|10:00 to 11:00||Tony Yue Yu (Caltech, USA)||
Generalizing GKZ secondary fan using Berkovich geometry
Gelfand-Kapranov-Zelevinski introduced the notion of secondary fan in the study of the Newton polytopes of discriminants and resultants. It also controls the geometric invariant theory for toric varieties. We propose a generalization of the GKZ secondary fan to general Fano varieties using ideas from Berkovich geometry and Mori theory. Furthermore, inspired by mirror symmetry, we propose a synthetic construction of a universal family of Kollár-Shepherd-Barron-Alexeev stable pairs over the toric variety associated to the generalized secondary fan. This generalizes the families of Kapranov-Sturmfels-Zelevinski and Alexeev in the toric case. We gave a detailed construction and proved the stability in the case of del Pezzo surfaces. This is joint work with Hacking and Keel.
|11:30 to 12:30||Piotr Tourkine (LAPTh, France)||
The tropical limit of string theory and Feynman integrals
Besides its goal as a theory of quantum gravity, from the quantum field theory (QFT) standpoint, string theory is a remarkable mathematical tool that allows to compute certain quantities in QFT (that is, roughly, particle’s physics) more efficiently than using standard QFT approach.
In this talk, I will describe how scattering amplitudes in QFT, which are quantities that characterise the interactions between particles, are recovered using the so called tropical limit of string theory scattering amplitudes. The idea, which has been known since the onset of string theory, is that when strings become point-like, they reproduce particle’s dynamics. In https://arxiv.org/abs/1309.355, I have shown how that limit is actually described by a tropical limit, which allowed me to derive some practical results on scattering amplitudes at higher order and reformulate the sum over Feynman diagrams as an integral over the tropical moduli space of weighted graphs of https://arxiv.org/abs/0907.3324.
I will try to convey the essential ideas without assuming knowledge in quantum field theory or string theory, and conclude with open questions and perspectives.
|14:30 to 15:30||Michael Joswig (TU Berlin, Germany)||
Optimization and Tropical Combinatorics (Lecture 2)
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.
|16:00 to 17:00||Helge Ruddat (Johannes Gutenberg University, Germany)||
Tropical Quantum Field Theory, Mirror Polyvector Fields and Multiplicities of Tropical Curves
We introduce algebraic structures on the polyvector fields of an algebraic torus that serve to compute multiplicities in tropical and log Gromov-Witten theory while also connecting to the mirror symmetry dual deformation theory of complex structures. Most notably, these structures include a tropical quantum field theory and an L∞-structure. The latter is an instance of Getzler's gravity algebra, and the l2-bracket is a restriction of the Schouten-Nijenhuis bracket. I will explain its relationship to string topology. The work is joint with Travis Mandel.
|18:00 to 19:00||Greg Blekherman (Georgia tech, USA)||
Algebraic and Convex Geometry of Sums of Squares on Varieties (Lecture 2)
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 optimization.