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

**October 30
**

**Brooke Ullery
**

Harvard University

**Title**: Measures of
irrationality for algebraic varieties

**Abstract:**In algebraic geometry, a smooth curve is said to be rational if it is isomorphic to $P^1$, the projective line. More generally, the gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. There are a few different definitions that attempt to generalize the notion of gonality even further to higher dimensional varieties. The intuition is that the higher these numbers, the further the variety is from being rational. We will discuss these measures of irrationality and various methods of calculating and bounding them. We will mainly focus on the examples of hypersurfaces and, more generally, complete intersections in projective space. All of these terms will be defined and there will be lots of accessible examples!

**November 2 (Friday at 3:10pm - Carver 305 -
Colloquium / Discrete Math seminar)
**

**Emily Sergel
**

University of Pennsylvania

**Title**: Parking functions,
schedules, and the Delta Conjecture

**Abstract:**A parking function is a kind of labeled lattice path. The Shuffle Conjecture states that a certain enumeration of parking functions is closely related to an important symmetric function basis called the Macdonald polynomials. The Delta Conjecture is a generalization with similar ties to Macdonald polynomials. We discuss the combinatorics of parking functions and the notion of schedules as they pertain to these two settings.

Joint work with Jim Haglund.

**November 6
**

**Marta D'Elia
**

Sandia National Laboratories

**Title: **Nonlocal models in
computational science and engineering: challenges
and applications

**Abstract: **Nonlocal
continuum theories such as peridynamics and nonlocal
elasticity can capture strong nonlocal effects due
to long-range forces at the mesoscale or microscale.
For problems where these effects cannot be
neglected, nonlocal models are more accurate than
classical Partial Differential Equations (PDEs) that
only consider interactions due to contact. However,
the improved accuracy of nonlocal models comes at
the price of a computations cost that is
significantly higher than that of PDEs.

In this talk I will present nonlocal models and
the Nonlocal Vector Calculus, a theory that allows
one to treat nonlocal diffusion problems in almost
the same way as PDEs. Furthermore, I will present
current open challenges related to the numerical
solution of nonlocal problems and show how we are
currently addressing them. Specifically, I will
describe an optimization-based local-nonlocal
coupling strategy and briefly introduce a technique
to improve the performance of Finite Element (FE)
approximations.

The goal of local-nonlocal coupling methods is
to combine the computational efficiency of PDEs with
the accuracy of nonlocal models. These couplings are
imperative when the size of the computational domain
or the extent of the nonlocal interactions are such
that the nonlocal solution becomes prohibitively
expensive to compute, yet the nonlocal model is
required to accurately resolve small scale features.
Our approach formulates the coupling as a control
problem where the states are the solutions of the
nonlocal and local equations, the objective is to
minimize their mismatch on the overlap of the
nonlocal and local domains, and the controls are
virtual volume constraints and boundary conditions.
I will present consistency and convergence studies
and, using three-dimensional geometries, I will also
show that our approach can be successfully applied
to challenging, realistic problems.

Finally, I will introduce a new concept of
nonlocal neighborhood that helps improving the
performance of FE methods and show how our approach
allows for fast assembling in two- and
three-dimensional computations.

**November 13
**

**Catherine Searle
**

Wichita State University

**Title**:

**Abstract:**

**April 9
**

**Mark Lewis
**

Cornell University

**Title**:

**Abstract:**

__ __

__Past__

**September 7 (Friday at 3:10pm - Carver 305 -
Colloquium / Discrete Math seminar)
**

**Shira Zerbib
**

University of Michigan

**Title**: Colorful coverings of
polytopes -- the hidden topological truth behind
different colorful phenomena

**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.