GRK Retreat 2025 Schedule

Location:

Talks:

Splitting Problem of Vector Bundles

1. Splitting topological vector bundles (Fabian Rodatz)
2. Interlude: projective modules (Marcus Zibrowius)
3. Splitting algebraic vector bundles (Tariq Syed)
4. Moral support: Jan Hennig

Abstract: Can we split off some trivial component from any vector bundle? We will present a leisurely introduction to the concept of vector bundles, both in topology and in algebraic geometry, present precise variations of the above question, and illustrate how homotopy theory can help provide partial answers.

Chow Groups

1. Intersection Theory (Pengcheng Zhang)
2. Chern Classes (Alexander Ziegler)
3. Slides


4. Abstract: Given two algebraic curves in the projective plane, Bézouts theorem predicts the number of their intersection points, assuming that the intersections are always transversal. One can ask this question for the nature of intersections of subvarieties in an ambient variety X generally, and the natural setting for this type of problem is the so-called Chow ring CH*(X) of X. We will introduce the Chow ring with a view towards intersection theory and then explore distinguished elements of CH*(X) called Chern classes. The Chern classes will give us an idea of why we should think of this topic as algebraic topology.

Algebraic Geometry: A Journey Through History

1. Birth (Daan van Sonsbeek)
2. Early developments (Dzoara Nunez)
3. Towards schemes (Vivien Picard)

Abstract: Algebraic geometry is one of the deepest and broadest fields of study within the realm of algebra. In this presentation, we go on a journey through its history: from the early, more concrete geometry of the Greeks, all the way to the abstract yet elegant language of schemes.

Rationality à la Igusa

1. (Vicente Monreal)
2. Generalizing Igusa's rationality result (Immi Halupczok)
3. Definable sets in ℚ_p (Blaise Boissonneau)

Abstract: Counting the number of solutions modulo p^m of a polynomial equation f=0 with integer coefficients leads to the notion of the Poincaré series of f. Igusa showed that this series is a rational function. Later, this result was greatly generalized using model theoretic methods. We will give an introduction to this topic.

Zeta Functions

1. Introduction to zeta functions (Clotilde Gauthier)
2. The Hasse-Weil zeta function for groups (Max Gheorghiu)
3. The probabilistic zeta function (Alena Ramona Meyer)

Abstract: We give an introduction to zeta functions and their applications in different areas of group theory. In the first talk, we define the subgroup zeta function and the representation zeta function, and examine their role in detecting polynomial subgroup/representation growth. The second talk will focus on the Hasse-Weil zeta functions, which are a tool to count some particular representations over finite fields. Lastly, in the third talk, we look at the probabilistic zeta function and show how it can be used in probabilistic group theory and to obtain information about the structure of the group.

Gong Talks:

1. On bases of Chow groups of flag varieties (Erik Barinaga)
2. What is the length of a potato? (Immi Halupczok)
3. Counting conjectures (Gaëtan Mancini)
4. Canonical models of algebraic groups (Otto Overkamp)
5. Filtered Techniques for Noncommutative Rings (Julian Reichardt)
6. Chow Rings of Fano Quiver Modulis (Pengcheng Zhang)
7. Everything you do has already been done (Blaise Boissonneau)

Poster Session:

1. On Chow groups of Deligne--Lusztig varieties (Erik Barinaga)
2. Motives of Nullcones of Quiver Representations (Lydia Gösmann, Markus Reineke)
3. A double coset zeta function (Doris Grothusmann)
4. The Alperin–McKay Conjecture (Gaëtan Mancini)
5. Classification of Horikawa surfaces with T-singularities (Vicente Monreal)