## PRACTICAL INFORMATIONS

• The Graduate Colloquium will take place from November 24 on the afternoon to Friday 26 in the morning.
• All talks will take place at the Faculty of Science of the University of Neuchâtel, 11 Rue Émile-Argand, 2000 Neuchâtel.
• We will have a social dinner on Thursday 25 November.
• Talks should last 45 minutes, and be aimed at PhD Students in any area of mathematical research. Typically, this means focusing on explaining the general context of a problem rather than giving its proof.
• Contact for the organizers: [email protected]

## Abstracts

#### Alexander Kolpakov: Space vectors forming rational angles

We classify all sets of nonzero vectors in $\mathbb{R}^3$ such that the angle formed by each pair is a rational multiple of $\pi$. The special case of four-element subsets lets us classify all tetrahedra whose dihedral angles are multiples of $\pi$, solving a 1976 problem of Conway and Jones: there are $2$ one-parameter families and $59$ sporadic tetrahedra, all but three of which are related to either the icosidodecahedron or the $B_3$ root lattice. The proof requires the solution in roots of unity of a $W(D_6)$-symmetric polynomial equation with $105$ monomials (the previous record was only $12$ monomials).

This is joint work with Kiran S. Kedlaya, Bjorn Poonen, and Michael Rubinstein.

#### Giulia Gaggero: Multivariate cryptography

In this talk, I will give an outlook of what multivariate cryptography is. First of all, we look at the concept of digital signature. Then we recall some algebraic tools, further, we look at the main invariants that are used in the crypto-analysis of a multivariate digital signature scheme. Given a system in a polynomial ring with $n$ variables over a field $K$,  we introduce the solving degree of the system and, in order to estimate it, we look at the Castelnuovo-Mumford regularity of the ideal generated by the system, the last fall degree and the degree of regularity of the system.

#### Joe Brendel: From Archimedes to exotic Lagrangian submanifolds

I will start by a very beautiful (and very old) observation by Archimedes: Slices of equal height on a two-sphere have equal surface area. This will serve as my excuse to tell you some things about symplectic geometry and Hamiltonian dynamics and draw many pictures. If time permits, I will mention some recent progress on classification questions of Lagrangian submanifolds in symplectic geometry.

#### Leonard Tschanz: The Steklov problem on graphs with boundary

In this presentation, I will introduce the Steklov problem on graphs with boundary and give an example of question that I am interested in.  After that, I will speak about a technique allowing us to associate a manifold to the graph and a technique to associate a graph to a manifold and use them in order to answer the question.

#### Livio Ferretti: On finitely generated subgroups of the mapping class group

This talk will be a gentle introduction to some topics in the theory of mapping class groups, i.e. the group of isotopy classes of self-diffeomorphisms of an orientable surface. After recalling the basics, we will concentrate in particular on subgroups generated by finitely many Dehn twists, discuss some known results and open problems and see how those groups naturally arise in geometrically interesting contexts

#### Rik Voorhaar: Randomized algorithms: A revolution in numerical linear algebra

Over the past decade randomized algorithms have emerged in numerical linear algebra as a faster and more parallelizable way to perform many important operations in linear algebra, such as solving linear systems or computing matrix decompositions. Virtually all of numerical mathematics is based around numerical linear algebra, therefore this development can have a tremendous impact on the efficiency of numerical algorithms in general.
I will explain in particular how randomized algorithms can speed up low-rank matrix decompositions. I will show that despite the fact that these algorithms are randomized, we can still derive very good performance guarantees.

#### Corentin Bodart: Groups with and without rational cross-sections

Given a group $G$ and a finite generating set $S$, we know all elements of $G$ can be written as words over the alphabet $S$. Something we might want is some "nice" subset of special words such that each element of the group correspond to a unique special word. Such subsets are what we call rational cross-sections.

#### Antoine Bourquin: How can random switching between environments be so fun?

In this talk, we will present a model that takes into account the random switching between some environments. For example, switching between a cold and a warm environment. One of our goals will be to present some counter-intuitive results when the switches occur. In particular, depending of the switch, it's not always the same species that will die...

#### Yacine Aoun: Will your message get to Joe Biden ?

Suppose that you have an important information that you would like the president of USA to know. You have two choices: either you try to contact him directly (which has a very low probability of happening) or you send your message to somebody else (a friend, a Swiss politician…) who will try to reach Joe Biden for you. Which option shall you choose to have a higher probability of getting your message through ? In order to tackle this problem, I will use a well-established model of statistical physics: Bernoulli percolation. I will define the model, review some basic results and then present some new results that will provide an answer for the question in the title (i.e. I will provide sharp asymptotics of the two-point connectivity function in the subcritical regime).
Joint work with Dmitry Ioffe, Sébastien Ott and Yvan Velenik.

#### Jérémy Colombo: A brief overview on Markov Chains and some properties

Let’s discover Markov Chain! We will start with basic discrete Markov Chains and their properties on a finite state space, supported and motivated by some out-of-the-box examples. Then we will focus on continuous Markov Chains and their properties on an infinite state space, always supported by direct examples. We will finish with a large class of Markov Chains motivated by Stochastic Differential Equations, and we will see how this kind of Markov Chains brings new problems (and how we can hope to solve them).

#### Flavio Salizzoni: Sum-rank metric codes

The sum-rank metric is recently drawing a lot of attention among the Coding Theory's community, especially since many applications have been discovered over the last decade (reliable and secure multishot network coding, rate-diversity optimal space-time codes, and PMDS codes for repair in distributed storage). Moreover, the sum-rank metric is a natural generalization of both the Hamming and the rank metric and so it allows us to find a link between these two theories. In fact, many properties of these two well-known metrics can be extended to the sum-rank case. In this talk, we will introduce all these metrics, we will discuss their main properties and we will give a complete characterization of the optimal anticodes in the sum-rank metric case.

#### Arina Voorhaar: On the Newton Polytope of the Morse Discriminant

A famous classical result by Gelfand, Kapranov and Zelevinsky provides a combinatorial description of the vertices of the Newton polytope of the A-discriminant (the closure of the set of all non-smooth hypersurfaces defined by polynomials with the given support A). Namely, it gives a surjection from the set of all convex triangulations of the convex hull of the set A with vertices in A (or, equivalently, the set of all possible combinatorial types of smooth tropical hypersurfaces defined by tropical polynomials with support A) onto the set of vertices of this Newton polytope. In my talk, I will discuss a similar problem for the Morse discriminant — the closure of the set of all polynomials with the given support A which are non-Morse if viewed as polynomial maps. Namely, for a 1-dimensional support set A, there is a surjection from the set of all possible combinatorial types of so-called Morse tropical polynomials onto the vertices of the Newton polytope of the Morse discriminant.

#### Damaris Meier: Uniformization of metric surfaces

The classical uniformization theorem states that every simply connected Riemann surface is conformally diffeomorphic to $D$, $\mathbb{C}$ or $S^2$. The uniformization problem for metric spaces now asks to find conditions on a given metric space $X$, homeomorphic to some model space, under which there exists a homeomorphism from the model space to $X$ with good geometric and analytic properties. Towards the existence of such a map under minimal conditions on a two-dimensional metric surface $X$, we will encounter three different uniformization results and consider applications, for example in geometric group theory.