ENG 
ESP Session S31 - Mathematical-Physical Aspects of Toric and Tropical Geometry
Talks
Monday, July 12, 11:00 ~ 12:00 UTC-3
On the classification of symplectic DQ-algebroids
Paul Bressler
University of the Andes, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.
We show that every symplectic manifold admits a canonical DQ-algebroid quantizing the structure sheaf and classify symplectic DQ-algebroids by adapting the method of P. Deligne.
Monday, July 12, 12:15 ~ 13:15 UTC-3
The rational cohomology of $\mathcal{A}_g$
Melody Chan
Brown University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
I'll discuss recent work using tropical techniques to find new rational cohomology classes in moduli spaces $\mathcal{A}_g$ of abelian varieties, building on previous joint work with Soren Galatius and Sam Payne relating the cohomology of moduli spaces of curves $\mathcal{M}_g$ to homology in Kontsevich's graph complexes. I will try to present a broad view.
Joint work with Madeline Brandt (Brown University), Juliette Bruce (University of California, Berkeley), Margarida Melo (Università Roma Tre), Gwyneth Moreland (Harvard University) and Corey Wolfe (Tulane University).
Monday, July 12, 13:30 ~ 14:30 UTC-3
Decomposition of the skeleton of a tropical linear space
Luciá Lopez de Medrano
UNAM, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.
The skeletons of tropical varieties have been the subject of several studies. In this talk, we present a different approach to the combinatorial study of the skeleton of a tropical linear space. In particular, we show that in the case of realizables tropical linear spaces, the k-dimensional skeleton is an arrangement of k-dimensional tropical linear spaces. The main tool used in this study are tropical modifications. Work in progress with Aubin Arroyo.
Joint work with Aubin Arroyo (UNAM, Mexico).
Monday, July 12, 14:45 ~ 15:45 UTC-3
KP Solitons from Tropical Limits
Yelena Mandelshtam
UC Berkeley, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
We study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. We compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces and we present an algorithm that finds a soliton solution from a rational nodal curve.
Joint work with Daniele Agostini (MPI MiS Leipzig), Claudia Fevola (MPI MiS Leipzig) and Bernd Sturmfels (MPI MiS Leipzig and UC Berkeley).
Tuesday, July 13, 11:00 ~ 12:00 UTC-3
The spine of the $T$-graph of the Hilbert scheme of points
Diane Maclagan
University of Warwick, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.
The torus $T$ of projective space also acts on the Hilbert scheme of subschemes of projective space, and the $T$-graph of the Hilbert scheme has vertices the fixed points of this action, and edges the closures of one-dimensional orbits. In general this graph depends on the underlying field. I will discuss joint work with Rob Silversmith, in which we construct of a subgraph, which we call the spine, of the $T$-graph of $\mathrm{Hilb}^N(\mathbb A^2)$ that is independent of the choice of infinite field. A key technique is an understanding of the tropical ideal, in the sense of tropical scheme theory, of the ideal of the universal family of an edge in the spine.
Joint work with Rob Silversmith (Northeastern, USA).
Tuesday, July 13, 12:15 ~ 13:15 UTC-3
Topological SYZ fibrations with discriminant in codimension two.
Ilia Zharkov
Kansas State University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
To date only for K3 surfaces (trivial) and the quintic threefold (due to M. Gross) the discriminant can be made to be in codimension two. I will outline the source of the problem and how to resolve it in much more general situations using phase and over-tropical pairs-of-pants. Joint project with Helge Ruddat. If time permits, I'll explain an application to lifting tropical cycles from the SYZ base to "holomorphic" and "Lagrangian" type objects in the torus fibrations.
Joint work with Helge Ruddat (University of Mainz, Germany), Grigory Mikhalkin (University of Geneva, Switzerland), Gabe Kerr (Kansas State University, USA) and Ludmil Katzarkov (University of Miami, USA & Higher School for Economics, Russia).
Tuesday, July 13, 13:30 ~ 14:30 UTC-3
Irrational toric varieties and the secondary polytope
Frank Sottile
Texas A&M University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Classical toric varieties come in two flavours: Normal toric varieties are given by rational fans in ${\mathbb R}^n$. A (not necessarily normal) affine toric variety is given by finite subset $A$ of ${\mathbb Z}^n$. When $A$ is homogeneous, it is projective. Applications of mathematics have long studied the positive real part of a toric variety as the main object, where the points $A$ may be arbitrary points in ${\mathbb R}^n$. For example, in 1963 Birch showed that such an irrational toric variety is homeomorphic to the convex hull of the set $A$.
Recent work showing that all Hausdorff limits of translates of irrational toric varieties are toric degenerations suggested the need for a theory of irrational toric varieties associated to arbitrary fans in ${\mathbb R}^n$. These are ${\mathbb R}^n_>$-equivariant cell complexes dual to the fan. Among the pleasing parallels with the classical theory is that the space of Hausdorff limits of the irrational projective toric variety of a finite set $A$ in ${\mathbb R}^n$ is homeomorphic to the secondary polytope of $A$.
This talk will sketch this story of irrational toric varieties. It represents work with Garcia-Puente, Zhu, Postinghel, Villamizar, and Pir.
Joint work with Luis David Garcia-Puente (Sam Houston State, USA), Ata Pir (TAMU, USA), Elisa Postinghel (Loughborough, UK), Nelly Villamizar (Swansea, UK) and Chungang Zhu (Dalian Univ. of Tech., China).
Tuesday, July 13, 14:45 ~ 15:45 UTC-3
Series with support in cones, discriminants and polyhedra
Fuensanta Aroca
UNAM, México - This email address is being protected from spambots. You need JavaScript enabled to view it.
Roots of polynomials with coefficients series is several variables may be expressed as series with support in strongly convex rational cones. We will discuss the relationship of these cones with the discriminant of the projection.