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Events for May – September 2022

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May. 16, 2022 9am (Zoom)	Probability Vlad-Dumitru Margarint (CU Boulder) X The Schramm-Loewner Evolution (SLE) is a family of random planar fractal curves, introduced by Schramm in 2000. These random fractal curves are proved to describe the scaling limits of a number of discrete models that are of great interest in planar Statistical Physics. In this mini-course, I will give an overview of some introductory aspects of this theory, focusing the exposure on the intuition behind the results rather than their proofs. I will also give a short sketch of the proof of Rohde-Schramm Theorem on the existence of the SLE trace, an overview of a technique used to approximate uniformly the SLE traces, and a short description of a simultaneously growing Multiple SLE model, for N=2 curves and beyond. A Mini-course on Schramm-Loewner Evolution
May. 18, 2022 9am (Zoom)	Probability Vlad-Dumitru Margarint (CU Boulder) X The Schramm-Loewner Evolution (SLE) is a family of random planar fractal curves, introduced by Schramm in 2000. These random fractal curves are proved to describe the scaling limits of a number of discrete models that are of great interest in planar Statistical Physics. In this mini-course, I will give an overview of some introductory aspects of this theory, focusing the exposure on the intuition behind the results rather than their proofs. I will also give a short sketch of the proof of Rohde-Schramm Theorem on the existence of the SLE trace, an overview of a technique used to approximate uniformly the SLE traces, and a short description of a simultaneously growing Multiple SLE model, for N=2 curves and beyond. A Mini-course on Schramm-Loewner Evolution
May. 20, 2022 9am (Zoom)	Probability Vlad-Dumitru Margarint (CU Boulder) X The Schramm-Loewner Evolution (SLE) is a family of random planar fractal curves, introduced by Schramm in 2000. These random fractal curves are proved to describe the scaling limits of a number of discrete models that are of great interest in planar Statistical Physics. In this mini-course, I will give an overview of some introductory aspects of this theory, focusing the exposure on the intuition behind the results rather than their proofs. I will also give a short sketch of the proof of Rohde-Schramm Theorem on the existence of the SLE trace, an overview of a technique used to approximate uniformly the SLE traces, and a short description of a simultaneously growing Multiple SLE model, for N=2 curves and beyond. A Mini-course on Schramm-Loewner Evolution
May. 25, 2022 1:30pm (Math 3…	Diversity Howy Jordan (UC Boulder) X You're invited to a show n tell n monch time with Howy! What we now call the autism spectrum is a cluster of social, cultural, and developmental disabilities popularized by Hans Asperger and Leo Kanner. Because of the cultural and social components, the presentation of autism varies widely from person to person, including gendered characteristics, which has led to widespread misunderstanding and misdiagnosis. In this talk we'll discuss the most common identifying characteristics of the autism spectrum, co-diagnostics, and autism's place in the modern neurodiversity and disability rights movements. Food will be served: expect Papa John's pizza and Krispey Kreme donuts. This is a joint seminar with the Gagnon Student Seminar. A hybrid option will be available over Zoom. Neurodiversity & Autism Self-Advocacy Networks
May. 25, 2022 1:30pm (Math 3…	Grad Student Seminar Howy Jordan (UC Boulder) X You're invited to a show n tell n monch time with Howy! What we now call the autism spectrum is a cluster of social, cultural, and developmental disabilities popularized by Hans Asperger and Leo Kanner. Because of the cultural and social components, the presentation of autism varies widely from person to person, including gendered characteristics, which has led to widespread misunderstanding and misdiagnosis. In this talk we'll discuss the most common identifying characteristics of the autism spectrum, co-diagnostics, and autism's place in the modern neurodiversity and disability rights movements. Food will be served: expect Papa John's pizza and Krispey Kreme donuts. This is a joint seminar with the Diversity Committee. A hybrid option will be available over Zoom. Neurodiversity & Autism Self-Advocacy Networks
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?

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