EPFL Postdoc Days

• Pierre Marion (Dynamics and learning algorithms)

Title : Two stories on deep linear networks

Abstract : Understanding the non-convex optimization dynamics of learning algorithms is a key component of modern theoretical machine learning research. In this talk, we discuss two related facets of the optimization dynamics of deep linear networks. The first result regards the maximal learning rate to ensure stable learning. We show that it is upper-bounded and explain this by a lower-bound on the sharpness of minimizers, which grows linearly with depth. Second, we study the properties of the minimizer found by gradient flow, which is the limit of gradient descent with vanishing learning rate, starting from a small-scale initialisation. We show that the learned weight matrices are approximately rank-one and that their singular vectors align. This implies an implicit regularization towards flat minima: the sharpness of the minimizer is no more than a constant times the lower bound.

• Sergej Monavari (Arithmetic geometry)

Title : A modern view on Enumerative Geometry

Abstract : Enumerative geometry deals with counting geometric objects satisfying prescribed constraints. While classically its focus was on finding explicit solutions to each counting problem via ad hoc methods, the modern point of view is to study the underlying structures of such counting problems, uncovering their deep relations with all other areas of Mathematics and Physics. In this talk, we will "walk" through a series of problems which illustrates the complexity of the subject, and how changing perspective could solve and implementing ideas from String Theory could solve a long-standing problem. 

• Barbora Hudcová (Discrete dynamical systems)

Title : Complexity and Computational Capacity of Cellular Automata

Abstract : Cellular automata are fascinating models of complex behaviour that have been widely studied ever since they were famously used by Jon Von Neumann to construct the first universal self-replicating system. In this talk, we will briefly go over some important results that have been obtained since, focusing mainly on the following question: is there a connection between systems with fascinating dynamics and those that can compute hard tasks?

• Baptiste Cerclé (Random geometry)

Title : A small promenade through two-dimensional (probabilistic) Conformal Field Theory

Abstract : Informally speaking, a Conformal Field Theory (CFT) is a model for random functions defined in the physics literature and that enjoys a very strong symmetry: conformal invariance. In order to give a mathematically rigorous meaning to such a notion, several approaches have been developed in the mathematics community, such as representation theory, vertex operator algebra, geometric quantization or probability theory. In this talk we will discuss how this last perspective allows to make sense of a celebrated CFT, Liouville theory, and sketch some of its connections with the other approaches.

• Andrew Mc Rae (Continuous optimization)

Title : Nonconvex optimization landscapes and overparametrization in statistical estimation

Abstract : In non-convex optimization, local optima can cause serious difficulties. However, for many practical problems, non-convex approaches work very well empirically. My work attempts to explain this phenomenon by showing that many problems arising in practice have “benign landscapes”: every local optimum is global (or at least is “good enough" for the application). I will present results and analysis ideas for a variety of problems in group synchronization and low-rank matrix sensing, all of which fall into the category of low-rank matrix optimization and are thus amenable to techniques from convex matrix optimization. Many of these nonconvex landscapes benefit from “overparametrization,” that is, optimizing over more degrees of freedom (in these cases determined by the matrix rank) than required by the application.

• Petru Constantinescu (Analytic number theory)

Title : Counting geodesics and binary quadratic forms: an analytic number theory viewpoint

Abstract : The study of closed geodesics on the modular surface is a rich and beautiful subject at the confluence of number theory, geometry, dynamics and quantum chaos. In this talk, I aim to provide a gentle tour of the history and briefly explain the connections with all these different areas. We start with the work of Gauss on binary quadratic forms, make a detour into hyperbolic geometry, explain how analytic number theory comes into play, and maybe end with a view on modern research.

• David Wallauch-Hajdin (PDE)

Title : Wave equations: From d’ Alembert to Blowup

Abstract : This talk will provide a short overview of some aspects of wave equations. We will begin in the year 1747 and end up somewhere in this century. Along the way, we will first encounter the linear wave equation and gradually progress to a simple nonlinear variant. On this example, some basic but interesting phenomena, which occur in the study of nonlinear evolution equations, will be discussed. Finally, we will have a look at self similar blowup and, if time permits, shortly glance at the prototypical example of a geometric wave equation.

• Domenico Valloni (Algebraic geometry) 

Title: Rational points on algebraic varieties

Abstract : In this talk I will give an introduction to the study of rational points on algebraic varieties, especially local-to-global principles. I will explain what is known in the case of curves, and then I will move on to the case of surfaces. If time permits, I will also explain how positive characteristic methods can give new insights in characteristic zero. 

• Guillaume Blanc (Random geometry)

Title : Poisson processes of lines and random geometry

Abstract : The goal of this talk is to present the Aldous-Kendall random metric model, which is constructed out of a random collection of lines with a speed limit. Picturing these as roads in Euclidean space, on which one can drive while respecting the speed limits, we are then interested in the "driving time" metric induced by this random road network. We will see that the random metric space obtained in this way has nice fractal properties, and that its geodesics are also interesting objects to study.

• Jaeyun Yi (Probability and PDE)

Title : Chaotic nature in stochastic PDEs. 

Abstract : We will introduce stochastic PDEs and then present some qualitative properties of them, such as intermittency, multifractality, and turbulence. As basic examples, we first discuss the stochastic heat equation and the Kardar-Parisi-Zhang (KPZ) equation, which describe diffusion in random environments or the growth of random surfaces. If time permits, we will also discuss stochastic PDEs arising from fluid dynamics. 

• Sebastian Schlegel Mejia (Arithmetic geometry)

Title : Hall algebras: overthinking linear algebra

Abstract : There are various algebraic invariants that can be used to study abelian categories, such as the category of vector spaces or modules over ring. A well-known invariant is the Grothendieck group, which is an abelian group, that, in some sense, is the universal additive invariant of the category (the dimension of a vector space is an example of such an invariant). However, a draw-back of the Grothendieck group it that doesn’t see non-trivial extensions of objects. 

On the other hand, the Hall algebra is another invariant of certain abelian categories which encode information about non-trivial extensions. Hall algebras have found applications in quantum groups and modern enumerative geometry. In my talk I will explain the above and introduce Hall algebras from scratch and, time permitting, explain how they were useful in my research.