This page lists mini-courses organized by the VSM as well as other types of lectures (with or without ECTS) particularly designed for student members of the VSM.

## Special Lectures

**Mini-Course: A gentle introduction to geometric measure theory (with Fabian Rupp, University of Vienna – June 24 to 28, 2024)**

**Abstract**

Please find the abstract here.

**Time slots**Monday, June 24, 11.30 am to 1 pm, SR4, Oskar-Morgenstern-Platz 1 (OMP1)

Tuesday, June 25, 11.30 am to 1 pm, SR4, OMP1

Wednesday, June 26, 11.30 am to 1 pm, SR4, OMP1

**Thursday, June 27, 11.30 am to 1 pm, SR4, OMP1**

Friday, June 28, 11.30 am to 1 pm, SR4, OMP1

**Mini-Course: Introduction to random planar geometry (with William Da Silva, University of Vienna – April 22 to 26, 2024)**

**Abstract**

The study of random geometry originates from theoretical physics where it appeared in the pionneering work of Polyakov as a model of string theory. It roughly comes down to the seemingly naive question: How to pick a surface at random?

The aim of this mini-course is to present some important ideas and powerful tools in random geometry. We will focus on random plane trees and their scaling limits. One of the objectives along the way is to provide the basics of excursion theory and to describe some properties of the continuum random tree.

If time permits, we will see how some useful knowledge on trees can be leveraged to approach planar maps.

**Time slots**Monday, April 22, 9.45 to 11.15 am, SR4, Oskar-Morgenstern-Platz 1 (OMP1)

Tuesday, April 23, 9.45 to 11.15 am, SR4, OMP1

Wednesday, April 24, 9.45 to 11.15 am, SR4, OMP1

**Thursday, April 25, 9.45 to 11.15 am, SR6, OMP1**

Friday, April 26, 9.45 to 11.15 am, SR4, OMP1

**Mini-Course: Functions of bounded variation and their applications (with Wojciech Górny, University of Vienna – January 22 to 26, 2024)**

**Abstract**Variational problems with linear growth are currently the

subject of intense mathematical research and play a distinguished role

in applications. In particular, they commonly arise in image processing

and denoising, in materials science, and in phase transitions. They are

also connected with geometric problems involving the minimisation of the

perimeter, including geometric curvature flows.

The mathematical treatment of linear-growth functionals requires setting

the problem in the space of functions of bounded variation, and

discussing weak normal traces, and weak integration by parts formulas.

The goal of this course is to provide an introduction to these

functional analytical tools, and to presenting some applications. In

particular, we will discuss some examples of minimisations and

degenerate partial differential equations including the

Rudin-Osher-Fatemi model of image denoising, the least gradient problem,

and the total variation flow.

**Lecture 1:**Functions of bounded variation and their properties**Lecture 2:**Compactness of BV functions; first look at their use in

variational problems**Lecture 3:**Anzellotti pairings and weak integration by parts formulas**Lecture 4:**Total variation denoising**Lecture 5:**Least gradient problem and total variation flow

**Time slots**Monday, January 22, 11.30 am to 1 pm, SR12, Oskar-Morgenstern-Platz 1 (OMP1)

Tuesday, January 23, 11.30 am to 1 pm, SR6, OMP1

Wednesday, January 24, 11.30 am to 1 pm, SR6, OMP1

**Thursday, January 25, 11.30 am to 1 pm, SR3, OMP1**

Friday, January 26, 11.30 am to 1 pm, SR12, OMP1

**Mini-Course: Introduction to Optimal Transport (with Chiara Rigoni, University of Vienna – October 17 to 24, 2023)**

**Abstract**

The main goal of this course is to provide the participants with the techniques necessary to understand the current research in optimal transport, and the tools which are most useful for its applications.

We will start by introducing the Kantorovich and Monge problems and we will then discuss some issues related to the existence of optimal transport maps. Thus we will introduce the Wasserstein spaces and present some applications of this theory (e.g., in metric geometry- by introducing the notion of synthetic curvature for metric measure spaces- and in the theory of Mean Field Games). If time permits, we could also discuss how these tools can be applied in the theory of evolution of PDEs, in particular in the theory of gradient flows.

**Time slots**Tuesday, October 17, 9.45 to 11.15 am, SR6, Oskar-Morgenstern-Platz 1 (OMP1)

Thursday, October 19, 9.45 to 11.15 am, SR13, OMP1

Friday, October 20, 9.45 to 11.15 am, SR6, OMP1

**Monday, October 23, 9.45 to 11.15 am, SR6, OMP1**

Tuesday, October 24, 9.45 to 11.15 am, SR6, OMP1

**Mini-Course: Approximation of functions from scattered data (with Frank Filbir, Helmholtz Munich & Technische Universität München – June 26 to 30, 2023)**

**Abstract**Please find the abstract here.

**Time slots**

Monday, June 26, 9.45 to 11.15 am, SR6, Oskar-Morgenstern-Platz 1 (OMP1)

Tuesday, June 27, 9.45 to 11.15 am, SR6, OMP1

Wednesday, June 28, 9.45 to 11.15 am, SR3, OMP1

Thursday, June 29, 9.45 to 11.15 am, SR6, OMP1

Friday, June 30, 9.45 to 11.15 am, SR6, OMP1

**Mini-Course: Introduction to mathematics of statistical learning theory (with Daniel Bartl, University of Vienna – May 2 to 5, 2023)**

**Abstract**

Statistical learning theory is a framework for machine learning that plays a key role in many areas of science, such as finance, health

management, and industry. We introduce the basic concepts and discuss them with examples. Participants will be equipped with the essential statistical techniques needed to develop an understanding in which cases learning procedures can be successful, and in which cases phenomena such as overlearning might occur.

**Time slots**

Tuesday, May 2, 9.45 to 11.15 am, SR1, Oskar-Morgenstern-Platz 1 (OMP1)

Tuesday, May 2, 11.30 am to 13.00 pm, SR1, OMP1

Wednesday, May 3, 9.45 to 11.15 am, SR3, OMP1

Thursday, May 4, 9.45 to 11.15 am, SR3, OMP1

Friday, May 5, 9.45 to 11.15 am, SR6, OMP1

**Mini-Course: String Theory (with Pavel Safronov, The University of Edinburgh – April 25 to April 29, 2022)**

**Abstract**String theory has its origin in physics as an attempt to provide a theory of quantum gravity. However, it has also greatly influenced pure mathematics from algebraic geometry (mirror symmetry) to representation theory. In this week-long mini-course I will present an overview of what superstring theory is about and describe some interesting mathematics behind anomaly cancellation, topological strings and some other topics.

**Time slots**

Monday, April 25, 9.45 to 11.15, SR3, Oskar-Morgenstern-Platz 1 (OMP1)

Tuesday, April 26, 9.45 to 11.15 am, SR6, OMP1

Wednesday, April 27, 11.30 am to 1.00 pm, SR5, OMP1

Thursday, April 28, 11.30 am to 1.00 pm, SR3, OMP1

Friday, April 29, 9.45 to 11.15 am, SR6, OMP1

Further information and materials for preparing before the mini-course can be found here.

***

**Mini-Course: A Gentle Introduction to Metric Geometry (with Chiara Rigoni, University of Vienna) – March 29 to April 7, 2022**

**Abstract**Aim of this short course is to introduce the main concepts and ideas that lead to the definition of Alexandrov spaces. More in detail, we will:

• introduce the definition of metric space, focusing our attention on examples and first properties of these spaces;

• study the length structures induced by metrics;

• quantify the distance between two metric spaces, introducing the so-called Gromov-Hausdorff distance;

• find a way to speak about lower bounds on the curvature of metric spaces and present the definition of Alexandrov spaces.

**Time slots**

Tuesday, March 29, 9.45 to 11.15 am; SR 15; Oskar-Morgenstern-Platz 1 (OMP1)

Thursday, March 31, 9.45 to 11.15 am, SR 15, OMP1

Friday, April 1, 1.15 to 2.45 pm, SR 15, OMP1

Tuesday, April 5, 9.45 to 11.15 am, SR 10, OMP1

Thursday, April 7, 9.45 to 11.15 am, SR 10, OMP1

***

**Mini-Course: Cross-diffusion systems (with Ayman Moussa, Sorbonne Université) – ****April 1 to April 5, 2019**

**Abstract**Cross-diffusion systems were introduced 40 years ago as a tool of modeling in the context of population dynamics. At the mathematical level, these systems happen to be surprisingly hard to study ; global (weak) solutions for the most simple of those were (only) built in 2006 even though they have been studied since the eighties.

We will start this lecture with a short introduction to present the model of Shigesada-Kawasaki-Teramoto (SKT, the first instance of cross-diffusion in the literature) and present (without proof) the existing results and the open questions related to it. Then, in the first part of the lecture, we will focus on the Kolmogorov equation, studying existence and uniqueness with the help of the duality lemma. In the second part of the lecture we will establish a global existence result for SKT systems, using a generalization of the entropy introduced by Chen and Jüngel in 2006, the duality lemma and a specific approximation procedure.

***

**Mini-Course: Evolutionary equations with singular nonlinear terms (with Giulio Schimperna, Università di Pavia) – March 25 to March 28, 2019**

**Abstract**

After recalling a number of basic notions from convex analysis (maximal monotone operators, subdifferentials), I will focus on some mathematical aspects of evolutionary partial differential equations characterized by the presence of singular (i.e., very fastly growing) terms. I will discuss existence of solutions, regularity properties, uniqueness (or non-uniqueness), and long-time behavior.

Moreover, I will try to illustrate the theory by applying the results to some specific problems related to physical models from the theory of phase transitions.

***

**Mini-Course: Combinatoris, groups and algebraic varieties (with Fernando Rodriguez Villegas, ICTP Trieste) – November 5 to November 14, 2018**

**Abstract**

In this course we will discuss how techniques from enumerative combinatorics and the representation theory of finite groups can be used to study the geometry of certain algebraic varieties. Examples will include some character varieties as well as some related quiver varieties. We will introduce all the basic theory needed from scratch.

**Speaker**

Professor Fernando Rodriguez Villegas is a Senior Research Scientist at the Abdus Salam International Centre for Theoretical Physics (ICTP) in Trieste, Italy. His research interests lies in number theory, in the study of L-functions (particularly those coming from hypergeometric motives), character varieties, combinatorics of quiver representation, etc.

***

**Mini-Course: Spatial Discretizations of Generic Dynamical Systems (with Pierre-Antoine Guihéneuf, IMJ-PRG, Paris 6) – November 20 to November 24, 2017**

**Abstract:** How is it possible to read dynamical properties of a system on numerical simulations? The computer working with finite numerical precision, it replaces the dynamics f by a spatial discretization f_N (where stands for the numerical accuracy, e.g. the number of binary digits). We will be interested in the dynamical behaviour of the finite maps f_N for a generic system f and N going to infinity (mainly among sets of homeomorphisms or C^{^1}-diffeomorphisms). These lectures will be enlightened by numerical simulations, and will also be the occasion to understand some tools of generic dynamics.

***

**Mini-Course: Topological Methods in Aperiodicity (with James J. Walton) – June 7 to June 9, 2017**

***

**Mini-Course: An Introduction to Noncommutative Geometry and Quantum Groups (with Réamonn Ó Buachalla) – May 29 to June 2, 2017**

***

**Mini-Course: A tale of two phenomena: Fixed Point Property and Minimal Dynamical Systems – May 16 to May 19, 2017**

***

**Mini-Course: Dynamics of the unimodal interval family (with Ana Anusic) – May 3 to May 5, 2017**

***

**Mini-Course: Permutation groups and transformation semi-groups (with Prof. Peter Cameron) – March 20 to March 24, 2017**

#### VO Theory of Distributions

Giovanni di Fratta, Dirk Praetorius / TU Wien

#### VO Non-local operators

Markus Faustmann / TU Wien