# Scientific

## Self-organization and pattern selection in run-and-tumble processes

I will report on a simple model for collective self-organization in colonies of myxobacteria. Mechanisms include only running, to the left or to the right at fixed speed, and tumbling, with a rate depending on head-on collisions. We show that variations in the tumbling rate only can lead to the observed qualitatively different behaviors: equidistribution, rippling, and formation of aggregates. In a second part, I will discuss in somewhat more detail questions pertaining to the selection of wavenumbers in the case where ripples are formed, in particular in connection with recent progress on the marginal stability conjecture for front invasion.

## A simple stochastic model for cell population dynamics in colonic crypts

The questions of how healthy colonic crypts maintain their size under the rapid cell turnover in intestinal epithelium, and how homeostasis is disrupted by driver mutations, are central to understanding colorectal tumorigenesis. We propose a three-type stochastic branching process, which accounts for stem, transit-amplifying (TA) and fully differentiated (FD) cells, to model the dynamics of cell populations residing in colonic crypts. Our model is simple in its formulation, allowing us to estimate all but one of the model parameters from the literature. Fitting the single remaining parameter, we find that model results agree well with data from healthy human colonic crypts, capturing the considerable variance in population sizes observed experimentally. Importantly, our model predicts a steady-state population in healthy colonic crypts for relevant parameter values. We show that APC and KRAS mutations, the most significant early alterations leading to colorectal cancer, result in increased steady-state populations in mutated crypts, in agreement with experimental results. Finally, our model predicts a simple condition for unbounded growth of cells in a crypt, corresponding to colorectal malignancy. This is predicted to occur when the division rate of TA cells exceeds their differentiation rate, with implications for therapeutic cancer prevention strategies.

## Spaces of geodesic triangulations of surfaces

In 1962, Tutte proposed a simple method to produce a straight-line embedding of a planar graph in the plane, known as Tutte's spring theorem. It leads to a surprisingly simple proof of a classical theorem proved by Bloch, Connelly, and Henderson in 1984, which states that the space of geodesic triangulations of a convex polygon is contractible. In this talk, I will introduce spaces of geodesic triangulations of surfaces, review Tutte's spring theorem, and present this short proof. It time permits, I will briefly report the recent progress in identifying the homotopy types of spaces of geodesic triangulations of general surfaces.

## Around Artin's primitive root conjecture

In this talk we will first discuss this soon to be 100 years old conjecture, which states that the set of primes for which an integer \(a\) different from \(-1\) or a perfect square is a primitive root admits an asymptotic density among all primes. In 1967 Hooley proved this conjecture under the Generalized Riemann Hypothesis.

After that, we will look into a generalization of this conjecture, where we don't restrain ourselves to look for primes for which \(a\) is a primitive root but instead elements of an infinite subset of \(\mathbb{N}\) for which \(a\) is a generalized primitive root. In particular, we will take this infinite subset to be either \(\mathbb{N}\) itself or integers with few prime factors.

## On the eigenvalues of the graphs D(5,q)

In 1995, Lazebnik and Ustimenko introduced the family of q-regular graphs D(k,q), which is defined for any positive integer k and prime power q. The connected components of the graph D(k, q) have provided the best-known general lower bound on the size of a graph for any given order and girth to this day. Furthermore, Ustimenko conjectured that the second largest eigenvalue of D(k, q) is always less than or equal to 2\sqrt{q}, indicating that the graphs D(k, q) are good expanders. In this talk, we will discuss some recent progress on this conjecture. This includes the result that the second largest eigenvalue of D(5, q) is less than or equal to 2\sqrt{q} when q is an odd prime power. This is joint work with Vladislav Taranchuk.

## Bounds on the Number of Solutions to Thue Equations

In 1909, Thue proved that when $F(x,y) \in \mathbb{Z}[x,y]$ is irreducible, homogeneous, and has degree at least 3, the inequality $|F(x,y)| \leq h$ has finitely many integer-pair solutions for any positive $h$. Because of this result, the inequality $|F(x,y)| \leq h$ is known as Thue’s Inequality and much work has been done to find sharp bounds on the number of integer-pair solutions to Thue’s Inequality. In this talk, I will describe different techniques used by Akhtari and Bengoechea; Baker; Bennett; Mueller and Schmidt; Saradha and Sharma; and Thomas to make progress on this general problem. After that, I will discuss some improvements that can be made to a counting technique used in association with "the gap principle’’ and how those improvements lead to better bounds on the number of solutions to Thue’s Inequality.

## Understanding adversarial robustness via optimal transport perspective.

In this talk, I will present the recent progress of understanding adversarial multiclass classification problems, motivated by the empirical observation of the sensitivity of neural networks by small adversarial attacks. From the perspective of optimal transport theory, I will give equivalent reformulations of this problem in terms of 'generalized barycenter problems' and a family of multimarginal optimal transport problems. These new theoretical results reveal a rich geometric structure of adversarial learning problems in multiclass classification and extend recent results restricted to the binary classification setting. Furthermore, based on this optimal transport approach I will give the result of the existence of optimal robust classifiers which not only extends the binary setting to the general one but also provides shorter proof and an interpretation between adversarial training problems and related generalized barycenter problems.

## Sofic groups are surjunctive

In this talk, which is based on Benjamin Weiss' Sofic groups and dynamical systems, I will give the definition of sofic groups which can be considered as common generalizations of amenable and residually finite groups, and discuss examples and some of their basic properties. Finally, we will give Weiss' alternative proof Gromov's theorem stating that sofic groups are surjunctive. This theorem settles Gottschalk's conjecture for sofic groups.

## The eighth moment of $\Gamma_1(q)$ L-functions

In this talk, I will discuss my on-going joint work with Xiannan Li on an unconditional asymptotic formula for the eighth moment of $\Gamma_1(q)$ L-functions, which are associated with eigenforms for the congruence subgroups $\Gamma_1(q)$. Similar to a large family of Dirichlet L-functions, the family of $\Gamma_1(q)$ L-functions has a size around $q^2$ while the conductor is of size $q$. An asymptotic large sieve of the family is available by the work of Iwaniec and Xiaoqing Li, and they observed that this family of harmonics is not perfectly orthogonal. This introduces certain subtleties in our work.

## Twisted moments of characteristic polynomials of random matrices

In the late 90's, Keating and Snaith used random matrix theory to predict the exact leading terms of conjectural asymptotic formulas for all integral moments of the Riemann zeta-function. Prior to their work, no number-theoretic argument or heuristic has led to such exact predictions for all integral moments. In 2015, Conrey and Keating revisited the approach of using divisor sum heuristics to predict asymptotic formulas for moments of zeta. Essentially, their method estimates moments of zeta using lower twisted moments. In this talk, I will discuss a rigorous random matrix theory analogue of the Conrey-Keating heuristic. This is ongoing joint work with Brian Conrey.