# Paolo Perrone: Category theory and applications

**Lecturer:**Paolo Perrone**Date:**Wednesday 11:00 - 12:30**Room:**MPI MiS, A3 02**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Keywords:**Categories, functors, universal properties, applications**Prerequisites:**Basic mathematical knowledge (linear algebra, basic analysis)**Remarks:**Participants can give a lecture too (if they want).

## Abstract

This course is meant as an introduction to the concepts and methods of category theory from a rather *applied* point of view. We will review the basic ideas, such as:

- Categories, functors, natural transformations. What it means for a construction to be "functorial", or "natural", and why it is important;
- The Yoneda lemma, a.k.a. how objects are determined purely by the way they interact with one another. How to reason in terms of diagrams, and what why it is helpful;
- Universal properties, limits, colimits, why they are everywhere, and how they generalize constrained optimization.

Category theory has been central to the development of fields such as algebraic topology, algebraic geometry, and group theory. It has also many applications to general topology, analysis, and geometry. Recently, it has been applied to some subjects of more "applied" flavor, such as:

- Networks and complexity;
- Graphical models and statistical inference;
- Data structures and classes in computer science;
- ...and many more.

Depending on the interests of the participants, we will choose one or two of these particular applications to study in detail.

## Regular Lectures (Summer 2019)

**Artiﬁcial Neural Networks and Machine Learning: Theoretical Foundations II**- Nihat Ay
- Thursday, 11:15 - 12:45, MPI MiS A3 01

**Embedding theorems in complex geometry**- Judith Brinkschulte
- Tuesday, 11:15 - 12:45, University Augusteum A-314

**Continuum Mechanics**- Jan Burczak
- Wednesday, 9:15 - 10:45 and Friday, 9:15 - 10:45, University Augusteum A-314

**Modern Algebraic Geometry**- Claus Diem
- Monday, 11:15 - 12:45, University SG 3-13 and Friday, 13:15 - 14.45, University Augusteum A-314

**Introduction to Computer Algebra**- Christian Eder
- Wednesday 13:15 - 14:45, University HS-19 and Thursday 9:15 - 10:45, University SG 3-20

**Enumerative geometry**- Francesco Galuppi
- Wednesday, 09:30 - 11:00, and Thursday, 13:30 - 15:00, MPI MiS G3 10

**Stochastic variational inequalities**- Benjamin Gess
- Monday, 16:15 - 17:45, MPI MiS A3 01

**Schemes and Sheaves**- Jürgen Jost
- Monday, 13:30 - 15:00, MPI MiS A3 01

**Information and Complexity**- Jürgen Jost
- Friday, 13:30 - 15:00, MPI MiS A3 02

**Randomness in PDE–selected topics**- Felix Otto
- Thursday, 9:15 - 10:45, MPI MiS A3 01, starting May 16

**Category theory and applications**- Paolo Perrone
- Wednesday, 11:00 - 12:30, MPI MiS A3 02

**Introduction to Microlocal Analysis**- Angkana Rüland
- Thursday, 11:15 - 12:45, University, SG 311

**Introduction to percolation theory**- Artem Sapozhnikov
- Thursday, 13:15 - 14:45, University, Augusteum A 314

**Hodge Theory and Periods of Varieties**- Emre Sertöz
- May 6 - 10, MPI MiS G3 10, details see extra page

**Introduction to D-modules**- Christian Sevenheck
- Friday, 11:00 - 12:30, MPI MiS G3 10

**Invitation to Nonlinear Algebra**- Mateusz Michałek, Bernd Sturmfels
- June 4 - 14, MPI MiS, details see extra page

**Population Dynamics**- Florian Völlering
- Wednesday and Friday 9:00 - 11:00, University, SG 3-13

**IMPRS-Ringvorlesung**- Felix Otto, Peter Stadler (lectures), Tobias Ried (exercises)
- Lectures: Tuesday, 10:15 - 11:45, MPI MiS G3 10
- Exercises: Thursday, 15:30 - 17:00 (at 04.04.: 13:00 - 14:00), MPI MiS G3 10