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Events for June – October 2019

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Jun. 03, 2019 1pm (350)	Ulam Jonathan Belcher (CU Boulder) X The connection between Hochschild and cyclic cohomologies with generalized De Rham homology and index theories for arbitrary algebras has long been established by the work of Connes, Karoubi, Loday, Feigin, Tsygan, et al. Here we generalize these cohomology theories even further, essentially creating a theory that establishes a step-wise bridge between the two. Motivation for this construction comes from trying to generalize the Hochschild-Kostant-Rosenberg-Connes theorem to manifolds with boundary, and applications in tracial constructions in certain classes of pseudodifferential operators. Further geometric and topological interests of this theory include extending Chern-Weil theory to manifolds with boundary via pairings between bridge cohomology and higher K-theories. Bridge cohomology: a generalization of Hochschild and cyclic cohomologies with applications to Chern-Weil theory
Jun. 20, 2019 2pm (MATH 350)	Topology Henry Adams (Colorado State University) X This talk is an introduction to computational topology, as applied to data analysis and to sensor networks. The shape of a dataset often reflects important patterns within. Two such datasets with interesting shapes are a space of 3x3 pixel patches from optical images, which can be well-modeled by a Klein bottle, and the conformation space of the cyclo-octane molecule, which is a Klein bottle glued to a 2-sphere along two circles. I will introduce topological tools (persistent homology) for visualizing, understanding, and performing machine learning tasks on high-dimensional datasets. As a second application, I will describe how topology has been applied to coverage problems in mobile sensor networks. An introduction to applied topology
Jul. 25, 2019 2pm (MATH 350)	Probability Jamal Najim (Université de Marne La Vallée) X Consider a large linear system where $A_{n}$ is a $n \times n$ matrix with independent real standard Gaussian entries, $1n$ is a $n× 1$ vector of ones and with unknown the vector $x_{n}$ satisfying $xn = 1n + 1/αn n An xn$ We investigate the (componentwise) positivity of the solution $x_{n}$ depending on the scaling factor $α_{n}$ as the dimension $n$ goes to $∞$ . Such linear systems arise as solutions at equilibrium of large Lotka-Volterra systems of differential equations, widely used to describe large biological communities with interactions such as foodwebs for instance. In the domaine of positivity of the solution $x_{n}$ , we establish that the Lotka-Volterra system of differential equations whose solution at equilibrium is precisely $x_{n}$ is stable in the sense that its jacobian $J(xn) = diag(xn) ( -In + 1/αn n An xn)$ Positive solutions for large random linear systems
Aug. 15, 2019 3pm (MATH 350)	Probability Kyle Luh (Harvard) X Let $ε1, , εn$ be i.i.d.\ Rademacher random variables taking values $± 1$ with probability $1/2$ each. Given an integer vector $\boldsymbol{a} = (a_1, \dotsc, a_n)$, its concentration probability is the quantity $\rho(\boldsymbol{a}):=\sup_{x\in \Z}\Pr(\epsilon_1 a_1+\dotsb+\epsilon_n a_n = x)$. The Littlewood--Offord problem asks for bounds on $ρ(a)$ under various hypotheses on $a$ , whereas the \emph{inverse} Littlewood--Offord problem, posed by Tao and Vu, asks for a characterization of all vectors $a$ for which $ρ(a)$ is large. In this paper, we study the associated counting problem: How many integer vectors $a$ belonging to a specified set have large $ρ(a)$ ? The motivation for our study is that in typical applications, the inverse Littlewood--Offord theorems are only used to obtain such counting estimates. Using a more direct approach, we obtain significantly better bounds for this problem than those obtained using the inverse Littlewood--Offord theorems of Tao and Vu and of Nguyen and Vu. Moreover, we develop a framework for deriving upper bounds on the probability of singularity of random discrete matrices that utilizes our counting result. To illustrate the methods, we present the first `exponential-type' (i.e., $(-nc)$ for some positive constant $c$ ) upper bounds on the singularity probability for the following two models: (i) adjacency matrices of dense signed random regular digraphs, for which the previous best known bound is $O(n-1/4)$ due to Cook; and (ii) dense row-regular $0,1$ -matrices, for which the previous best known bound is $OC(n-C)$ for any constant $C>0$ due to Nguyen. This is joint work with Asaf Ferber, Vishesh Jain and Wojciech Samotij. The counting problem in inverse Littlewood-Offord theory
Aug. 27, 2019 11am (MATH 350)	Number Theory Katherine Stange X Apollonian circle packings -- more precisely, the collection of curvatures in such a packing -- are a particularly pretty example of the orbit of thin Kleinian group. For many such packings, the collection consists entirely of integers. Such orbits are conjectured to follow a local-global principle in the sense that, except for a congruence obstruction modulo 24, all sufficiently large integers are expected to appear. I will describe some of what I've learned and participated in concerning proving the positive density of curvatures in an Apollonian circle packing, and, more recently, in a prime component of such a packing. The methods and players include quadratic forms, the circle method, expander graphs and strong approximation. Some Apollonia
Aug. 27, 2019 2pm (Math 350)	Algebraic Geometry Fenglong You (University of Alberta) X Given a smooth projective variety X and a smooth divisor D \subset X, one can study the enumerative geometry of counting curves in X with tangency conditions along D. There are (at least) two theories associated to it: relative Gromov--Witten invariants of (X,D) and orbifold Gromov--Witten invariants of the r-th root stack X_{D,r}. For a sufficiently large r, Abramovich--Cadman--Wise proved that genus zero relative invariants are equal to genus zero orbifold invariants of root stacks (with a counterexample in genus 1). I will explain the precise relation in higher genus. If time permits, I will also talk about further results on structures of relative Gromov--Witten theory and mirror symmetry. This is based on joint works with Hisan-Hua Tseng, Honglu Fan and Longting Wu. Relative and orbifold Gromov--Witten invariants Sponsored by the Meyer Fund
Aug. 27, 2019 3pm (MATH 350)	Topology Emily Riehl (Johns-Hopkins University) X After sketching the idea of an infinite-dimensional category --- more precisely an (?,n)-category for n between 0 and ? --- I'll explain how one can do rigorous work with infinite-dimensional categories in a "model-independent" fashion, at least in the cases n = 0 or n = 1. The key idea is to reason inside an ?-cosmos, an axiomatization of the "universe" in which ?-categories live as objects. This is joint work with Dominic Verity. An introduction to model-independent ∞-category theory Sponsored by the Meyer Fund
Aug. 27, 2019 4pm (MATH 350)	Topology Emily Riehl (Johns-Hopkins University) X Some claim that "(∞,1)-category theory is just like 1-category theory except you put an `∞' in front of everything," but there is at least one case where this optimism is unwarranted: it's easy to state and prove the Yoneda lemma for ordinary categories but extremely complicated to even define the Yoneda embedding for an (∞,1)-category. In this talk we'll introduce left fibrations and cocartesian fibrations, which can be used respectively to model (homotopy coherent) functors of (∞,1)-categories valued in (∞,0)- or (∞,1)-categories. We then describe the construction of another homotopy coherent functor, the comprehension functor associated to a fixed cocartesian fibration, which "unstraightens" a homotopy coherent diagram into cocartesian fibration. We conclude by explaining how a special case of this construction provides the sought-for Yoneda embedding. This is joint work with Dominic Verity. The comprehension construction Sponsored by the Meyer Fund
Aug. 28, 2019 4pm (MATH 350)	Grad Student Seminar Mark Pullins (CU Boulder) X Perhaps it’s been a while since you’ve needed multivariable calculus, and you’ve forgotten parts of it. That’s fine. Except that during your MARC hours students are bombarding you with Calc 3 questions, and they’re starting to suspect you’re lost when you keep restating their questions back at them. Don’t fear! During this talk, I’ll attempt to cover the highlights of Calc 3 and frequent problem areas for students in the MARC. Since our goal is to be of maximum help to MARC students, we will focus on doing examples and sweep most theory issues under the rug. Calc 3 in the MARC (by Mark)
Sep. 03, 2019 10am (MATH 350)	Geometry/Analysis Taylor Klotz (CU Boulder) X A certain class of control systems exhibit a surprising property that addition of extra differential equations suddenly allow the system to be put into a linearized normal form. This is known as extended dynamic feedback linearization (EDFL). Vassiliou showed that a sufficient condition for a control system with symmetry to be EDFL is another type of linearization known as cascade feedback linearization (CFL), in which the system has two “hidden” linearizable systems. This talk will give an overview of the CFL process as well as some new existence results for CFL systems. Cascade Feedback Linearization
Sep. 03, 2019 2pm (MATH 350)	Lie Theory Richard Green (CU) X A Coxeter group is called a $Y$ -group if its Coxeter diagram $Y_{p q r}$ consists of a central node connected to three branches of lengths $p$ , $q$ , and $r$ . I will describe, in the special case $p=1$ , a diagram calculus for the generalized Temperley--Lieb algebra corresponding to this Coxeter group, and discuss how this relates to Kazhdan--Lusztig cells. The generalized Temperley--Lieb algebra of type E(1,q,r)
Sep. 04, 2019 4pm (MATH 350)	Grad Student Seminar Lucas Gagnon (CU Boulder) X A matroid is an abstraction of mathematical problems involving independence and overlap. Matroid structures occur in surprisingly disparate areas of mathematics including linear and abstract algebra, finite geometry, combinatorics, and topology. Moreover, general results about matroids and matroid invariants often specialize in a meaningful way to each of these cases. Unfortunately, matroids can also be frustratingly abstract and there are an absurd number of non-trivially equivalent definitions of a matroid, making matroid theory a challenging subject to begin learning about. In this talk, I will attempt to circumvent these obstacles in order to give an accessible and motivated introduction to matroids, with examples from some of the many areas of math in which they appear. Matroid Prime
Sep. 05, 2019 2pm (MATH 350)	Functional Analysis Robin Deeley (CU Boulder) X Many natural minimal homeomorphisms occur on homogeneous spaces (e.g., the Cantor set, the circle, etc). Floyd gave the first example of a minimal dynamical system on a nonhomogeneous space. I will discuss this example and further examples, which build on work of Gjerde and Johansen. C*-algebraic application will also be discussed. This is joint work with Ian Putnam and Karen Strung. Nonhomogeneous minimal dynamical systems
Sep. 10, 2019 11am (MATH 350)	Number Theory Ralph Heiner Buchholz (Defence Science and Technology Group, Australia) X I will discuss various ideas used to try to tackle one of the unsolved problems in Richard Guy's book UPINT. Two themes will be parametrised families of curves on the one hand and algebraic surfaces on the other. The status of D21 --- the search for a perfect triangle
Sep. 10, 2019 1pm (MATH 220)	Keith Kearnes (CU) Minimal Abelian Varieties of Algebras
Sep. 10, 2019 2pm (MATH 350)	Lie Theory Richard Green (CU) X [This is a continuation of last week's talk.] A Coxeter group is called a $Y$ -group if its Coxeter diagram $Y_{p q r}$ consists of a central node connected to three branches of lengths $p$ , $q$ , and $r$ . I will describe, in the special case $p=1$ , a diagram calculus for the generalized Temperley--Lieb algebra corresponding to this Coxeter group, and discuss how this relates to Kazhdan--Lusztig cells. The generalized Temperley--Lieb algebra of type E(1,q,r) [continued]
Sep. 10, 2019 2pm (MATH 220)	Jordan DuBeau (CU) X We prove a connection between Laver tables and rank-into-rank embeddings. Introduction to Laver tables 1
Sep. 11, 2019 4:03pm (MATH 3…	Grad Student Seminar Andrew Campbell (CU Boulder) X You may have heard that it takes seven riffle shuffles to randomize a standard deck of cards, and that there is even a proof to back this up. But what does it mean for a deck to be in random order? We will use a card trick from around the turn of the twentieth century to answer this question, study symmetric group algebras, and possibly mention connections to Hochschild homology. This talk is based on the work of Dave Bayer and Persi Diaconis. Using a hundred year old card trick to study group algebras and win money
Sep. 12, 2019 2pm (MATH 350)	Functional Analysis Philipp Schmitt (University of Copenhagen) X Formal deformation quantization was introduced as a conceptual approach to the quantization problem and is, by now, well understood: The existence and classification of formal star products can be deduced from Kontsevich's formality theorem. To make the theory more accessible to physics, strict quantization asks to replace the formal parameter by a real number (that plays the role of Planck's constant), in order to obtain a field of well-behaved algebras, such as Fréchet-*-algebras. Quantization is intimately related to symbol calculi for operators and therefore has many applictions, e.g. to index theory. In this talk, I would like to give a brief introduction to the theory of formal and strict quantization. I will mainly focus on the example of the 2-sphere and explain how to construct an equivariant strict star product on a subclass of analytic functions. Time permitting, I will sketch how to generalize this construction to arbitrary semisimple coadjoint orbits of semisimple connected Lie groups. Strict quantization of the 2-sphere
Sep. 17, 2019 1pm (MATH 220)	Keith Kearnes (CU) Minimal Abelian Varieties of Algebras 2
Sep. 17, 2019 2pm (MATH 220)	Jordan Dubeau (CU) X We finish proving a connection between Laver tables and rank-into-rank embeddings. Introduction to Laver tables 2
Sep. 17, 2019 2pm (MATH 350)	Lie Theory Tianyuan Xu (CU) X In 1985, Lusztig defined a function $a : W \to ℕ$ for any Coxeter group $W$ by using the Kazhdan-Lusztig basis of the Hecke algebra of $W$ . The $a$ -function has important connections with the cell representation theory of $W$ and its Hecke algebra, but is usually difficult to compute directly. However, it is known that an element has $a$ -value 1 if and only it it is a non-identity element with a unique reduced word, and that $W$ contains finitely many elements of $a$ -value 1 if and only the Coxeter diagram of $W$ satisfies certain conditions. In this talk, we describe a similar classification, in terms of Coxeter diagrams, of all Coxeter groups with finitely many elements of $a$ -value 2. We will show that elements of $a$ -value 2 are fully commutative in the sense of Stembridge, and our main tools for the classification include Viennot's heaps and certain so-called star operations on Coxeter groups. (Joint work with Richard Green.) Classification of a(2)-finite Coxeter groups
Sep. 17, 2019 4pm (MATH 350)	Topology Andre Davis (CU) Generalized Cohomology Theories and Spectra
Sep. 18, 2019 4pm (MATH 350)	Grad Student Seminar Carly Matson (CU Boulder) X Ravi Vakil's excellent book The Rising Sea opens with the following quote from David Mumford: "[Algebraic geometry] seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics! In one respect the last point is accurate..." In this talk we will look at some of the fundamentals of algebraic geometry from the more concrete perspective of polynomials and we will let this guide us to the land of abstraction. If time permits, there may be secret plotting. From Affine to Zariski: Demystifying Algebraic Geometry
Sep. 18, 2019 5pm (Math 350)	MathClub Yuhao Hu (CU Boulder postdoc) X Imagine a convex curve in the plane. Fitting the curve tightly between two parallel lines, the distance between those lines tells us the width of the curve in a particular direction. If such width does not depend on the angle in which you direct the parallel lines, the curve is said to have constant width. Analogously, we have the notion of bodies of constant width. Rolling a flat board over bodies of the same constant width creates an illusion of rolling over round balls of the same radius. Beyond such counter-intuitiveness, there are more interesting mathematical aspects related to/inspired by shapes of constant width. Shapes of Constant Width and Beyond
Sep. 19, 2019 4pm (MATH 350)	Topology Robert Bruner (Wayne State University) X A(1)-modules are relevant to the calculation of K-theory through the Adams spectral sequence. Many interesting cohomology theories correspond in this way to sub-Hopf algebras B of the Steenrod algebra. Margolis localizations allow us to understand B-modules through their localizations. The method is especially effective for A(1)-modules as there are only two localizations to consider. The category of A(1)-modules
Sep. 24, 2019 12pm (MATH 350)	Kempner Catherine Sulem (University of Toronto) X We consider the movement of a free surface of a two-dimensional fluid over a variable bottom. We assume that the bottom has a periodic profile and we study the water wave system linearized near a stationary state. The latter reduces to a spectral problem for the Dirichlet--Neumann operator in a fluid domain with a periodic bottom and a flat surface elevation. Bloch spectral decomposition is a classical tool to address problems in periodic geometries or equivalently differential operators with periodic coefficients. We show that the spectral problem admits a Bloch decomposition in terms of spectral band functions and their associated band-parametrized eigenfunctions. We find that, generically, the spectrum consists of a series of bands separated by spectral gaps which are zones of forbidden energies. Bloch theory and spectral gaps for linearized water waves
Sep. 24, 2019 4pm (MATH 350)	Topology Colton Grainger (CU) Lazard's Theorem
Sep. 25, 2019 4pm (MATH 350)	Grad Student Seminar Jonathan Quartin (CU Boulder) X Although last week’s talk is not strictly a prerequisite, I will continue in the spirit of peeking into the world of Algebraic Geometry via polynomials (this time over the complex numbers). We will focus our attention on homogeneous polynomials, which will land us in a space with so called “points at infinity”- Projective space. Algebraic Geometry: from affine to projective
Oct. 01, 2019 1pm (MATH 220)	Peter Mayr (CU) X We present a functorial construction which, starting from a congruence of finite index in an algebra, yields a new algebra built from the congruence blocks. As an application we show that supernilpotence of congruences is decidable. Algebras from congruences 1
Oct. 01, 2019 2pm (MATH 220)	Michael Wheeler (CU) Hyperprincipal Realizations of Ultrafilters, Part 1
Oct. 01, 2019 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X The categorification of the Hopf algebra of symmetric functions by the representation theory of the symmetric group is a fundamental result in combinatorial representation theory. While some of our favorite graded Hopf algebras have some representation theoretic interpretations via finite dimensional algebras, very few have come from a tower of groups (as in the symmetric group case). This talk gives a tour of recent successes in realizing graded Hopf algebras via towers of groups and their representation theory. We begin with the established case of symmetric functions in noncommuting variables and then quickly moving on to newer cases (including the Malvenuto—Reutenauer algebra). This work is joint with F. Aliniaeifard. Categorifying combinatorial Hopf algebras
Oct. 01, 2019 4pm (MATH 350)	Topology Juan Moreno (CU) Complex Oriented Cohomology Theories
Oct. 02, 2019 4pm (MATH 350)	Grad Student Seminar Mark Pullins (CU Boulder) X In the past two talks, we have explored the Zariski topology and projective space. In this talk, we will begin in familiar territory and review the Zariski topology and varieties. Then, we will introduce linear algebraic groups from a (hopefully) more intuitive perspective and explore the wonderful (and mildly surprising) structural properties gifted from the Zariski topology. Please note that the past two talks are in no way required! Linear Algebraic Groups: From Spaces to Groups
Oct. 02, 2019 5pm (Math 350)	MathClub Harrison Stalvey (CU Boulder) X Epistemology is the philosophy of knowledge. What is knowledge? Where does it come from? How does it develop? Two particular epistemologies are constructionism and constructivism, which view knowledge as a construction, not a discovery, even if it is influenced by social factors. Under the assumption that knowledge is constructed, it is natural to wonder about the psychology behind learning. We will explore some epistemology and psychology as they pertain to a concept central to mathematics: the function. In particular, we will survey the history of the concept as an illustration of constructionism. We will also discuss a particular constructivist learning theory that has been used to describe how an understanding of mathematics can develop in the mind of the learner. Then we will apply this learning theory to the concept of function. This talk will conclude with examples of various avenues of research in mathematics education and what that research might look like. Epistemology, Psychology, and the Concept of Function
Oct. 03, 2019 2pm (MATH 350)	Functional Analysis Mitsuru Wilson (CU Boulder) X This will be a series of two talks wherein I will give an introductory talk on pseudo-differential operators and their symbols in the first talk and I will present my recent results in the second talk. Pseudo-differential operators are a generalization of differential operators on the space of smooth functions on a manifold. One of the most fundamental constructions on a pseudo-differential operator is its symbol. Under some assumptions on the operator, then there is a one to one correspondence between the operators and the symbols. I generalized Michael Ruzhansky's construction of symbolic calculus on compact Lie groups to symbolic calculus of operators acting on the smooth sections of homogeneous vector bundles. In particular, I proved the formula for the parametrix of elliptic operators. Symbolic calculus on homogeneous vector bundles (Part I)
Oct. 08, 2019 1pm (MATH 220)	Peter Mayr (CU) Algebras from congruences 2
Oct. 08, 2019 2pm (MATH 350)	Lie Theory Lucas Gagnon (CU) X Classifying all conjugacy classes and/or irreducible representations in the family of unipotent upper triangular matrices $UTn(Fq)$ is a wild problem, meaning that it is an essentially hopeless endeavor. In this talk, I will take up the superficially similar question of normal subgroups in $UTn(Fq)$ and answer it with a bijection between normal subgroups and vector spaces over $𝔽_{q}$ which are parametrized by a family of Catalan-like combinatorial objects. Following this construction, I will show how supercharacter theory can turn our newfound understanding of the normal subgroup structure of $UTn(Fq)$ into a deeper representation theoretic understanding of this family of groups. Finding all normal subgroups of the unipotent upper triangular groups
Oct. 08, 2019 2pm (MATH 220)	Michael Wheeler (CU) Hyperprincipal Realizations of Ultrafilters, Part 2
Oct. 08, 2019 4pm (MATH 350)	Topology Andrew Stocker (CU) Complex Bordism, MU and Complex Orientations
Oct. 09, 2019 4pm (MATH 350)	Grad Student Seminar Sarah Arpin (CU Boulder) X Supersingular isogeny graphs were introduced as a hard problem into cryptography by Charles, Goren, and Lauter for the construction of cryptographic hash functions (CGL06). These are large expander graphs, and the hard problem is to find an efficient algorithm for routing between two vertices of the graph. In particular, we consider two related graphs that help us understand the big picture of the full supersingular $ℓ$ -isogeny graph: the `spine' $𝒮$ , which is the subgraph of $Gℓ(Fp)$ given by the j-invariants in $𝔽_{p}$ , and the graph $Gℓ(Fp)$ , in which both curves and isogenies must be defined over $𝔽_{p}$ . This is joint work with Catalina Camacho-Navarro, Kristin Lauter, Joelle Lim, Kristina Nelson, Travis Scholl, Jana Sotáková. https://arxiv.org/abs/1909.07779 No pre-requisites are required, but a willingness to take things on faith is requested. "Why, sometimes I've believed as many as six impossible things before breakfast." ~Lewis Carroll Adventures in Supersingularland: An Exploration of Supersingular Elliptic Curve Isogeny Graphs
Oct. 10, 2019 12:10pm (MATH …	Kempner Piotr M. Hajac (IMPAN and University of Colorado Boulder) X In topology, pushouts are formal recipes for collapsing and gluing topological spaces. For instance, shrinking the boundary circle of a disc to a point yields a sphere, shrinking the equator of a sphere to a point gives two spheres joined at the point, collapsing the boundary of a solid torus to a circle, or gluing two solid tori over their boundaries, produces a three-sphere. In noncommutative topology, such procedures are expressed in terms of pullbacks of C-algebras. It turns out that one can visualize a pullback of C-algebras of graphs as a pushout of these graphs thus providing much needed intuition to the abstract setting of operator algebras. The goal of this talk is to discuss how to make this visualization rigorous by conceptualizing abundant examples from noncommutative topology that lead to a new concept of morphisms of graphs. In particular, we replace the standard idea of mapping vertices to vertices and edges to edges by the more flexible idea of mapping finite paths to finite paths. (Based on joint works with Alexandru Chirvasitu, Sarah Reznikoff and Mariusz Tobolski.) From pushouts to pullbacks: a sample of noncommutative topology
Oct. 10, 2019 2pm (MATH 350)	Functional Analysis Mitsuru Wilson (CU Boulder) X This will be a series of two talks wherein I will give an introductory talk on pseudo-differential operators and their symbols in the first talk and I will present my recent results in the second talk. Pseudo-differential operators are a generalization of differential operators on the space of smooth functions on a manifold. One of the most fundamental constructions on a pseudo-differential operator is its symbol. Under some assumptions on the operator, then there is a one to one correspondence between the operators and the symbols. I generalized Michael Ruzhansky's construction of symbolic calculus on compact Lie groups to symbolic calculus of operators acting on the smooth sections of homogeneous vector bundles. In particular, I proved the formula for the parametrix of elliptic operators. Symbolic calculus on homogeneous vector bundles (Part II)
Oct. 15, 2019 4pm (MATH 350)	Topology Akira Tominaga (Tokyo University) Quillen's Theorem
Oct. 16, 2019 4pm (MATH 350)	Grad Student Seminar Erik Knutson (CU Boulder) X The Steenrod algebras are graded Hopf algebras over fields of prime order which arise in the study of topological invariants of spaces. The goal of this talk will be to learn just enough about homological algebra and the Steenrod algebra generated by the Steenrod squares over everyone's favourite two element field to play a little game using diagrammatic representations of Steenrod submodules. Playing in the Steenrod Sandbox
Oct. 16, 2019 5pm (Math 350)	MathClub Rob Tubbs (CU Boulder) X As an introduction to an important branch of number theory, diophantine approximation, we will examine several real numbers and show that they are irrational. We will also discuss whether they are surds, which are irrational numbers that are roots of rational numbers. An absurdist look at the nature of some peculiar decimals
Oct. 17, 2019 2pm (MATH 350)	Functional Analysis Shen Lu (CU Boulder) X For a fixed prime $p$ , a noncommutative solenoid as defined by Frederic Latremoliere and Judith Packer is a twisted group C*-algebra $Cleft( Z[1/p] × Z[1/p], σ )$ , where $Z[1/p] := j/pk ∈ Q : j ∈ Z, k ∈ N$ is an additive discrete group and $σ$ is a $T-$ valued group $2$ -cocycle (multiplier) on $Z[1/p] × Z[1/p]$ . In this talk, we first review the classification of all NC solenoids in terms of their defining multipliers using $K-$ theory. From there, we discuss two constructions for projective modules over the irrational NC solenoids and how they are related: one by constructing the Heisenberg equivalence bimodule of Rieffel utilizing the $p-$ adic numbers, and the other by writing the NC solenoids as direct limits of noncommutative tori. Noncommutative solenoids and their finitely generated projective modules
Oct. 22, 2019 11am (MATH 350)	Number Theory Min Ru (University of Houston) X In recent years, there have been some breakthroughs in extending Schmidt's subspace theorem to non-linear hypersurfaces, made by Corvaja-Zannier, and Evertse-Feretti, etc. In this talk, I will discuss my recent joint work with P. Vojta based on their works, as well as its applications. Schmidt's subspace theorem and its generalization
Oct. 22, 2019 12:10pm (MATH …	Kempner Jan Mycielski (CU Boulder) X A description of the structure of knowledge in the brains of animals, with an outline of the phylogeny of human intelligence. The theory supports some old ideas in the philosophy of mind and it may have applications in artificial intelligence. I will define finite algebraic structures, that I call mental models. They are intended to represent in a realistic way the descriptive component of knowledge (as opposed to its evaluating component). I will define the notions of mental object and of concepts in a mental model. Concepts can be real, like these representing objects and processes of everyday life or studied in science, or they may be imaginary, like these in tales and pure mathematics. Our definitions yields a precise form of Ockham's principle of economy of concepts. The theory suggests some ideas in ontology and pedagogy. Theories of the mind that ignore biology or brain science, or do not propose sufficiently clear notions of meaning, will be criticized. Mathematics of human intelligence and related ideas
Oct. 22, 2019 4pm (MATH 350)	Topology Jonathan Quartin (CU) The Moduli Stack of Formal Groups
Oct. 24, 2019 2pm (MATH 350)	Functional Analysis Paul Baum (Penn State University) X This talk will outline exactly how and why the Atiyah-Singer index theorem can be viewed as a corollary of Bott periodicity. The precise argument proceeds by introducing an abelian group K_0(.) obtained by considering pairs (M, E) where M is a closed Spin-c manifold and E is a a vector bundle on M. "Spin-c manifold" will be carefully defined. Most of the oriented manifolds which occur in practice are Spin-c. The above is joint work with Erik van Erp. ATIYAH-SINGER AS A COROLLARY OF BOTT PERIODICITY : PART 1 Sponsored by the Meyer Fund
Oct. 29, 2019 12:10pm (MATH …	Kempner Paul Baum X Let $Γ$ be a compact Hausdorff second countable topological group. ``second countable ``the topology of $Γ$ has a countable base" Irr( $Γ$ ) denotes the set of equivalence classes of irreducible representations of $Γ$ . The vector space for each irreducible representation of $Γ$ is a finite dimensional vector space over the complex numbers. Irr( $Γ$ ) is a countable set --- and we give it the discrete topology. Let $X$ be a locally compact Hausdorff topological space with a given (continuous) action of $Γ$ on $X$ . For each $x$ in $X$ , $Γ_{x}$ denotes the isotropy group of $x$ . If $x$ and $y$ are in the same orbit for the given action of $Γ$ on $X$ , then Irr( $Γ_{x}$ ) and Irr( $Γ_{y}$ ) are canonically in bijection. The usual quotient, denoted $X/Gamma$ , is obtained by collapsing each orbit (for the given action of $Γ$ on $X$ ) to a point. The extended quotient, denoted $X//Γ$ , is constructed by replacing the orbit of $x$ (for the given action of $Γ$ on $X$ ) by Irr( $Γ_{x}$ ). This talk will describe how extended quotient enters into the representation theory of Lie groups and reductive p-adic groups. For Lie groups this is due to Mackey-Higson-Afgoustidis. For reductive p-adic groups, this is due to Aubert-Baum-Plymen-Soll Extended quotient in representation theory or Non-commutative geometry impacts representation theory
Oct. 29, 2019 4pm (MATH 350)	Topology Kate Ponto (University of Kentucky) X The Lefschetz fixed point theorem gives an invariant that is zero when an endomorphism is homotopic to a map with no fixed points. The converse is not true unless we refine the Lefschetz number to a richer invariant called the Reidemeister trace. When you come to these invariants as a homotopy theorist (of a particular variety) you notice that the Reidemeister trace is a topological Hochschild homology invariant and is additive. This suggests that you should look for connections to algebraic K-theory since it is the universal home for additive invariants. In this talk I'll (start to) explain the connection between traces defined from fixed point invariants and traces defined for computing K-theory Traces in K-theory and fixed point theory Sponsored by the Meyer Fund
Oct. 30, 2019 3pm (MATH 350)	Algebraic Geometry Alicia Lamarche (University of South Carolina) X When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety X, to what extent can Db(X) be used as an invariant to answer rationality questions? In particular, what properties of Db(X) are implied by X being rational, stably rational, or having a rational point? On the other hand, is there a property of Db(X) that implies that X is rational, stably rational, or has a rational point? In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full ?etale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Matthew Ballard, Alexander Duncan, and Patrick McFaddin. Derived Categories, Arithmetic, and Rationality Questions Sponsored by the Meyer Fund
Oct. 30, 2019 4pm (MATH 350)	Grad Student Seminar Howie Jordan (CU Boulder) X Would you like to play a game? It's the day before Halloween, so let's get a little spooky. Open this door with the key of imagination, and step into a world beyond your usual understanding. A world where the axiom of choice fails, but not horrendously so. A world where every set of real numbers is Lebesgue measurable. A world where we can't turn a single sphere into two with the same volume or well-order every set. But it will all start with a simple game The goal of this talk is to provide an introduction to an alternative axiom of set theory, one which contradicts the axiom of choice but shares some of that axiom’s implications, as well as its own unique, interesting, and even paradoxical consequences. We will compare this axiom to choice, see how they contradict, and if time permits we may even discuss implications of our considerations for foundations of mathematics. You Have No Choice; It is Already Determined
Oct. 30, 2019 5pm (Math 350)	MathClub Sean O'Rourke (CU Boulder) X How many times do you have to shuffle a deck of cards to mix it up? The answer depends on what we mean by shuffling. I will discuss this question and explain how shuffling a deck of cards can be modeled mathematically. How many times should you shuffle a deck of cards?
Oct. 31, 2019 10am (Math 350)	Thesis Defenses Tyler Schrock X Conjugacy is the natural notion of isomorphism in symbolic dynamics. A major open problem in the field is that of finding an algorithm to determine conjugacy of shifts of finite type (SFTs). We consider several related computational problems restricted to k-block codes. We show verifying a proposed k-block conjugacy is in P, finding a k-block conjugacy is GI-hard, reducing the representation size of a SFT via a 1-block conjugacy is NP-complete, and recognizing if a sofic shift is a SFT is in P. Then we consider the complexity of isomorphism among finite groups, where we consider isomorphism between quotients of genus 2 groups. The centrally indecomposible genus 2 groups split into two cases: flat and sloped. We show determining isomorphism between quotients of genus 2 groups by non-central subgroups is in P and determining isomorphism between quotients of the genus 2 groups H^\flat_1 by central subgroups is in P. On the complexity of isomorphism in finite group theory and symbolic dynamics
Oct. 31, 2019 2pm (MATH 350)	Functional Analysis Paul Baum (Penn State University) X This talk will take the outline provided in Part 1 and fill in details so as to provide a complete proof. ATIYAH-SINGER AS A COROLLARY OF BOTT PERIODICITY : PART 2 Sponsored by the Simons Foundation
Oct. 31, 2019 3pm (MATH 350)	Probability Mei Yin (University of Denver) X Exponential random graphs are powerful in the study of modern networks. By representing a complex global configuration through a set of tractable local features, these models seek to capture a wide variety of common network tendencies. This talk will look into the asymptotic structure of weighted exponential random graphs and formulate a quantitative theory of phase transitions. The main techniques that we use are variants of statistical physics. Based on joint work with multiple collaborators. Perspectives on exponential random graphs

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