# Scientific

## The size function for imaginary cyclic sextic fields

The size function $h^0$ for a number field is analogous to the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. Van der Geer and Schoof conjectured that $h^0$ attains its maximum at the trivial class of Arakelov divisors if that field is Galois over $\mathbb{Q}$ or over an imaginary quadratic field. This conjecture was proved for all number fields with the unit group of rank $0$ and $1$, and also for cyclic cubic fields which have unit group of rank two. In this talk, we will discuss the main idea to prove that the conjecture also holds for totally imaginary cyclic sextic fields, another class of number fields with unit group of rank two. This is joint work with Peng Tian and Amy Feaver.

## Mixing times and representation theory.

Consider a poker game where you have to mix the deck of cards between two turns. How (many times) should you shuffle it to prevent any cheating? In this talk we will introduce the theory of mixing times, and explain how representation theory can be used to study card mixing and diffusions on other objects.

## Möbius function, an identity factory with applications

By using an identity relating a sum to an integral, we obtain a family of identities for the averages \(M(X)=\sum_{n\leq X} \mu(n)\) and \(m(X)=\sum_{n\leq X} \mu(n)/n\). Further, by choosing some specific families, we study two summatory functions related to the Möbius function, \(\mu(n)\), namely \(\sum_{n\leq X} \mu(n)/n^s\) and \(\sum{n\leq X} \mu(n)/n^s \log(X/n)\), where \(s\) is a complex number and \(\Re s >0\). We also explore some applications and examples when s is real. (joint work with O. Ramaré)

## Unbalanced Optimal Transport: Convex Relaxation and Dynamic Perspectives

I will try to present an overview of some results of unbalanced optimal transport for positive measures with different total masses, showing the crucial role of the so-called cone representation and of the corresponding homogeneous marginals. The cone perspective naturally arises in the convex-relaxation approach to optimal transport; in the more specific case of the Hellinger-Kantorovich (aka Fisher-Rao) metric, it provides a natural tool for representing solutions of the dual dynamical formulation via Hamilton-Jacobi equations, and it is very useful for studying the geodesic convexity of entropy type functionals. (In collaboration with M. Liero, A. Mielke, G. Sodini)

## 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.