**Michael Joswig**

*Technische Universität Berlin, Berlin, Germany*

**Title**: Optimization and Tropical Combinatorics

**Abstract**: We explore several classical problems in combinatorial optimization and how they can benefit from being viewed through the lens of tropical geometry. The includes parameterized versions of the shortest-path problem, applications to linear programming and phylogenetics.

**Greg Blekherman**

*Georgia Institute of Technology, Atlanta, USA*

**Title**: Algebraic and Convex Geometry of Sums of Squares on Varieties

**Abstract**: A polynomial with real coefficients is called nonnegative if it takes only nonnegative values. For example, any sum of squares of polynomials is obviously nonnegative. The study of the relationship between nonnegative polynomials and sums of squares is a classical area in real algebraic geometry. The lectures will be about the convex cones of nonnegative polynomials and sums of squares on a variety. Convex-geometric considerations will lead to new insights in algebraic geometry. The main questions we will consider are: when are all nonnegative polynomials sums of squares, and the number of squares needed to write a sum of squares. I will also introduce applications in matrix completion and optimizati

**Omid Amini**

*École Polytechnique, Palaiseau, France*

**Title**: Geometry of tropical varieties with a view toward applications

**Abstract**: These lectures provide an introduction to tropical geometry and the interactions between combinatorics and algebraic geometry, with focus on combinatorial aspects of Hodge theory.