# WS 2015/16

**Unless otherwise specified, all talks take place in room 119, Arnimallee 3 and begin at 16:15.**

# Schedule:

**26 October 2015: Jan Stevens (Göteborg)**

**Stably Newton non-degenerate singularities**

**Abstract: **

We discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. The answer is negative. The easiest example is the function x^p in characteristic p. Many singularities are stably Newton non-degenerate. An analysis of our methods leads to an example where they do not work. We conjecture that this function is in fact not stably equivalent to a non-degenerate function.

** 2 November 2015: J. Maurice Rojas (Texas A & M)**

**Tropical Geometry for Exponential Sums**

**Abstract: **We give extensions of non-Archimedean Newton polygons and non-Archimedean tropical varieties to certain exponential sums. This enables very natural extensions of Viro diagrams to simple polyhedral approximations of certain real analytic sets. We also show how the A-discriminant naturally extends to certain exponential sums.

We give numerous examples along the way and assume no background in tropical geometry.

**30 November 2015: Christian Sevenheck (Chemnitz)**

**Mirror symmetry for toric manifolds and non-commutative Hodge structures**

**Abstract: **Classical mirror symmetry expresses enumerative data like Gromov-Witten invariants of certain Calabi-Yau threefolds by Hodge theoretic information like period integrals on mirror families. As soon as one moves out of the Calabi-Yau-world, such a correspondence is unlikely to exist, due to the presence of irregular singularities in the quantum differential equations. Is has been noticed since long time that the "correct" mirror object in these cases is a function on a quasi-projective variety commonly known as Landau-Ginzburg model. In the talk, I will describe how to extend the category of Hodge structure to provide a framework for a mirror symmetry statement for the cases of weakly Fano toric manifolds and smooth complete intersections inside them. The main technical tools are Gauß-Manin systems of families of Laurent polynomials and intersection cohomologies of their fibreweise compactifications.

**14 December 2015: Slivia Sabatini (Köln)**

**12, 24 and beyond: from symplectic geometry to combinatorics**

**Abstract:** 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 smooth reflexive polytopes, and prove a generalization to every dimension of the so called "12" and "24" theorem, valid respectively in dimension 2 and 3. This is joint work with Leonor Godinho (Instituto Superior Tecnico, Lisbon) and Frederik von Heymann (Universität zu Köln).

**25 January: Alexander Zheglov (Moscow)**

**Torsion free sheaves on algebraic varieties and integrable systems.**

**Abstract: **There are two classical problems, common for Algebra, Integrable systems Theory and the Theory of Partial Differential equations. They were formulated and studied for the first time already by Schur and Burchnall-Chaundy in the beginning of 20th century. These problems are the problem to produce explicitely commuting families of differential operators and the problem of classification of commutative rings of partial differential operators.

One of approaches to these problems is to study the geometric spectral data of such rings, the most important part of which are torsion free sheaves with fixed Hilbert polynomial on the spectral variety, which appears to be almost always singular algebraic variety if its dimension is greater than 1.

In my talk I'm going to give an overview of old and recent results about the spectral data and to discuss some new approaches to solution of these problems. The talk is based on joint works with H.Kurke, D. Osipov, I.Burban and A.Mironov.

01.02.2016

**Hyunsuk Moon (KAIST, Daejeon, Korea)**

**Real rank geometry of Ternary forms**

**Abstract: **We study real ternary forms whose real rank equals the generic complex rank and their algebraic boundaries. And also we characterize the semialgebraic set of sum of powers representations with that rank. This talk will be mostly about ternary forms of degree 2 to 6. This is a joint work with Mateusz Michalek, Bernd Sturmfels, and Emanuele Ventura.

**08.02.2016**

**Lars Kastner (FU-Berlin)**

**The semi-stable reduction theorem**

**Absract:** This talk aims at giving a summary of the seminar on semi-stable reduction for toric geometers. After stating and explaining the theorem, I will define toroidal embeddings and elaborate on their involvement in the proof of the theorem.