Multiplicities is a broad audience seminar geared towards transversal topics whose aim is to highlight the diversity of mathematics and mathematicians.

It is especially intended for PhD students, interested in widening their horizons and is supported by the PRIDE Grant “Geometry, Probability and their Synergies (GPS)”.

## Calendar

April 2, 2019

Béatrice de Tilière, The Z-invariant Ising model via dimers

June 18, 2019

Diana Davis, Periodic paths on the pentagon

May 12, 2020

Olga Paris-Romaskevich, Triangle tiling billiards, interval exchange transformations with flips and movies

Mars 23, 2021

Persi Diaconis, Adding numbers and shuffling cards

May 11, 2021

Allison Henrich, Knotty Games

June 15, 2021

Lisa Sauermann, On the cap-set problem and the slice rank polynomial method

### Béatrice de Tilière

#### The Z-invariant Ising model via dimers

April 1, 2019    16:00 - 16:50
MNO 1.020, Campus Belval

Abstract. The Ising model is one of the most classical models of statistical mechanics. We shall present its relation to dimers, its Z-invariant version and some results obtained with C. Boutillier and K. Raschel.

### Diana Davis

#### Periodic paths on the pentagon

June 18, 2019    16:30 - 17:20
MNO 1.020, Campus Belval

Abstract. Mathematicians have long understood periodic trajectories on the square billiard table, which occur when the slope of the trajectory is rational. In this talk, I'll explain my joint work with Samuel Lelièvre on periodic trajectories on the regular pentagon, describing their geometry, symbolic dynamics, and group structure. The periodic trajectories are very beautiful, and some of them exhibit a surprising "dense but not equidistributed" behavior.

#### Triangle tiling billiards, interval exchange transformations with flips and movies

May 12, 2020  Postponed
MNO 1.020, Campus Belval

Abstract. T.B.A.

### Persi Diaconis

#### Adding numbers and shuffling cards

March 23, 2021    16:00 - 16:50
Zoom meeting

Abstract. When numbers are added in the usual way, carries'' appear along the way. If you are me (and hopefully you after this talk) it's natural to ask: How do the carries go? How many are there and, if you just had a carry, is it more or less likely that the next sum has a carry?'' Strangely, carries form a Markov chain with an amazing'' transition matrix. More strangely, this same matrix appears in the analysis of the usual way of riffle shuffling cards. I will explain the connection, tell you about the seven shuffles theorem'' and attempt to do it in English''.

### Alison Henrich

#### Knotty Games

May 11, 2021    16:00 - 16:50
Zoom meeting

Abstract. Far too many people in this world are under the mistaken impression that math can't be fun. The aim of much of my research with student and faculty collaborators, in part, is to provide yet another counterexample to this claim. Our work combines a delightfully visual mathematical subject, knot theory, with one of the most common sources of fun: games. Motivated by the fascinating work of Ayaka Shimizu on an unknotting operation called the region crossing change and research of Ryo Hanaki on unusual types of knot diagrams called pseudodiagrams, we have invented and explored several knot games. In this talk, we will play these games on knot diagrams, developing both our spatial intuition and our understanding of the structure of knots along the way.

De la recherche en mathématiques au lycée? L'exposé se poursuivra par une table ronde d'une heure (en français) consacrée à l'introduction de projets de recherche en mathématiques avec les élèves et étudiant·e·s. Cette table ronde est organisée dans le cadre d'une formation IFEN.

### Lisa Sauermann

#### On the cap-set problem and the slice rank polynomial method

June 15, 2021    16:00 - 16:50
Zoom meeting

Abstract. In 2016, Ellenberg and Gijswijt made a breakthrough on the famous cap-set problem, which asks about the maximum size of a subset of $$\mathbb{F}_3^n$$ not containing a three-term arithmetic progression. They proved that any such set has size at most $$2.756^n$$. Their proof was later reformulated by Tao, introducing what is now called the slice rank polynomial method. This talk will discuss Tao's proof of the Ellenberg-Gijswijt bound for the cap-set problem, as well as further applications of Tao's slice rank polynomial method.