Saturday, 21th of January, 2017 at TU Berlin

This conference series with a long tradition features talks in Group Theory, Geometry and Topology. It takes place twice a year, at varying locations.

There is no registration fee. Nonetheless, please register by email to Antje Schulz. Registration is mandatory for participation in the conference dinner.

Institut für Mathematik, TU Berlin, Lecture Hall: MA 004.

10:00 - 11:00

**Abstract:** We classify pairs of conjugacy classes in almost simple algebraic groups
whose product consists of finitely many classes. This leads to several
interesting families of examples which are related to a generalization of
the Baer-Suzuki theorem for finite groups. We also answer a question of
Pavel Shumyatsky on commutators of pairs of conjugacy classes in simple
algebraic groups. It turns out that the resulting examples are exactly those
for which the product also consists of only finitely many classes.
The original motivation is a conjecture of Arad and Herzog on finite
simple groups. This is joint work with R. Guralnick and P. Tiep.

11:30 - 12:30

**Abstract:** Mathematics finds itself divided and subdivided into hyper-specialized areas of study, each of them with its own internal beauty. However, what I find most fascinating is when one can build a bridge between two of these seemingly isolated theories.
For instance, symplectic geometry and combinatorics have a very strong connection, due to the existence of Hamiltonian torus actions. Such actions come with a map, called moment map, which "transforms" a compact symplectic manifold into a
convex polytope. Hence many combinatorial properties of (some special types of) polytopes can be studied using symplectic techniques.
In this talk I will focus on reflexive polytopes, and explain the so called "12" and "24" phenomenon in dimension 2 and 3 and its generalizations using symplectic geometry.

14:30 - 15:30

**Abstract:** The famous André-Bruck-Bose construction allows one to
represent any translation plane in a projective space (over a division
ring), through the observation that its kernel is a division ring. It is
a long-standing conjecture that the same result is true for *translation
generalized quadrangles*, the 4-gonal analogues of projective planes in
the theory of buildings. Translation generalized quadrangles that
satisfy the conjecture are usually called *linear*.
In this talk, we survey some of the known results on this conjecture,
and we include results of the speaker which were written up only
recently. One of these results says that any translation generalized
quadrangle can be ideally embedded in a translation quadrangle which
itself is linear, so which can be represented in a projective space over
a division ring. It follows that there is a well-defined notion of
“characteristic” for these objects. One can then show that each
translation quadrangle in positive characteristic indeed *is* linear.

16:00 - 17:00

**Abstract:** Chern-Schwarz-Macpherson (CSM) classes are one way to extend the notion of Chern classes to singular and non-complete varieties. Matroids are an abstraction of the notion of independence in mathematics. In this talk, I will provide a combinatorial analogue of these classes for matroids, motivated by the situation for hyperplane arrangements. In this setting, CSM classes are polyhedral fans which are Minkowski weights. One goal in doing this is to express matroid invariants such as, the characteristic polynomial, h-vector, and conjecturally Speyer's g-polynomial, as invariants from algebraic geometry. But also these combinatorial CSM classes can be used to study the complexity of more general objects such as subdivisions of matroid polytopes and tropical varieties.
This is based on joint work with Lucia Lopez de Medrano and Felipe Rincon and also work in progress with Alex Fink and David Speyer.

At 18:00 we will have a joint dinner (self-paid) at a restaurant near the conference venue. Pre-registration is mandatory.

**Organizers:** Michael
Joswig and Linus Kramer.