Abstract: The work discussed is joint with André Belotto da Silva, Michael Temkin and Jaroslaw Wlodarczyk, and with Ming Hao Quek and Bernd Schober.
I will continue a long struggle to explain to non-experts why resolution of singularities in characteristic zero works.
Abstract: In this talk we introduce a geometric refinement of Gromov-Witten invariants for P1-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. We’ll give certain invariance properties of the correlated invariants and provide some computations in the case of P1-bundles over an elliptic curve. Such invariants are expected to play a role in the degeneration formula for reduced Gromov-Witten invariants for abelian and K3 surfaces.
Abstract: We give an overview of the arithmetic geometry of toric varieties developed by Burgos, Philippon and Sombra in 2014. In particular we discuss the classification of continuous toric metrics on toric line bundles both in the Archimedean and in the non-Archimedean case. Then we report on partial results generalizing this classification on toric vector bundles using the theory of buildings.
This is work in progress with Burgos, Kaveh and Mallik.
Abstract: We present a method to explicitly compute the tropicalization of a non-archimedean lattice. This method is based on computing the entropy vector of the lattice in question and producing a polyhedral complex whose support is the image of the lattice under coordinate-wise valuation. Each lattice gives rise to a natural probability distribution on the tropical linear space associated to the vector space it spans. We call these measure Gaussian distributions on the tropical linear spaces.
Abstract: Self-dual point configurations have been studied by Dolgachev and Ortland, Coble and many others. We investigate them in the context of self-dual matroids: Can every identically self-dual matroids be realised by a self-dual point configuration?
We classify identically self-dual matroids of rank up to five and determine the dimension of their (self-dual) realization spaces. Self-dual point configurations are parametrized by the self-dual Grassmannian, a subvariety of the Grassmannian Gr(n,2n). When considering degenerations of the point configurations, the tropicalization of this subvariety becomes important. We analyze the case n=3 in detail and point towards the difficulties for n=4.
This talk is based on joined work with Sachi Hashimoto, Bernd Sturmfels and Raluca Vlad.
Abstract: The matroid Grassmannian is the moduli space of oriented matroids; this is an important combinatorial analogue of the ordinary oriented real Grassmannian. Thirty years ago MacPherson showed us that understanding the homotopy type of this space can have significant implications in manifold topology, such as providing combinatorial formulae for the Pontrjagin classes. In some easy cases, the matroid Grassmannian is homotopy equivalent to the oriented real Grassmannian, but in most cases we have no idea whether or not they are equivalent. This question is known as MacPherson's conjecture. I'll show that one of the important homotopical structures of the oriented Grassmannians has an analogue on the matroid Grassmannian: the direct sum monoidal product (which gives rise to topological K-theory) is E-infinity.
Abstract: Hurwitz numbers count branched morphisms between Riemann surfaces with fixed numerical data. While a classical invariant, having been introduced in the 19th century, Hurwitz numbers are an active topic of study, among others due to their interplay with Gromov-Witten theory and their role in mirror symmetry. In recent work of Chapuy and Dołęga a non-orientable generalisation of Hurwitz numbers was introduced, so-called b-Hurwitz numbers. These invariants are a weighted enumeration of maps between non-orientable surfaces weighted by a power of a parameter b. This parameter should be viewed as measuring the non-orientability of the involved covers. For b=0, one recovers classical Hurwitz numbers, while b=1 represents a non-weighted count of non-orientable maps yielding so-called twisted Hurwitz numbers. In this talk, we derive a combinatorial model of twisted Hurwitz numbers via tropical geometry and employ it to derive a wide array of new structural properties.
This talk is based on joint work with Hannah Markwig.
Abstract: The talk is devoted to several real and tropical enumerative problems. We suggest new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from appropriate signed enumeration of real algebraic curves of genus 1 and 2. These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the genus zero case. It turns out that two different rules of signs in the enumeration surprisingly lead to the same collection of refined invariants.
The proof of the latter fact uses the tropical counterparts of the invariants under consideration.
This is a joint work with Eugenii Shustin.
Abstract: We introduce the notion of tropical toric vector bundles on tropical toric varieties using valuated matroids. Our motivation comes from the work of Di Rocco, Jabbusch, and Smith, where the authors associated realizable matroids to a toric vector bundle. We introduce the notion of decomposability of tropical toric vector bundle and study splitability on tropical projective line. In contrast to classical situation, we show that tropical toric bundles on tropical projective line do not split necessarily. We also study semistability of such bundle.
This is a joint work with Diane Maclagan.
Abstract: The Chow ring of the toric variety of the Bergman fan of a matroid is a ring which resembles the cohomology ring of a smooth projective variety. When the matroid is realizable, it coincides with the cohomology ring of de Concini and Procesi's wonderful compactification of the corresponding hyperplane arrangement complement. We study the K-rings of toric varieties of Bergman fans. We show that they resemble the K-rings of smooth projective varieties, and when the matroid is realizable, they coincide with the K-rings of wonderful compactifications.
Joint work with Shiyue Li, Sam Payne, and Nick Proudfoot.
Abstract: The SYZ conjecture concerns degenerations of complex Calabi-Yau manifolds and was proposed as a geometric explanation of mirror symmetry. Kontsevich and Soibelman introduced a non-archimedean approach to this conjecture, and more recently, Yang Li's work has connected the non-archimedean approach with the original SYZ conjecture.
In this talk, I will explain the key concepts of the non-archimedean approach and present recent developments in the context of hypersurfaces.
This is based on a projects in collaboration with Rolf Andreasson, Jakob Hultgren, Mattias Jonsson and Nick McCleerey.
Abstract: Polyhedral homotopies were originally introduced by Huber and Sturmfels nearly 30 years ago, and have since become a staple strategy for solving polynomial systems. Main topic of the talk is a generalisation thereof with special emphasis on the matroid theoretic challenges that arise along the way.
Building on ideas of Jensen, Leykin, and Yu, we will discuss two distinct types of tropical homotopies: First, we will discuss how to use tropical points to construct homotopies for solving systems of polynomial equations. Second, we will discuss how to compute tropical points using homotopies for intersecting balanced polyhedral complexes.
Centerpiece of the talk are systems of parametrized polynomial equations, and we will focus two main cases:
Vertically parametrized polynomial systems are systems in which parameters are shared between equations but always bound to the same monomial. These are for example the steady state equations of chemical reaction networks or they arise in the computation of ED or ML degrees.
Horizontally parametrized polynomial systems are systems in which parameters are shared between monomials but always bound to the same equation. These are prominently studied using the theory of Khovanskii bases and Newton Okounkov bodies.
Abstract: One of the most prominent examples of a moduli space is the Grassmannian which parametrizes all linear spaces.
Moreover, its tropicalization parametrizes all tropicalized linear spaces. In my talk I we will look at a related moduli space, namely the moduli space of flags of two linear spaces of dimension d and d+1. We will also look at its tropical moduli space and investigate this space from a matroidal point of view.
This talk is based on joined work with Philipp Jell, Hannah Markwig and Felipe Rincón
Abstract: I will present a (cohomological) Fourier-Mukai transform for tropical Abelian varieties and give some applications, including a (generalized) Poincaré formula (for non-degenerate line bundles on tropical Abelian varieties).
Based on joint work with Soham Ghosh.
Abstract: We present recent work with Mario Kummer and Raluca Vlad in real non-archimedean geometry. A curve of genus g is maximal Mumford (MM) if it has g+1 ovals and g tropical cycles. We construct full-dimensional families of MM curves in the Hilbert scheme of canonical curves. This rests on first-order deformations of graph curves whose graph is planar.
Abstract: Positroids are a well-behaved class of realizable matroids, in bijection with many combinatorial objects. In particular, they are uniquely defined by the value of the rank function on cyclic intervals. Essential sets form a natural family of rank conditions on cyclic intervals uniquely defining a positroid. However, the minimal information is contained in a subset of the essential sets, called core. I will show how to construct these families and explain how essential sets and core encode properties of the positroid.
Based on joint work with Fatemeh Mohammadi.
