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Events for May – September 2019

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May. 01, 2019 3pm (Math350)	Math Phys Ravishankar Sundararaman (Rensselaer Polytechnic Institute) X Density-Functional Theory (DFT) is the workhorse for computational chemistry and materials science. It relies on exactly mapping intractable many-body problems, which involve computing energies in terms of (classical or quantum) configurations of several particles, on to a problem that involves only the much lower-dimensional density of particles. Practical DFT requires a computable approximate functional that maps density to energy with sufficient accuracy. In this talk, I will review the general mathematical formalism of DFT, covering both the ubiquitous (quantum) electronic DFT and the less well-known classical DFT of liquids. I will outline the hierarchy of functional approximations that have proven successful in both these fields and discuss the challenge of capturing non-local correlations accurately. I will end with a hypothesis: an old technique in classical DFT named the Percus rank-2 approximation that quickly became analytically intractable, combined with modern machine learning methods, may hold the key to accurate non-local functionals for both quantum and classical DFT. Functional approximations for Density-Functional Theory Sponsored by the Meyer Fund
May. 01, 2019 3pm (MATH 350)	Topology Ravishankar Sundararaman (Rensselaer Polytechnic Institute) X Density-Functional Theory (DFT) is the workhorse for computational chemistry and materials science. It relies on exactly mapping intractable many-body problems, which involve computing energies in terms of (classical or quantum) configurations of several particles, on to a problem that involves only the much lower-dimensional density of particles. Practical DFT requires a computable approximate functional that maps density to energy with sufficient accuracy. In this talk, I will review the general mathematical formalism of DFT, covering both the ubiquitous (quantum) electronic DFT and the less well-known classical DFT of liquids. I will outline the hierarchy of functional approximations that have proven successful in both these fields and discuss the challenge of capturing non-local correlations accurately. I will end with a hypothesis: an old technique in classical DFT named the Percus rank-2 approximation that quickly became analytically intractable, combined with modern machine learning methods, may hold the key to accurate non-local functionals for both quantum and classical DFT. Functional approximations for Density-Functional Theory Sponsored by the Meyer Fund
May. 01, 2019 4pm (MATH 350)	Grad Student Seminar Yu Wang (CU Boulder) X Symplectic manifolds are a special kind of manifolds which have rich applications in physics and mathematics. Roughly speaking, a symplectic manifold is a smooth manifold equipped with a symplectic form, which makes mathematicians only need to study global properties of the manifold. Combining with a compatible almost complex structure, it will give a math version of mirror symmetry conjecture from string theory. Most modern geometry are built up on Kahler manifolds and Calabi-Yau manifolds that can be gotten from giving more analytic structures to a symplectic manifold. In the talk, I am going to roughly talk about basic properties of symplectic manifolds, complex manifolds and Kahler manifolds. Eventually, I am gonna mention a very famous theorem called Hodge decomposition of Kahler manifolds and a famous conjecture named “Homological Mirror Symmetry”. From symplectic manifolds to Kahler manifolds
May. 02, 2019 3pm (MATH 350)	Probability Steven N. Evans (Departments of Statistics and Mathematics, University of California at Be…) X The $α$ -Lipschitz minorant of a function is the greatest $α$ -Lipschitz function dominated pointwise by the function, should such a function exist. I will discuss this construction when the function is a sample path of a (2-sided) L\'evy process. The contact set is the random set of times when the sample path touches the minorant. This is a stationary, regenerative set. I will provide a description of the excursions of the sample path away from the contact set that is analogous to It\^o's theory for the excursions of a Markov process away from some point in the state space. In particular, I will elucidate the probabilistic structure of both a ``generic'' excursion and the special excursion that straddles the time zero. Lipschitz minorants of L\'evy processes
May. 06, 2019 2pm (MATH 350)	Lie Theory Julianne Rainbolt (Saint Louis University) X A possible Supercharacter Theory based on the degrees of the irreducible characters of a finite group will be discussed. A proof that this Supercharacter Theory applies for certain families of finite groups but does not apply for other families will be outlined. Supercharacter Theory and Degrees of the Irreducible Characters of a Group
May. 06, 2019 3:30pm (MATH 3…	Topology Jonathan Wise (CU Boulder) Ansatz
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 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 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 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

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