# Department of Mathematics Colloquium

 The ISU Department of Mathematics Colloquium is organized by Pablo Raúl Stinga (stinga@iastate.edu) Fall 2018 Tuesdays 4:10 p.m. in Hoover 1213 - Tea and cookies starting at 3:45 p.m. in Carver 404 _________________________________________________________________________________________________________ October 23 Eric Weber Iowa State University Title: The Kaczmarz algorithm: theory and applications Abstract: The Kaczmarz algorithm is an iterative method for solving systems of linear equations that was introduced by Stefan Kaczmarz in 1937. The algorithm is now enjoying a resurgence in interest, as it has been found useful in data science applications. It also has remarkably deep connections to complex and harmonic analysis. We shall introduce the algorithm, demonstrate some of its features, and present some of its applications. Towards the end of the talk, we shall outline these deep connections to complex and harmonic analysis. Upcoming Abstract: The topological KKMS theorem, a powerful extension of Brouwer's Fixed-Point theorem, was proved by Shapely in 1973 in the context of game theory. We prove a colorful and polytopal generalization of the KKMS Theorem, and show that our theorem implies some seemingly unrelated results in discrete geometry and combinatorics involving colorful settings. For example, we apply our theorem to provide a new proof of the Colorful Caratheodory Theorem due to Barany. We further apply our theorem to obtain a new upper bound on the piercing numbers in colorful $d$-interval families, extending results of Tardos, Kaiser and Alon for the non-colored case. Finally, we apply our theorem to questions regarding fair division. Joint with Florian Frick. September 11 Maja Taskovic University of Pennsylvania Title: On the relativistic Landau equation Abstract: In 1936, Landau derived a model for a dilute hot plasma where fast moving particles interact via Coulomb interactions. Instead of tracking every particle separately, which would lead to a large number of equations, the dynamics is being described by the particle density function. This model, also known as the Landau equation, does not include the effects of Einstein's theory of special relativity. However, when particle velocities are close to the speed of light, which happens frequently in a hot plasma, then relativistic effects become important. A model that captures these effects, the relativistic Landau equation, was derived by Budker and Beliaev in 1956. We study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. The difficulty of the problem lies in the extreme complexity of the kernel of the collision operator. We present a new decomposition of such kernel. This is then used to prove the global Entropy dissipation estimate, the propagation of any polynomial moment for a weak solution, and the existence of a true weak solution for a large class of initial data. This is joint work with Robert M. Strain. September 18 Krishna B. Athreya Iowa State University Title: David Harold Blackwell Abstract: David Harold Blackwell was a great mathematician, statistician that belonged to the African American Community from Central Illinois. He was born in 1919 and died in 2010. He got his PhD in mathematics from University of Illinois in 1941 under the great American mathematician J. L. Doob. David Blackwell worked in many areas in mathematics. These include Probability Theory, Game Theory, Information Theory and Bayesian Analysis. He was the first African American to be elected to full professorship at UC Berkeley. He was a member of the US Academy of Sciences, Fellow of the UK Royal Statistical Society, and received many other honors. In this talk we shall outline some of his major research contributions. September 25 Caroline Terry University of Chicago Title: A stable arithmetic regularity lemma in finite-dimensional vector spaces over fields of primer order Abstract: Arithmetic combinatorics studies the additive and multiplicative structure of subsets of groups, especially finite abelian groups such as cyclic groups of prime order, or finite dimensional vector spaces over finite fields. Insipired by Szemerèdi's regularity lemma, arithmetic regularity lemmas are tools used in arithmecit combinatorics to produce group theoretic analogues of results from graph theory. The arithmetic regularity lemma for $F_p^n$ (first proved by Green in 2005) states that given $A\subseteq F_p^n$, there exists $H\leq F_p^n$ of bounded index such that $A$ is Fourier-uniform with respect to almost all cosets of $H$. In general, the growth of the codimension of H is required to be of tower type depending on the degree of uniformity, and must also allow for a small number of non-uniform elements. The main result of this talk is that, under a natural model theoretic assumption called stability, the bad bounds and non-uniform elements are not necessary. Specifically, we present an arithmetic regularity lemma for $k$-stable sets $A\subseteq F_p^n$, where the bound on the codimension of the subspace is only polynomial in the degree of uniformity, and where there are no non-uniform elements. This result is a natural extension to the arithmetic setting of the work on stable graph regularity lemmas initiated by Malliaris and Shelah. This is joint work with Julia Wolf. October 2 Tuncay Aktosun The University of Texas at Arlington Title: Determining the shape of a human vocal tract from speech sounds Abstract: The elementary units for human speech are called phonemes, and the utterance of each phoneme by a person is governed by a particular shape of that person's vocal tract. A mathematical description is presented for the shape of the vocal tract during the creation of each phoneme, which corresponds to the direct problem. Then, a corresponding inverse problem is analyzed; namely, the determination of the shape of the human vocal tract from the sound pressure measurements at the mouth associated with an uttered phoneme. The talk is based on joint work with P. Sacks of Iowa State University.  October 9 Naiomi Cameron Clark College Title: Inversion generating functions for signed pattern avoiding permutations Abstract: We consider the classical Mahonian statistics on the set $B_n(\Sigma)$ of signed permutations in the hyperoctahedral group $B_n$ which avoid all patterns in $\Sigma$, where $\Sigma$ is a set of patterns of length two.  In 2000, Simion gave the cardinality of $B_n(\Sigma)$ in the cases where $\Sigma$ contains either one or two patterns of length two and showed that $\left|B_n(\Sigma)\right|$ is constant whenever $\left|\Sigma\right|=1$, whereas in most but not all instances where $\left|\Sigma\right|=2$, $\left|B_n(\Sigma)\right|=(n+1)!$.  We answer an open question of Simion by providing bijections from $B_n(\Sigma)$ to $S_{n+1}$ in these cases where $\left|B_n(\Sigma)\right|=(n+1)!$.  In addition, we extend Simion's work by providing a combinatorial proof in the language of signed permutations for the major index on $B_n(21, \bar{2}\bar{1})$ and by giving the major index on $D_n(\Sigma)$ for $\Sigma =\{21, \bar{2}\bar{1}\}$ and $\Sigma=\{12,21\}$.  The main result of this paper is to give the inversion generating functions for $B_n(\Sigma)$ for almost all sets $\Sigma$ with $\left|\Sigma\right|\leq2.$ October 16 Vlad Vicol Courant Institute Title: Nonuniqueness of weak solutions to the Navier-Stokes equation Abstract: We prove that distributional solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Hölder continuous weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations. This is a joint work with Tristan Buckmaster.