# Mathematics of Data

**Lecturers:**Christiane Görgen and Sara Kališnik Verovšek**Date:**Wednesday 13:30 - 14:30**Room:**MPI MiS, A3 01**Target audience:**MSc students, PhD students, Postdocs**Language:**English**Keywords:**Algebraic Statistics, Topological Data Analysis**Prerequisites:**Undergraduate degree in Mathematics, linear algebra, and probability theory.**Remarks:**Almost all lectures will be self-contained.

## Abstract:

This lecture gives an introduction to how tools from algebraic geometry and topology can be used to tackle problems in data analysis and statistics.

The first half of the course will focus on what is known as *Algebraic Statistics*: see e.g., Pistone et al. (2001), Pachter and Sturmfels (2005), and Sullivant (2017). To set the scene, the first two lectures will illustrate some classical statistical theory such as linear models and Bayesian vs frequentist approaches to model selection on small-scale examples. We continue by treating topics such as exponential families and (Gaussian) graphical models and show how their properties can be naturally described using the language of (toric) ideals and varieties. We show that questions of equivalence for conditional independence models can be solved using polytopes.

The second half of the course will start with an intuitive introduction to topology, including homotopy equivalent spaces, homology groups, and homotopy groups. We will then move to the realm of data analysis: given only a dataset, i.e. a finite sampling from a space, what can we say about the space's shape (which may be reflective of patterns within the data)? To study the shape we consider different ways of building geometric objects (simplicial complexes) on point clouds and studying their properties. The main technique we cover is persistent homology; we describe its theoretical underpinnings and discuss examples of how it has been used on real-life data. Lastly, we explain how another popular method called 'mapper' works and show some of its applications.

## Regular Lectures (Winter 2017/2018)

**An introduction to rough paths**- Benjamin Fehrman
- Date: Tuesday 10:30 - 12:00, MPI MiS G3 10

**Stochastic scalar conservation laws**- Benjamin Gess
- Date: Thursday 11:15 - 12:45, MPI MiS G3 10

**Mathematics of Data**- Christiane Görgen and Sara Kališnik Verovšek
- Date: Wednesday 13:30 - 14:30, MPI MiS A3 01

**Analysis of discrete structures from the neurosciences and other fields**- Jürgen Jost
- Date: Friday 13:30 - 15:00, MPI MiS A3 01

**Riemann surfaces, Teichmüller theory and harmonic maps**- Enno Keßler and Ruijun Wu
- Date: Friday 09:30 - 11:00, MPI MiS A3 02

**Regularity theory for Optimal Transportation**- Felix Otto
- Date: Wednesday 09:15 - 11:00, MPI MiS A3 01

**Random functions on the hypercube**- Artem Sapozhnikov
- Date: Friday 15:15 - 17:00, Leipzig University SG 2-14

**Representation Theory and Complex Geometry**- Emre Sertöz and Jacinta Torres
- Date: Thursday 15:15 - 16:30, MPI MiS G3 10

**IMPRS-Ringvorlesung: Partial Differential Equations**- Benjamin Gess, Felix Otto, László Székelyhidi
- Thursday, 09:00 - 11:00, MPI MiS E1 05 (Leibniz-Saal)