### L'échantillonnage compressé

Gabriel Peyré
A l'occasion du centenaire de la naissance de Claude Shannon, la SMF, la SMAI et le CIRM organisent, à l'issue de la conférence SIGMA, une après-midi d'exposés grand public autour de l'oeuvre scientifique de Claude Shannon, de la théorie de l'information et de ses applications.

### Regularity of the optimal sets for spectral functionals. Part I: sum of eigenvalues

Susanna Terracini
In this talk we deal with the regularity of optimal sets for a shape optimization problem involving a combination of eigenvalues, under a fixed volume constraints. As a model problem, consider $\min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\ |\Omega|=1\Big\},$ where $\langle_i(\cdot)$ denotes the eigenvalues of the Dirichlet Laplacian and $|\cdot|$ the $d$-dimensional Lebesgue measure. We prove that any minimizer $_{opt}$ has a regular part of the topological boundary which is relatively open and $C^{\infty}$ and...

### Multiple ergodic theorems: old and new - Lecture 2

Bryna Kra
The classic mean ergodic theorem has been extended in numerous ways: multiple averages, polynomial iterates, weighted averages, along with combinations of these extensions. I will give an overview of these advances and the different techniques that have been used, focusing on convergence results and what can be said about the limits.

### Möbius randomness and dynamics six years later

Peter Sarnak
There have many developments on the disjointness conjecture of the Möbius (and related) function to topologically deterministic sequences. We review some of these highlighting some related arithmetical questions.

### Collective dynamics in life sciences - Lecture 1. Collective dynamics and self-organization in biological systems: challenges and some examples

Pierre Degond
Lecture 1. Collective dynamics and self-organization in biological systems : challenges and some examples. Lecture 2. The Vicsek model as a paradigm for self-organization : from particles to fluid via kinetic descriptions Lecture 3. Phase transitions in the Vicsek model : mathematical analyses in the kinetic framework.

### Spacetime replication of continuous-variable quantum information

Barry Sanders
Combining the relativistic speed limit on transmitting information with linearity and unitarity of quantum mechanics leads to a relativistic extension of the no-cloning principle called spacetime replication of quantum information. We introduce continuous-variable spacetime-replication protocols, expressed in a Gaussian-state basis, that build on novel homologically constructed continuous-variable quantum error correcting codes. Compared to qubit encoding, our continuous-variable solution requires half as many shares per encoded system. We show an explicit construction for the five-mode case...

### The unitary extension principle on LCA groups

Ole Christensen
The unitary extension principle (UEP) by Ron & Shen yields a convenient way of constructing tight wavelet frames in L2(R). Since its publication in 1997 several generalizations and reformulations have been obtained, and it has been proved that the UEP has important applications within image processing. In the talk we will present a recent extension of the UEP to the setting of generalized shift-invariant systems on R (or more generally, on any locally compact abelian...

### Crystalline cohomology, period maps, and applications to K3 surfaces

Christian Liedtke
I will first introduce K3 surfaces and determine their algebraic deRham cohomology. Next, we will see that crystalline cohomology (no prior knowledge assumed) is the "right" replacement for singular cohomology in positive characteristic. Then, we will look at one particular class of K3 surfaces more closely, namely, supersingular K3 surfaces. These have Picard rank 22 (note: in characteristic zero, at most rank 20 is possible) and form 9-dimensional moduli spaces. For supersingular K3 surfaces, we...

### Loop Grassmanians and local spaces

Ivan Mirkovic
The loop Grassmannians of reductive groups will be reconsidered as a construction in the setting of “local spaces” over a curve. The notion of a local space is a version of the fundamental structure of a factorization space introduced and developed by Beilinson and Drinfeld. The weakening of the requirements formalizes some well-known examples of “almost factorization spaces.” The change of emphases leads to new constructions. The main example will be generalizations of loop Grassmannians...

### Geometric Langlands correspondence and topological field theory - Part 2

David Ben-Zvi
Kapustin and Witten introduced a powerful perspective on the geometric Langlands correspondence as an aspect of electric-magnetic duality in four dimensional gauge theory. While the familiar (de Rham) correspondence is best seen as a statement in conformal field theory, much of the structure can be seen in the simpler (Betti) setting of topological field theory using Lurie's proof of the Cobordism Hypothesis. In these lectures I will explain this perspective and illustrate its applications to...

Fabrice Voitus

### Lipschitz embedding of complex surfaces

Walter Neumann
Pham and Teissier showed in the late 60’s that any two plane curve germs with the same outer Lipschitz geometry have equivalent embeddings into $\mathbb{C}^2$. We consider to what extent the same holds in higher dimensions, giving examples of normal surface singularities which have the same topology and outer Lipschitz geometry but whose embeddings into $\mathbb{C}^3$ are topologically inequivalent. Joint work with Anne Pichon. Keywords: bilipschitz - Lipschitz geometry - normal surface singularity - Zariski...

### Continuous (semi-)frames revisited

Jean-Pierre Antoine
We start by recalling the essential features of frames, both discrete and continuous, with some emphasis on the notion of frame duality. Then we turn to generalizations, namely upper and lower semi-frames, and their duality. Next we consider arbitrary measurable maps and examine the standard operators, analysis, synthesis and frame operators, and study their properties. Finally we analyze the recent notion of reproducing pairs. In view of their duality structure, we introduce two natural partial...

### Wavelet-based multifractal analysis of dynamic infrared thermograms and X-ray mammograms to assist in early breast cancer diagnosis

Alain Arneodo
Breast cancer is the most common type of cancer among women and despite recent advances in the medical field, there are still some inherent limitations in the currently used screening techniques. The radiological interpretation of X-ray mammograms often leads to over-diagnosis and, as a consequence, to unnecessary traumatic and painful biopsies. First we use the 1D Wavelet Transform Modulus Maxima (WTMM) method to reveal changes in skin temperature dynamics of women breasts with and without...

### Integrable probability - Lecture 3

Ivan Corwin
A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent...

### Arc spaces and singularities in the minimal model program - Lecture 2

Tommaso De Fernex
The space of formal arcs of an algebraic variety carries part of the information encoded in a resolution of singularities. This series of lectures addresses this fact from two perspectives. In the first two lectures, we focus on the topology of the space of arcs, proving Kolchin's irreducibility theorem and discussing the Nash problem on families of arcs through the singularities of the variety; recent results on this problem are proved in the second lecture....

### Hodge theory and syzygies of the Jacobian ideal

Alexandru Dimca
Let $f$ be a homogeneous polynomial, defining a principal Zariski open set $D(f)$ in some complex projective space $\mathbb{P}^n$ and a Milnor fiber $F(f)$ in the affine space $\mathbb{C}^{n+1}$. Let$f_0, . . . , f_n$ denote the partial derivatives of $f$ with respect to $x_0, . . . , x_n$ and consider syzygies $a_0f_0 + a_1f1 + a_nf_n = 0$, where $a_j$ are homogeneous polynomials of the same degree $k$. Using the mixed Hodge structure...

### Vertex degrees in planar maps

Michael Drmota
We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even infinitely)-dimensional central limit theorem....

### Beyond Endoscopy and elliptic terms in the trace formula

James Greig Arthur
Beyond endoscopy is the strategy put forward by Langlands for applying the trace formula to the general principle of functoriality. Subsequent papers by Langlands (one in collaboration with Frenkel and Ngo), together with more recent papers by Altug, have refined the strategy. They all emphasize the importance of understanding the elliptic terms on the geometric side of the trace formula. We shall discuss the general strategy, and how it pertains to these terms.

Uli Wagner

### Twisted equivariant $\mathrm{K}$-theory and topological phases

Yosuke Kubota
The classification of topological phases in each Altland-Zirnbauer symmetry class is related to one of 2 complex or 8 real $\mathrm{K}$-theory by Kitaev. A more general framework, in which we deal with systems with an arbitrary symmetry of quantum mechanics specified by Wigner’s theorem, is introduced by Freed and Moore by using a generalization of twisted $\mathrm{K}$-theory. In this talk, we introduce the definition of twisted $\mathrm{K}$-theory in the sense of Freed-Moore for $C^²$-algebras, which...

### Markov Chain Monte Carlo Methods - Part 1

Christian P. Robert
In this short course, we recall the basics of Markov chain Monte Carlo (Gibbs & Metropolis sampelrs) along with the most recent developments like Hamiltonian Monte Carlo, Rao-Blackwellisation, divide & conquer strategies, pseudo-marginal and other noisy versions. We also cover the specific approximate method of ABC that is currently used in many fields to handle complex models in manageable conditions, from the original motivation in population genetics to the several reinterpretations of the approach found...

### Parametrized model order reduction for component-to-system synthesis

Anthony Patera
Parametrized PDE (Partial Differential Equation) Apps are PDE solvers which satisfy stringent per-query performance requirements: less-than or approximate 5-second problem specification time; less-than or approximate 5-second problem solution time, field and outputs; less-than or approximate 5% solution error, specified metrics; less-than or approximate 5-second solution visualization time. Parametrized PDE apps are relevant in many-query, real-time, and interactive contexts such as design, parameter estimation, monitoring, and education. In this talk we describe and demonstrate a PDE...

Frédéric Nataf

### Unramified graph covers of finite degree

Winnie Li
Given a finite connected undirected graph $X$, its fundamental group plays the role of the absolute Galois group of $X$. The familiar Galois theory holds in this setting. In this talk we shall discuss graph theoretical counter parts of several important theorems for number fields. Topics include (a) Determination, up to equivalence, of unramified normal covers of $X$ of given degree, (b) Criteria for Sunada equivalence, (c) Chebotarev density theorem. This is a joint work...

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