The category 3Cob has closed oriented surfaces as objects and 3-dimensional cobordisms, i.e. 3-dimensional compact oriented manifolds (possibly with boundary) with canonical orientation preserving (reversing) identification of the incoming (outgoing) boundary. The composition is defined in terms of gluing. We present this category using a diagrammatic language similar to the language of standard surgery presentation of closed, orientable, connected 3-manifolds, save that besides framed links we use wedges of circles in our diagrams.
We will explain how to interpret such a diagram as an arrow of 3Cob and give an outline of the composition calculus for diagrams. This is a joint work with Jovana Nikolic and Mladen Zekic.
The four dimensional ellipsoid embedding function of a toric symplectic manifold M measures when a symplectic ellipsoid embeds into M. It generalizes the Gromov width and ball packing numbers. In 2012, McDuff and Schlenk computed this function for a ball. The function has a delicate structure known as an infinite staircase. This implies infinitely many obstructions are needed to know when an embedding can exist. Based on work with McDuff, Pires, and Weiler, we will discuss the classification of which Hirzebruch surfaces have infinite staircases. We will focus on the part of the argument where symplectic embeddings are constructed via almost toric fibrations.
We extend extreme value statistics to independent data with possibly very different distributions. In particular, we present novel asymptotic normality results for the Hill estimator, which now estimates the positive extreme value index of the average distribution. Due to the heterogeneity, the asymptotic variance can be substantially smaller than that in the i.i.d. case. As a special case, we consider a heterogeneous scales model where the asymptotic variance can be calculated explicitly. The primary tool for the proofs is the functional central limit theorem for a weighted tail empirical process. A simulation study shows the good finite-sample behavior of our limit theorems. We present an application to assess the tail heaviness of earthquake energies. This is joint work with Yi He (Univ. of Amsterdam).
Despite the non-convex optimization landscape, over-parametrized shallow networks are able to achieve global convergence under gradient descent. The picture can be radically different for narrow networks, which tend to get stuck in badly-generalizing local minima. Here we investigate the cross-over between these two regimes in the high-dimensional setting, and in particular investigate the connection between the so-called mean-field/hydrodynamic regime and the seminal approach of Saad & Solla. Focusing on the case of Gaussian data, we study the interplay between the learning rate, the time scale, and the number of hidden units in the high-dimensional dynamics of stochastic gradient descent (SGD). Our work builds on a deterministic description of SGD in high-dimensions from statistical physics, which we extend and for which we provide rigorous convergence rates.
Given a symplectic manifold, one can ask what Lagrangian submanifolds it contains. I will discuss this question for one of the simplest examples of a non-trivial symplectic manifold, namely the cotangent bundle of the 2-sphere. Specifically, I will present a result about monotone Lagrangian tori as objects in the Fukaya category. If time permits, I will also discuss the problem of classifying Lagrangian tori up to Hamiltonian isotopy.
Em poucos anos, as ferramentas de Inteligência Artificial tornaram-se essenciais para a atividade dos diversos operadores de justiça. A crescente utilização da Inteligência Artificial no contexto das profissões jurídicas trouxe também a urgência da sua regulação. A apresentação oferece um panorama dos instrumentos de regulação já existentes e em preparação no plano internacional e comparado, ao mesmo tempo que descreve algumas das mais importantes aplicações de Inteligência Artificial atualmente usadas na área do Direito.
Motivated by physics, especially string theory and quantum field theory, we take a promenade in the landscape of Calabi-Yau geometries. This subject of Calabi-Yau varieties has been a fruitful cross-fertilization between mathematics, physics and computer science. We will take a computational algebro-geometric and data science driven perspective and focus on the explicit constructions and the various databases which have emerged over the decades. Finally, we discuss some recent developments in using neural networks and machine-learning to study such geometries.
This mini-course is aimed at beginning Ph.D. students interested in the physics and mathematics of algebraic geometry. No coding background is needed. The lectures will involve some live coding demonstrations however.
The search for the Theory of Everything has led to superstring theory, which then led physics, first to algebraic/differential geometry/topology, and then to computational geometry, and now to data science. With a concrete playground of the geometric landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machine-learning can help explore problems of interest to theoretical physics and to pure mathematics. At the core of our programme is the question: how can AI help us with mathematics?