Seminars

Um Esquema para Resolução de Equações Diferenciais Parciais em Domínios Discretos

Abstract: Apresentamos um método para resolução de uma classe de EDPs parabólicas em triangulações com curvatura. O nosso método se aproveita de uma técnica chamada “separação de Strang” combinada com otimização convexa. Applicamos o nosso método várias EDPs lineares e não lineares que aparecem em problems de difusão e propagação de frentes na área de processamento geométrico. Esse é um trabalho colaborativo com Oded Stein e Justin Solomon.

Live @ https://www.youtube.com/live/R3oTnrrKyFU

Matroids and Idylls

Abstract: A (classical) matroid abstracts the notion of linear independence in vector spaces. An idyll $F$ is an algebraic object that generalizes fields and hyperfields. In this talk, we present basic concepts of Baker-Bowler's theory of $F$-matroids and give examples. For instance, matroids over the Krasner hyperfield are related to vector spaces over the field with one element. We exemplify how this relation leads to a more "classical" problem. This is a joint work with Manoel Jarra and Oliver Lorscheid.

Auction Design using Value Prediction with Hallucinations

Abstract: We investigate a Bayesian mechanism design problem where a seller seeks to maximize revenue by selling an indivisible good to one of n buyers, incorporating potentially unreliable predictions (signals) of buyers’ private values derived from a machine learning model. We propose a framework where these signals are sometimes reflective of buyers’ true valuations but other times are hallucinations, which are uncorrelated with the buyers’ true valuations. Our main contribution is a characterization of the optimal auction under this framework. Our characterization establishes a near-decomposition of how to treat types above and below the signal. For the one buyer case, the seller’s optimal strategy is to post one of three fairly intuitive prices depending on the signal, which we call the “ignore”, “follow” and “cap” actions.

Beyond Nelson's Theorem

Abstract: A classic theorem by Nelson characterizes when a representation of the universal enveloping algebra on a Hilbert space by unbounded operators integrates to a representation of the corresponding simply connected Lie group: this happens if and only if a certain Laplacian in the enveloping algebra acts by an essentially self-adjoint operator. I will present a framework for generalizing this to general *-algebras. Starting with a class of "integrable" representations, the goal is to identify a C*-algebra, called *-hull, which has the same representations as the chosen integrable representations. A key result in this setting is an induction theorem, which produces a C*-hull for a graded algebra, given a C*-hull for the degree zero part. I will finish speaking about ongoing work that aims at generalizing this to enveloping algebras of Lie algebras. Here the only result that is proven concerns the tangent Lie algebroid of a simply connected manifold, where the corresponding Lie groupoid is the pair groupoid of the manifold.

Classification problems for geometric structures and relative algebroids

Abstract: I will introduce the notion of a relative algebroid to explain the appearance of Lie theory behind generic classification problems in differential geometry. We will see that this framework unifies both Lie algebroids as well as (formal) partial differential equations. I'll show through examples how these relative algebroids describe typical existence and classification problems in differential geometry, where a geometric structure is subject to a PDE with symmetries. This is joint work with Rui Loja Fernandes.

Betti maps and orbits of parabolic automorphisms of hyperkähler manifolds (joint with Serge Cantat)

Abstract: Let f be a parabolic automorphism of a hyperkahler manifold (that is, f preserves a lagrangian fibration). In a joint work with Misha Verbitsky we proved that the orbits of f are dense (in the euclidean topology) in a general fiber of p, as a consequence of Hodge theory. It turns out that (assuming Bakker's recent proof of Matsushita's conjecture) the problem can essentially be reduced to several results about the so-called Betti map, by Andre-Corvaja-Zannier-Gao and Voisin, at least in the projective case. This approach yields more precise results: for example as soon as one proves that the rank of the Betti map is maximal, one deduces not only the density of orbits in the general fiber, but also the existence of a dense set of finite orbits. I shall focus on a recent joint work with S. Cantat, where we provide a "dynamical" proof of maximality of the rank of the Betti map in the hyperkahler case. I shall also explain how the non-projective case reduces to the projective one by using degenerate twistor deformations, developed by Verbitsky and Soldatenkov.