With Joris Koefler, Elke Neuhaus and Niharika Paul we are reading the first five chapters of 3264 and all that: A second course in Algebraic Geometry [EH] also following the curated notes by Rainer Sinn [S]. We meet Mondays, Wednesdays and Fridays from 1pm to 3pm in G310 starting November 21st until December 19th. The plan is flexible but we attempt to cover the following. 

1. Nov 21 - Nov 28: Background and basics led by Niharika (Chapters 1 and 2 in S). 

2. Dic 1 - Dic 3: Chow rings led by Joris (Chapter 3 in S and Chapter 1 in EH).

3. Dic 5- Dic 12: Grassmannians led by Elke (Chapter 4 in S and Chapters 3 and 4 in EH).

4. Dic 15 - Dic 19: Chern classes led by Veronica (Chapter 5 in S and EH).

Zack Greenberg is organising a Cluster Algebras course for which Shelby Cox and I are running the exercises and discussion sessions.  The discussion sessions take place from May 12th to May 23rd on Mondays and Fridays at 14:30.  See the MPI website for the detailed schedule.

Part I: Definitions

1. Cluster algebras from triangulated surfaces 

2. General cluster algebras 

3. Cluster algebras from Grassmannians

Part II: Combinatorial Models

1. Finite type classification

2. Folding

Part III: Cluster Algebras meets Tropical Geometry (Discussion sessions)

1. Introduction to (positive) tropicalizations

2. Totally positive tropical Grassmannian

3. Totally positive tropical cluster varieties 

With Anna-Laura Sattelberger we are organizing a reading group to understand the connection between representation theory and D-modules. This was an important development which allowed not only to solve open problems but also gave rise to new research directions in various fields of mathematics. For example, it helped solving the Kazhdan-Lusztig conjecture (on the side of representation theory) and pushed the study of perverse sheaves (on the side of algebraic analysis). More concretely, our main goal is to understand the Beilinson-Bernstein correspondance, which relates D-modules on a flag variety with the representations of its corresponding Lie algebra, and its consequences. We also aim to make this abstract topic approachable and hands-on by engaging with concrete examples and computations.

Main reference: Hotta, Takeuchi, Tanisaki: D-Modules, perverse sheaves and representation theory. Birkhäuser, 2008.

Format: One talk per week on Wednesdays 10am-12m and example and computations session on Wednesdays 1pm-2pm within the period March 12-April 23,2025.

Schedule: 

1. "A quick reminder on D-modules" by Anna-Laura Sattelberger on March 12 (parts of 1.1, 1.2, 1.4, 2.1, 2.2, and 2.3 in HTT)

2. "A primer on semisimple Lie Algebras and their representations" by Veronica Calvo Cortes on March 19 (9.1-9.8 in HTT)

3. "Flag varieties, Borel-Weil-Bott theorem and equivariant vector bundles" by Veronica Calvo Cortes on March 26 (9.9-9.11 in HTT)

4. "Beilinson-Bernstein correspondence" by Irina Bobrova on April 9 (Chapter 11 in HTT)

5. "Deligne’s Riemann-Hilbert correspondence" by Nicolas Weiss on April 16 (parts of Chapters 4 and 5 in HTT)

6. "D-modules of highest weight modules and Kazhdan-Lusztig polynomials" by Enya Hsiao on April 23 (Chapter 12 in HTT)

We are reading the paper "Matroids and Amplitudes" by Thomas Lam. For details see the MPI website. 

Together with Francesca Zaffalon we are in charge of Part I: Combinatorics.