# Colloquium

The colloquium of the Center for Mathematical Sciences, Lund University, normally runs once a month, Wednesdays from 14.15 until 15.15 in the Hörmander or Gårding lecture halls. It is aimed at the entire Centre for Mathematical Sciences with overview talks by renowned experts about exciting mathematical topics. The purpose of our colloquium is twofold: firstly, it is to provide an inspiring overview of a specific field of mathematics, secondly, it is to bring together students and staff from the entire department and to serve as the proverbial waterhole where contacts are made and maintained. For more information, see the guidelines for colloquium speakers.

The colloquium is organized by Dragi Anevski, Ida Arvidsson, Magnus Goffeng, Gustavo Jasso and Tony Stillfjord. Feel free to contact any one of us for questions or suggestions for colloquia speakers. See also the information for suggesting colloquium speakers.

## Colloquia, Spring 2024

### January 17 at MH:Gårding

#### Speaker

Elena Celledoni (Norwegian University of Science and Technology)

#### Title

Deep learning from the point of view of numerical analysis

#### Abstract

A large amount of progress made in deep learning has been based on heuristic explorations, but there is a growing effort to mathematically understand the structure in existing deep learning methods and to design new approaches preserving (geometric) structure in neural networks.

The optimal control point of view to neural networks offers an interpretation of deep learning from a dynamical systems and numerical analysis perspective and opens the way to mathematical insight.

We discuss some interesting directions of current research in structure preserving deep learning. Some deep neural networks can be designed to have desirable properties such as invertibility and group equivariance, or can be adapted to problems of manifold value data. Neural networks with inbuilt stability properties yield converging schemes when used for solving inverse problems in imaging.

### February 21 at MH:Gårding

#### Speaker

Erik Lindström (Lund University)

#### Title

Methods for the inference of stochastic processes and their applications

#### Abstract

We provide a brief introduction to stochastic dynamical systems, in particular stochastic differential equations, generalized state space models and the more recent class of sparse statistical jump models. We cover probabilistic properties and their importance for forecasting, before touching upon computational algorithms making parameter and latent state inference feasible.

The usefulness of these models will be illustrated with a range of applications including river flow, mathematical finance, the power system and natural language processing.

### March 13 at MH:Gårding

#### Speaker

Hedvig Kjellström (KTH Royal Institute of Technology)

#### Title

Embodiment of AI - why is it important and what are the implications?

#### Abstract

As brought forward already by Kant in the 1700's, our embodiment affects not only how we can perceive our environment or act upon it, but also our entire experience of the world around us. Human perception and cognition has developed together with our embodiment so as to help us survive and obtain goals in collaboration with other humans. Given the enormous capabilities of humans, it is interesting to study and take inspiration from human perception, cognition and learning when developing AI systems. This also aligns with early AI work by Hubert Dreyfus and classical computer vision approaches by David Marr.

In this talk I will go through work in my group addressing embodied AI from different perspectives, including affordances, interpreting human communication, and continuous, online learning.

### April 10 at MH:Hörmander

#### Speaker

Roghayeh Hajizadeh (Linköping University)

#### Title

Optimization of Snow Removal in Cities

#### Abstract

Removing snow in a city is an unavoidable task in Nordic countries like Sweden. A number of streets in an area need to be cleared of snow by a limited number of vehicles and the tours for the vehicles must be planned in order to minimize the time and/or cost. Since the amount of snow can vary significantly from one year to another, the plans/tours of one year cannot be used for the next year. Hence, new tours need to be planned each time.

There are different relevant specifics of the urban snow removal problem. For instance, there are different types of streets which need different numbers of sweeps in order to remove the snow. In addition, some tasks must be done before other tasks can be started. This leads to precedence constraints. Furthermore, each vehicle needs a certain time to switch from a task to another task. The problem can be formulated as a huge time-indexed mixed integer programming which often is not directly solvable in practice. Hence, we study different relaxations and heuristics to find feasible solutions and improve the bounds on the optimal objective function values.

### May 8 at MH:Gårding

#### Speaker

Aad van der Vaart (Delft University of Technology)

#### Title

Gaussian processes in Bayesian statistics: review and some recent results

#### Abstract

A nonparametric Bayesian statistical method models an infinite-dimensional unknown parameter of interest by a `prior' probability distribution and next obtains a `posterior' distribution over the unknowns using Bayes's rule. The approach is quite elegant, and popular in applied settings, for instance in inverse problems or machine learning, when it is desired to obtain not only a reconstruction (or best guess) of the unknowns, but also a quantification of the uncertainty in this reconstruction. In this talk we discuss the validity of the approach and its determinants from a non-Bayesian point of view, focusing on Gaussian processes, which are a first choice as priors for functions. We explain ways to formalise `validity', and review some theoretical results on posterior distributions resulting from such priors when used to model a regression function or density function, or a functional parameter in an inverse problem described by a differential equation. We review the role of the small ball probability of the Gaussian process as a determinant of the contraction rate of the posterior distribution, the importance of the length scale of the process, and the accuracy of credible sets for uncertainty quantification. We present recent results on approximating a posterior distribution by distributed computing, and on using linear methods to solve inverse problems resulting from some nonlinear partial differential equations.

### May 29 at MH:Hörmander

#### Speaker

Catharina Stroppel (University of Bonn)

#### Title

From Platonic solids to Springer fibers

#### Abstract

In this talk I want to give a small tour starting from Platonic solids and explain how one might naturally construct spaces/manifolds/varieties which arise in representation theory, more precisely Springer theory and sketch why representation theorists care. From that spaces we will construct a Fukaya type category. The talk is for a general audience.