## April – Mai 2018

Staged trees are a new and exciting class of statistical models that generalise the well known Bayesian Networks. In this talk I will explain the algebraic and statistical properties of these models and characterise the case in which they are toric varieties.

Maximum entropy probability distributions are important for information theory and relate directly to exponential families in statistics. Having the property of maximizing entropy can be used to define a discrete analogue of the classical continuous Gaussian distribution. We present a parametrization of such a density using the Riemann Theta function, use it to derive fundamental properties and exhibit strong connections to the study of abelian varieties in algebraic geometry. This is joint work with Carlos Améndola (TU Munich).

In this seminar, I am going to talk mostly about Ollivier (Coarse) Ricci curvature in metric measure spaces and specifically graphs. Before coming to that point, I describe some intuitions behind this notion which comes from ricci curvature in Riemannian manifolds. Although as one of the discrete generalizations of ricci curvature , Ollivier type keeps fewer properties of Riemannian manifolds , it is somewhat simpler to present and has wide range of examples. This curvature, as an edge-based measure, is a part of relatively new approach for analyzing networks and can encode some important properties of the network.

This seminar is a requisitory introduction of an ongoing project under the supervision of Professor Jost.

Vortrag in der Reihe: Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

We prove well-posedness for singular semilinear SPDEs

on a smooth bounded domain $D$ in $mathbb{R}^n$ of the form

[

dX(t) + AX(t),dt + beta(X(t)),dt

i B(t,X(t)),dW(t),, qquad X(0)=X_0,.

]
The linear part is associated to a

linear coercive maximal monotone operator $A$ on $L^2(D)$, while

$beta$ is a (multivalued) maximal monotone graph

everywhere defined on $mathbb{R}$

on which no growth nor smoothness conditions are required. Moreover,

the noise is given by a cylindrical Wiener process on a Hilbert space $U$,

with a stochastic integrand $B$

taking values in the Hilbert-Schmidt operators from $U$ to $L^2(D)$:

classical Lipschitz-continuity hypotheses for the diffusion coefficient are assumed.

A comparison with the corresponding deterministic equation

and possible generalizations are discussed.

This study is based on a joint work with Carlo Marinelli (University College London).

We study Bayesian networks based on max-linear structural equations as introduced by Gissibl and Klüppelberg (2015) and provide a summary of their independence properties. In particular we emphasize that distributions for such networks are never faithful to the independence model determined by their associated directed acyclic graph unless the latter is a polytree, in which case they are always faithful. In addition, we consider some of the basic issues of estimation and discuss generalized maximum likelihood estimation of the coefficients, using the concept of a generalized likelihood ratio for non-dominated families as introduced by Kiefer and Wolfowitz. Particular emphasis will be placed on the use of max-times algebra in the formulation and analysis of such models. The lecture is based on joint work with Gissibl and Klüppelberg.

Neural networks can be trained to perform functions, such as classifying images. The usual description of this process involves keywords like neural architecture, activation function, cost function, back propagation, training data, weights and biases, and weight-tying.

In this talk we will define a symmetric monoidal category Learn, in which objects are sets and morphisms are roughly “functions that adapt to training data”. The back propagation algorithm can then be viewed as a strong monoidal functor from a category of parameterized functions between Euclidean spaces to our category Learn.

This presentation is algebraic, not algorithmic; in particular it does not give immediate insight into improving the speed or accuracy of neural networks. The point of the talk is simply to articulate the

various structures that one observes in this subject—including all the keywords mentioned above—and thereby get a categorical foothold for further study. For example, by articulating the structure in this way, we find functorial connections to the well-established category of lenses in database theory and the much more recent category of compositional economic games.

Moves on manifolds

A classical question in topological combinatorics asks if there exists a (finite) set of moves that suffices to relate any two triangulations of a manifold with each other. In the first part of this talk, I will provide a survey of classical results that are known for manifolds with and without boundary, which includes theorems by Alexander and Pachner. In the second part, I want to focus on so-called balanced triangulations, which are triangulations admitting a minimal vertex coloring . For the case of balanced manifolds without boundary, Izmestiev, Klee and Novik recently constructed a set of moves, so-called cross-flips, meeting the above conditions. For the case of balanced manifolds with boundary, we show that any two such PL homeomorphic manifolds can be connected by a sequence of shellings and inverses such that balancedness is preserved in each step.

Any unknown notions will be explained during the talk.This is joint work with Lorenzo Venturello.

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY