**Convex geometry - discrete and computational**

29 June - 10 July 2015

The aim of the summer school is to introduce contemporary methods from discrete and computational convex geometry to a broad audience ranging from graduate students to postdocs and junior faculty. The central theme is the computation with (non)linear convex bodies by way of structural insights. The courses will be supplemented with problem sessions and guest lectures.

**Speakers**

**Alexander Barvinok (U Michigan): Integer points in polyhedra**

Some highlights of the course are: polyhedra, valuations, Euler characteristic and polarity; exponential valuations (discrete and continuous) as extensions of volumes and the number of integer points to unbounded polyhedra; Brion's Theorem, cone decompositions, continued fractions and their extensions; Ehrhart polynomial and its ramifications; efficient algorithms for counting integer points.

**Greg Blekherman (Georgia Tech): Convex geometry of nonnegative polynomials**

The course will be about the convex geometry of the cones of nonnegative polynomials and sums of squares. Simple geometric considerations, such as facial structure, symmetries, and structure of the dual cone will lead to new insights in real algebraic geometry.**Jesus de Loera (UC Davis): Convex geometry arising in optimization**

This course will be about the many fascinating problems in convex geometry that arise in the context of convex and discrete optimization. The following selected topics (some of many possible) will be covered: The combinatorial geometry of augmentation algorithms in optimization (e.g., geometry of the simplex method); global optimization of polynomials: integration and summation methods; Caratheodory, Helly, and Radon theorems in optimization.**Frank Vallentin (U Köln): Packings, coverings, and embeddings **

It will be shown how to use semidefinite optimization in discrete geometry. Covered topics will include: Finding upper bounds for geometric packing problems, constructing optimal lattice sphere coverings, and computing low-distortion embeddings of finite metric spaces into Euclidean spaces.

**Alexander I. Bobenko (TU Berlin): Variational methods in discrete differential geometry**

**Michael Joswig (TU Berlin): On the tropical geometry of the interior-point method in linear programming**

**Martin Grötschel (ZIB): Optimization, Applications, Convex Geometry**

**Günter M. Ziegler (FU Berlin): Polytopes - Some Examples from Berlin**

**Student talks**

Charu Goel: Some applications of sum of squares representations of even symmetric forms

Bennet Goeckner: A non-partitionable Cohen-Macaulay complex

Robert Thijs Kozma: New Density Bounds and Optimal Ball Packings for Hyperbolic Space

Zeljka Stojanac: Tensor theta norms

Zafeirakis Zafeirakopoulo: Polyhedral Omega: A linear Diophantine system solver

Judit Abardia: The role of the Rogers-Shephard inequality in the characterization of the difference body**Abstracts**

**Schedule *Update***

The first week of lectures will take place in Room H 0110, TU Berlin, Strasse des 17. Juni 135, 10623 Berlin.

The second week of lectures will take place in the Lecture Hall ZIB, Takustraße 7, 14195 Berlin.

**Registration and Application**

** Application is closed**.**Application Deadline: Friday, 13 March 2015.**

Recommendation Letter: Sunday, 15 March 2015

Applications have to include

- letter of interest

- curriculum vitae

- letter of recommendation

Each text preferably no longer than one page.

**Organizers**

Martin Henk (TU Berlin) & Raman Sanyal (FU Berlin)

Please direct scientific questions about the school to the organizers and administrative questions to This email address is being protected from spambots. You need JavaScript enabled to view it..

Supported by BMS, SFB/TRR „Discretization in Geometry and Dynamics“ and RTG „Methods for Discrete Structures“