Seminários, para a disseminação informal de resultados de investigação, trabalho exploratório de equipas de investigação, actividades de difusão, etc., constituem a forma mais simples de encontros num centro de investigação de matemática.

O CAMGSD regista e publica o calendário dos seus seminários há bastante tempo, servindo páginas como esta não só como um método de anúncio dessas actividades mas também como um registo histórico.

This talk focuses on generalizations of the exclusion process whose hydrodynamic limits are sub-diffusive equations. After recalling some known results in dimension 1, I will present in detail the partial exclusion process in random environment. This is a system of random walks where the random environment is obtained by assigning random maximal occupancies to each site of the Euclidean lattice. I will show that, when assuming that the maximal occupancies are heavy tailed and i.i.d., the hydrodynamic limit of the particle system (in any dimension greater than 1) is the fractional-kinetics equation.

This talk is based on partly ongoing projects in collaboration with A. Chiarini (Padova), F. Redig (TU Delft) and F. Sau (ISTA).

In this talk, a mathematically rigorous approach toward geometric supergravity will be discussed which, in the physical literature, is usually known as the Castellani-D'Auria-Fré approach. To this end, using tools from supergeometry, the notion of a super Cartan geometry will be introduced. Interestingly, in order to consistently incorporate the anticommutative nature of fermionic fields, the ordinary category of supermanifolds needs to be generalized in a physically consistent way leading to the notion of so-called enriched supermanifolds. We then apply this formalism to discuss a geometric formulation of (generalized) pure Anti-de Sitter supergravity with N=1,2 supersymmetry in D=4 modified by an additional Holst term. In this context, we will also talk about so-called picture changing operators (PCO) and how they can be implemented in a mathematically rigorous way. Finally, an outlook will be given for applications of this formalism to (loop) quantum supergravity and the description of quantum supersymmetric black holes.

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.

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.