### Beyond uniform hyperbolicity 2019 specification - lecture 2

Vaughn Climenhaga
Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to specification" using a decomposition of the space of finite-length orbit segments, and then survey...

### Rough volatility from an affine point of view

Christa Cuchiero
We represent Hawkes process and their Volterra long term limits, which have recently been used as rough variance processes, as functionals of infinite dimensional affine Markov processes. The representations lead to several new views on affine Volterra processes considered by Abi-Jaber, Larsson and Pulido. We also discuss possible extensions to rough covariance modeling via Volterra Wishart processes. The talk is based on joint work with Josef Teichmann.

### L-space knots in twist families and satellite L-space knots

Kimihiko Motegi
Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $n > 0$ or $g(K_n) / g_4(K_n) \to 1$ as $n \to \infty$, then either the winding number of $K$ about...

### An introduction to BSDE

Peter Imkeller
Backward stochastic differential equations have been a very successful and active tool for stochastic finance and insurance for some decades. More generally they serve as a central method in applications of control theory in many areas. We introduce BSDE by looking at a simple utility optimization problem in financial stochastics. We shall derive an important class of BSDE by applying the martingale optimality principle to solve an optimal investment problem for a financial agent whose...

### Distributions of Frobenius of elliptic curves #3

Nathan Jones
In the 1970s, S. Lang and H. Trotter developed a probabilistic model which led them to their conjectures on distributional aspects of Frobenius in $GL_2$-extensions. These conjectures, which are still open, have been a significant source of stimulation for modern research in arithmetic geometry. The present lectures will provide a detailed exposition of the Lang-Trotter conjectures, as well as a partial survey of some known results. Various questions in number theory may be viewed in...

### Additive combinatorics methods in fractal geometry - lecture 2

Peter Varju
In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter.

### Data-driven wildfire behavior modelling: focus on front level-set data assimilation

Mélanie Rochoux
A front data assimilation system named FIREFLY has been developed at CERFACS in collaboration with the University of Maryland to better estimate the environmental conditions (biomass properties, near-surface wind). We discuss the sequential application of the ensemble Kalman filter (EnKF) in FIREFLY for correcting in a spatially-distributed way, input parameters in order to better track the fire front position. In particular, using a polynomial chaos surrogate to mimic the wildfire spread model in the EnKF...

### Linear and fractional response: a survey

When a dynamical system admitting a natural (SRB) measure is perturbed, it is natural to ask how the SRB measure responds to the perturbation. In the tamest cases, this response is linear, and the derivative of the SRB measure with respect to the parameter can be expressed as a sum of decorrelations (involving the derivative of the system with respect to the parameter). In more subtle situations - for example, systems with bifurcations, or observables...

### A classification of gapped Hamiltonians in $d=1$

Sven Bachmann
A quantum phase transition is commonly referred to as a point in a family of gapped Hamiltonians where the spectral gap closes. In the absence of a general perturbation theory for quantum spin systems in the thermodynamic limit, I will discuss necessary, and sufficient, conditions for a transition, and present explicit constructions of paths of uniformly gapped Hamiltonians in one dimension.

### Multi-time distribution of periodic TASEP

Jinho Baik
We consider periodic TASEP with periodic step initial condition, and evaluate the joint distribution of the locations of m particles. For arbitrary indices and times, we find a formula for the multi-time, multi-space joint distribution in terms of an integral of a Fredholm determinant. We then discuss the large time limit in the so-called relaxation scale. The one-point distributions for other initial conditions are also going to discussed. Based on joint work with Zhipeng Liu...

### Automorphism groups of low complexity subshift - Lecture 2

Samuel Petite
An automorphism of a subshift $X$ is a self-homeomorphism of $X$ that commutes with the shift map. The study of these automorphisms started at the very beginning of the symbolic dynamics. For instance, the well known Curtis-Hedlund-Lyndon theorem asserts that each automorphism is a cellular automaton. The set of automorphisms forms a countable group that may be very complicated for mixing shift of finite type (SFT). The study of this group for low complexity subshifts...

### Logique épistémique dynamique

François Schwarzentruber
On introduira la logique modale épistémique avec des exemples (enfants sales, etc.). On abordera la notion de logique modale et de structure de Kripke. On évoquera le problème de satisfiabilité et la méthode de tableau. Ensuite, nous verrons comment mettre à jour un modèle de Kripke. Nous verrons d'abord les annonces publiques. Puis nous verrons comment modéliser quelques actes de communications à l'aide des modèles d'actions. Une démonstration des enfants sales et du puzzle des...

### Traffic flow models with non-local flux and extensions to networks

Simone Göttlich
We present a Godunov type numerical scheme for a class of scalar conservation laws with nonlocal flux arising for example in traffic flow modeling. The scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme and also allows to show well-posedness of the model. In a second step, we consider the extension of the non-local traffic flow model to road networks by defining appropriate conditions at junctions. Based on the proposed numerical scheme...

### Betti Langlands in genus one

We will report on an ongoing project to understand geometric Langlands in genus one, in particular a version that depends only on the topology of the curve (as appears in physical descriptions of the subject). The emphasis will be on the realization of the automorphic and spectral categories as the center/cocenter of the affine Hecke category. We will mention work with D. Ben-Zvi and A. Preygel that accomplishes this on the spectral side, then focus...

### Large-time behavior in (hypo)coercive ODE-systems and kinetic models

Anton Arnold
In this talk we discuss the convergence to equilibrium in conservative-dissipative ODE-systems, kinetic relaxation models (of BGK-type), and Fokker-Planck equation. This will include symmetric, non-symmetric and hypocoercive evolution equations. A main focus will be on deriving sharp decay rates. We shall start with hypocoercivity in ODE systems, with the ”hypocoercivity index” characterizing its structural complexity. BGK equations are kinetic transport equations with a relaxation operator that drives the phase space distribution towards the spatially local...

### Beyond Bowen’s specification property (II) - lecture 1

Daniel J. Thompson
These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand,...

### Delocalization of eigenfunctions and quantum chaos - lecture 2

Nalini Anantharaman

### The uniformization of the moduli space of abelian 6-folds

Gavril Farkas
The general principally polarized abelian variety of dimension at most five is known to be a Prym variety. This reduces the study of abelian varieties of small dimension to the beautifully concrete theory of algebraic curves. I will discuss recent breakthrough on finding a structure theorem for principally polarized abelian varieties of dimension six as Prym-Tyurin varieties associated to covers with $E_6$-monodromy, and the implications this uniformization result has on the geometry of the moduli...

### On the proximity of additive and multiplicative functions

Jean-Marie De Koninck
Given an additive function $f$ and a multiplicative function $g$, let $E(f,g;x)=\#\left \{ n\leq x:f(n)=g(n) \right \}$ We study the size of $E(f,g;x)$ for those functions $f$ and $g$ such that $f(n)\neq g(n)$ for at least one value of $n> 1$. In particular, when $f(n)=\omega (n)$ , the number of distinct prime factors of $n$ , we show that for any $\varepsilon >0$ , there exists a multiplicative function $g$ such that \$E(\varepsilon ,g;x)\gg \frac{x}{\left...

### Graphons and graphexes as limits of sparse graphs - lecture 2

Christian Borgs
Graphons and graphexes are limits of graphs which allow us to model and estimate properties of large-scale networks. In this pair of talks, we review the theory of dense graph limits, and give two alterative theories for limits of sparse graphs – one leading to un- bounded graphons over probability spaces, and the other leading to bounded graphons (and graphexes) over sigma-finite measure spaces. Talk I, given by Jennifer, will review the general theory, highlight...

### Expected topology of a random subcomplex in a simplicial complex

Jean-Yves Welschinger
I will explain how to bound from above and below the expected Betti numbers of a random subcomplex in a simplicial complex and get asymptotic results under infinitely many barycentric subdivisions. This is a joint work with Nermin Salepci. It complements previous joint works with Damien Gayet on random topology.

### Introduction

Pierre Pudlo
​L'intérêt pour l'intelligence artificielle (IA) s'est considérablement accru ces dernières années et l'IA a été appliquée avec succès à des problèmes de société. Le Big Data, le recueil et l’analyse des données, la statistique se penchent sur l’amélioration de la société de demain. Big Data en santé publique, dans le domaine de la justice pénale, de la sécurité aéroportuaire, des changements climatiques, de la protection des espèces en voie de disparition, etc. ​ ​C'est sur...

### The H-Principle and Turbulence

László Székelyhidi
It is well known since the pioneering work of Scheffer and Shnirelman that weak solutions of the incompressible Euler equations exhibit a wild behaviour, which is very different from that of classical solutions. Nevertheless, weak solutions in three space dimensions have been studied in connection with a long-standing conjecture of Lars Onsager from 1949 concerning anomalous dissipation and, more generally, because of their possible relevance to the K41 theory of turbulence. In recent joint work...

### Algorithmic aspects of embeddability: higher-dimensional analogues of graph planarity

Uli Wagner

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