Events

One of the most fundamental examples of non-linear dynamics is given by the class of unimodal interval maps. It is the simplest setting in which one can study the behavior of a critical orbit and the profound impact it has on the geometry of the system. By the works of Sullivan, McMullen and Lyubich, we have a complete renormalization theory for these maps, and as a result, their dynamics is now very well understood. In this talk, we discuss the extension of this theory to a higher dimensional setting--namely, to properly dissipative diffeomorphisms in dimension two. Using the notion of non-uniform partial hyperbolicity, we identify what it means for such maps to be "unimodal." Then we show that properly dissipative infinitely renormalizable unimodal diffeomorphisms have a priori bounds (a certain uniform control on their geometry that holds at arbitrarily small scales). This is based on a joint work with S. Crovisier, M . Lyubich and E. Pujals.

Let D be a smooth rational ample divisor in a smooth projective surface X. In this talk, we will present a simple uniform recursive formula for (primary) Gromov-Witten invariants of O_X(-D). The recursive formula can be used to determine such invariants for all genera once some initial data is known. The proof relies on a correspondence between all-genus Gromov–Witten invariants and refined Donaldson–Thomas invariants of acyclic quivers. In particular, the corresponding BPS invariants are expressed in terms of Betti numbers of moduli spaces of quiver representations. This is a joint work with Pierrick Bousseau.

Motivated by crowd-sourcing applications, we consider a model where we have partial observations from a bivariate isotonic n x d matrix with an unknown permutation \pi^* acting on its rows. Focusing on the twin problems of recovering the permutation \pi^* and estimating the unknown matrix, we introduce a polynomial-time procedure achieving the minimax risk for these two problems, this for all possible values of n, d and all sampling efforts. Along the way, we establish that in some regimes, recovering the permutation \pi^* is considerably simpler than estimating the matrix.

Optimal mass transportation is a versatile tool to prove sharp geometric inequalities in Euclidean spaces. This method works also in the case of more general metric-measure spaces satisfying a curvature condition. In this talk I will present some recent results in the setting of CD(0,N) spaces satisfying a volume growth condition at infinity. This is a joint work with Alexandru Kristaly and Francesca Tripaldi.

This cycle of talks wants to highlight how ideas from tropical geometry have contributed not only to the solution, but also to the development of enumerative geometric problems regarding moduli spaces of curves, and maps from curves to curves. We will spend a little of time reviewing the origins of this story, i.e. the development of tropical Hurwitz numbers as combinatorial analogues for the classical Hurwitz numbers. We will discuss a more recent interpretation that views tropical Hurwitz numbers as the natural computation for the intersection number of the double ramification cycle with an element of the log Chow ring of the moduli space of curves (called in this case the branch polynomial, as it is presented as a piecewise polynomial function on the moduli spaces of tropical curves) which is determined by the tropical moduli space of covers of the projective line. We will see that from the tropical perspective analogous piecewise polynomial functions may be associated to $k$-DR cycles (cycles arising from spaces of twisted pluri-differentials), thus giving rise to $k$-analogues of Hurwitz numbers (called leaky Hurwitz numbers) that enjoy many of the algebro-combinatorial properties of Hurwitz numbers - such as piecewise polynomiality and wall crossings. We will present some work in progress which intends to incorporate descendants into these pictures. Tropical algorithms are developed that give rise to some intruiguingly simple formulas in the case when one point is fully ramified. The material presented is based on many years of joint work with several people, including Paul Johnson, Hannah Markwig, Dhruv Ranganathan and Johannes Schmitt.

Sigma invariants are geometric invariants associated to a finitely generated group that can be used to determine the homological and homotopical finiteness properties of coabelian subgroups. The Sigma invariants for right angled Artin groups have been computed by Meier, Meinert and VanWyck. In this talk we will see how to generalize their computations to some Artin groups. This is a joint work with Ruben Blasco, José Ignacio Cogolludo and Marcos Escartín.

This talk is devoted to the study of the equation $u u_x - u_{yy}=f$ in the domain $(x_0,x_1)\times (-1,1)$, in the vicinity of the shear flow profile $u(x,y)=y$. This equation serves as a toy model for more complicated fluid equations such as the Prandtl system. The difficulty lies in the fact that we are interested in changing sign solutions. Hence the equation is forward parabolic in the region where $u>0$, and backward parabolic in the region $u<0$. The line $u=0$ is a free boundary and an unknown of the problem. Unexpectedly, we prove that even when the data (i.e. the source term $f$ or the boundary data) are smooth, existence of strong solutions of the equation fails in general. This phenomenon is already present at the linear level, and linked to the existence of singular profiles for the homogeneous linearized equation. In fact, we prove that strong solutions exist (both for the linearized and for the nonlinear system) if and only if the data satisfy a finite number of orthogonality conditions, whose purpose is to avoid the presence of singular profiles in the solution. A key difficulty of our work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow. This is a joint work with Frédéric Marbach and Jean Rax.

Let us consider metric spaces in which the triangular inequality is strengthened to d(x,z) ≤ max{d(x,y), d(y,z)}, that is, ultrametric spaces. These spaces, interesting in themselves, play an important role in the study of the geometry at infinity of various mathematical objects. The topological properties induced by the strengthened triangle inequality do not facilitate the study of the analytic properties of these spaces. The aim of this colloquium is to give an introduction to Berkovich's analytification, which provides a way to study the geometric properties of ultrametric spaces by analytical means.

As a part of Langlands duality, certain equations were found in two different areas of mathematics. They are known as Baxter’s TQ systems and the QQ type systems, as they trace back to Baxter’s study on integrable models in the 1970s. During the same decade, similar systems of equations were discovered in the area of ordinary differential equations (ODE) by Sibuya, Voros and others. Today, this remarkable correspondence is realized as a duality between representation theory of nontwisted quantum affine algebras (QAA) and the theory of opers for their Langlands dual Lie algebras. We are interested in this duality when the roles of the affine Lie algebra and its dual are exchanged. When the nontwisted QAA is of type BCFG, its dual will be a twisted QAA. To exchange their roles amounts to studying representations of twisted QAAs. In this talk, we will begin by reviewing this story. We will explain the representation theory of Borel subalgebras of twisted QAAs and the expected relationship between twisted and nontwisted types. We will establish TQ systems and QQ^~ systems for twisted QAAs.

We consider axisymmetric solutions without swirl of the 3D Navier-Stokes equations originating from circular vortex filaments at initial time. In the case of a single filament, we construct an asymptotic expansion of the viscous vortex ring in the high Reynolds number regime, where the kinematic viscosity is small compared to the circulation of the vortex. We then show that the unique solution of the axisymmetric Navier-Stokes equations remains close to our approximation over a long time interval, during which the vortex ring moves along its symmetry axis at a speed that was predicted by Kelvin in 1867. To prove that, we introduce self-similar variables located at the (unknown) position of the ring, and we control the evolution of the perturbations using an energy functional related to Arnold's variational characterization of steady states for the 2D Euler equations. This talk is based on joint work with Vladimir Sverak.

Around 1996 Yoccoz and Birkeland introduced an endogenous lagged integral equation to explain the approximately periodic but chaotic lemming population fluctuations in the Arctic ecosystem. The evidence of cyclical but unpredictable behavior of fluctuations is also a well-known feature of livestock product prices (hog cycle) that has attracted the attention of economists for a long time. I will discuss deterministic and random versions of the population-market model proposed by Arlot, Marmi, and Papini in Arlot et al. (2019). The model is obtained by coupling the Yoccoz–Birkeland integral equation with a demand- and supply-dependent price dynamics as in Bélair and Mackey (1989). In the random model, we introduce a stochastic component into the price equation inspired by the Black-Scholes market model and prove the existence of a random attractor and a random invariant measure. We numerically compute the fractal dimension and the entropy of the random attractor and prove its convergence to the deterministic one when the volatility of the market equation tends to zero. We also numerically analyze in detail the dependence of the attractor on the choice of the temporal discretization parameter. We implement several statistical distances to quantify the similarity between the attractors of the discretized systems and the original ones. In particular, following a work by Cuturi (2013), we use the Sinkhorn distance. This distance is a discrete and penalized version of the optimal transport distance between two measures, given a transportation cost matrix. The work on the random dynamics and the investigation of the dependence on the discretization is joint with R. Ceccon and G. Livieri.

The intersection theory of the moduli space of K3 surfaces polarized by a lattice is a subject of recent interest because of its deep connections with a wide variety of mathematics, including the intersection theory of moduli spaces of curves and the study of modular forms. Oprea and Pandharipande conjectured that the tautological rings of these moduli spaces of K3 surfaces are highly structured in a way that mirrors the picture for the moduli space of curves. I will explain how to compute the Chow ring of the moduli space of degree 2 quasipolarized K3 surfaces, which consequently proves the conjecture in this case. This is joint work with Dragos Oprea and Rahul Pandharipande.

We consider tautological bundles and their exterior and symmetric powers over the Quot scheme of zero dimensional quotients over the projective line. We prove several results regarding the vanishing of their higher cohomology, and we describe the spaces of global sections via tautological constructions. This is based on joint work with Alina Marian, Shubham Sinha and Steven Sam.