pm (MATH 3…	Grad Student Seminar Jon Kim (CU Boulder) X This GSS talk will focus on introducing and motivating the basic tools of algebraic geometry in affine space (and maybe projective space) using commutative rings and fields. We will introduce rational normal curves in affine space and show how we can use these tools to study them. This talk is aimed at people who are curious about what algebraic geometry is about and only knowledge of algebra will be assumed. A Friendly Introduction to Algebraic Geometry with Rational Normal Curves
Nov. 17, 2022 9am (MATH 350)	Algebraic Geometry Jonathan Quartin (Boulder) X In recent years, there have been attempts to use Algebraic Geometry to model routing problems in space communications. In this talk, I’ll introduce sheaves on graphs, and demonstrate their ability to capture the local nature of routing problems. Time permitting, I’ll discuss some work from my NASA summer internship, and NASA’s plans for future research into the applicability of Algebraic Geometry to routing. Sheaf-Theoretic Models for Communications
Nov. 29, 2022 11am (Math 350)	Number Theory Collective X We'll use the regular Number Theory seminar this week to plan a graduate-student reading group in number theory for next semester. The goal is to organize a low-commitment, accessible reading group for graduate students at all career stages. Everyone is welcome to join, and we are in the market for suggested topics and sources! Reading group organizational meeting
Nov. 29, 2022 1:25pm (Zoom)	Reinhard Pöschel (TU Dresden, Germany) X It is well-known that a function $f$ (of arity $n≥ 2$ ) preserves an equivalence or a quasiorder relation $ρ$ if (and only if) each unary polynomial function preserves $ρ$ which can be obtained from $f$ by substituting constants at all but one arguments. Thus the polymorphism clone Pol $ρ$ is completely determined by the endomorphism monoid End $ρ$ . We ask which other relations and which monoids of unary mappings may have the same property. We introduce so-called generalized quasiorders which play a central role for answering these questions (moreover, they lead to new algebraic notions generalizing, e.g., affine completeness). Join work with Danica Jakubikova-Studenovska (Kosice, Slovakia) and Sandor Radeleczki (Miskolc, Hungary) Generalized quasiorders
Nov. 30, 2022 3:30pm (Math350)	Math Phys Rebecah Storms (CU Boulder) Einstein Field Equations
Nov. 30, 2022 4:40pm (MATH 3…	Grad Student Seminar Ezzeddine El Sai (CU Boulder) X In this talk, we explore how similar ideas appear in different areas of mathematics. One of those recurring themes is the notions of convolution and the Fourier transform which originated in the context of analysis. We will see how these ideas appear in computing, representation theory, algebraic geometry, and category theory. Since these ideas appear at different levels of abstraction, we will be able to examine the power and limitations of abstraction in mathematics. Fourier analysis without the analysis

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