Michel Brion: Minimal rational curves on complete symmetric varieties

Abstract: Minimal rational curves are generalizations of lines, which play a important role in the geometry of uniruled varieties. Their tangent directions at a general point form the variety of minimal rational tangents (VMRT), a local invariant that has many applications to rigidity questions. The description of the families of minimal rational curves and their VMRTs is an open problem for almost homogeneous varieties. It has been solved for homogeneous varieties (Hwang-Mok), toric varieties (Chen-Fu-Hwang), and wonderful compactifications of adjoint semisimple groups (B.-Fu). The talk will present a solution to this problem for the class of complete symmetric varieties, which contains the two latter classes. In particular, the VMRTs are homogeneous varieties, and can be described in terms of the combinatorics of restricted root systems. This is joint work with Shinyoung Kim and Nicolas Perrin. 

Camilla Felisetti: On the intersection cohomology of vector bundles 

Abstract: Intersection cohomology is a topological notion adapted to the description of singular topological spaces, and the Decomposition Theorem for algebraic maps is a key tool in the subject. The study of the intersection cohomology of the moduli spaces of semistable bundles on Riemann surfaces began in the 80’s with the works of Frances Kirwan. Motivated by the work of Mozgovoy and Reineke, in joint work with Andras Szenes and Olga Trapeznikova, we give a complete description of these structures via a detailed analysis of the Decomposition Theorem applied to a certain map from parabolic bundles. We also give a new formula for the intersection Betti numbers of these moduli spaces, which has a clear geometric meaning. In the talk, I will give an introduction to the subject, and describe our results.

Mihai Fulger: Local positivity of divisors on abelian 3-folds

Abstract: Given $C$ a smooth projective curve of genus 3, let $J$ be its Jacobian, and let $\theta$ be the principal polarization on $J$. We are interested in the local positivity of $\theta$ at a point of $J$, e.g., the origin. In this case this is the computation of the infinitesimal Newton-Okounkov body of $\theta$ in the general case, that of quartic curves with 56 distinct bitangency points. We also discuss related problems for box product type divisors on a product of 3 curves, or on the product of a curve and a Jacobian surface. This is a joint project with Victor Lozovanu.

Daniel Greb: Varieties homeomorphic to projective spaces

Abstract: A famous result by Kodaira-Hirzebruch (in connection with analytic results obtained later by Yau) says that a projective manifold homeomorphic to complex projective space is itself complex projective space. In my talk, I will discuss what can be said if one starts with a singular variety homeomorphic to a complex projective space. 

Tamas Hausel: Higgs bundle tournaments

Abstract: I will discuss the combinatorics of tournaments attached to root systems and their relationship to certain Higgs bundles. We expect that the connection can be explained by the big algebra of the Lie algebra representation of highest weight $\rho$, the half sum of positive roots, which should describe the mirror of the bottom Lagrangian in the Higgs moduli space. Based on joint work with Mirko Mauri. 

Kiumars Kaveh: Toric vector bundles on toric schemes and Bruhat-Tits buildings

Abstract: I will talk about classification of toric vector bundles on toric schemes over a DVR in terms of piecewise affine maps to the (extended) Bruhat-Tits building of the general linear group. We also discuss how to recover equivariant Chern classes in this picture. This is a joint work with Chris Manon, Boris Tsvelikhovskiy and Ana Botero.

Patrick Kennedy-Hunt: The logarithmic Hilbert scheme and its tropicalisation 

Abstract:  A basic question is understanding how the Hilbert/ Quot scheme of a projective variety X changes when we degenerate X. The key to answering this question is restricting attention to subschemes/ sheaves that are transverse to a simple normal crossing divisor D on X. I will motivate and explain the Hilbert scheme of a pair (X,D), called the logarithmic Hilbert scheme, which is compact yet tracks this transverse geometry. On the other hand, I will discuss a tropical version of a Hilbert scheme with a close connection to the secondary fan of a polytope. I will then explain the close connection between logarithmic Hilbert/ Quot schemes and this tropical gadget.

Martina Lanini: GKM-Theory for cyclic quiver Grassmannians

Abstract : After recalling some background on Goresky-Kottwitz-MacPherson (GKM) version of the Localization Theorem for equivariant cohomology, and some of the applications of such a result to (equivariant) Schubert calculus and geometric representation theory, I will explain how it is possible -and why it is desirable- to extend such techniques to the quiver Grassmannian setting. This is joint work with Alex Puetz.

Diane Maclagan: Tropical vector bundles

Abstract: Tropicalization replaces a variety by a combinatorial shadow that preserves some of its invariants.  When the variety is a subspace of projective space the tropical variety is determined by a (valuated) matroid.  I will review this, and discuss a resulting definition for a tropical vector bundle in the context of tropical scheme theory.  This is joint work with Bivas Khan.

Chris Manon: Convex algebraic geometry of projectivized toric vector bundles

Abstract:  I'll give an overview of some recent work on the geometry of projectivized toric vector bundles.  A toric vector bundle is a vector bundle over a toric variety equipped with an action by the defining torus of the base.  As a source of examples, toric vector bundles and their projectivizations provide a rich class of spaces that still manage to admit a combinatorial characterization.  Using results about the Cox ring, I'll describe some techniques to compute the nef cone and Newton-Okounkov bodies of a projectivized toric vector bundle in terms of certain matroid data.   I'll compute these objects for irreducible toric vector bundles of rank $n$ on $\mathbb{P}^n$, and assess Fujita's freeness and ampleness conjectures in this case.  This is joint work with Courtney George.

Ernesto Carlo Mistretta: Semiample Vector bundles and Iitaka fibrations

Abstract: We will construct an Iitaka fibration for vector bundles, and prove that even for semiample vector bundles this does not behave as in the line bundle case. Then we will discuss some properties of semiampleness for vector bundles and some geometric interpretation of those.

Jonas Stelzig: Pluripotential homotopy theory 

Abstract: By results of Sullivan from the 70's, one may model (finite type, simply connected) spaces up to rational homotopy equivalence by their commutative differential graded algebras of (piecewise-linear) differential forms. For example, one may use this algebra to calculate the homotopy groups modulo torsion, or certain higher operations Massey products'. 

Inspired by this, and recent construction by Angella, Tomassini and Tardini, we discuss a the following invariant of complex manifolds: The equivalence class of the bigraded, bidifferential algebra of C-valued differential forms, modulo a notion of quasi-isomorphism that tracks information on existence and uniqueness of the d-dbar-equation $x=\partial\bar\partial y$. 

The goal of this talk, which may at first glance seem a little far from the topic of the conference, is to give a gentle introduction to this circle of ideas and to show how it leads to some open problems which might well be approachable with techniques closer to the hearts and minds of the audience.

Hendrik Süß: On the stability of tangent bundles on rational surfaces

Abstract: Given a smooth projective variety X one my ask whether its tangent bundle is slope stable with respect to any polarisation. For toric surfaces it has been proved in joint work with Milena Hering and Benjamin Nill that this is the case if and only if X is an iterated blowup of the projective plane. One could conjecture that the same holds more generally for rational surfaces or at least for those among them which admit at least a C*-action. In my talk I will report on interesting challenges connected to this question and on some recent progress obtained in joint work with Valentin Boboc. 

Frederik Witt: Exceptional sequences of line bundles on toric and flag varieties.

Abstract: Beilinson gave a semi-orthogonal decomposition of the bounded derived category of the projective space by exhibiting a full exceptional sequence of line bundles. We discuss a combinatorial classification of such sequences on toric varieties of Picard rank $2$ and the flag varieties FL(1,ℓ,ℓ+1) based on joint work with Klaus Altmann and Andreas Hochenegger.

Milena Wrobel: Intrinsic Grassmannians and their geometry

Abstract: We introduce the notion of intrinsic Grassmannians which generalizes the well-known weighted Grassmannians introduced by Corti and Reid. An intrinsic Grassmannian is a normal projective variety whose Cox ring is defined by the Plücker ideal I_{k,n} of the Grassmannian Gr(k, n). 

For k = 2, we give a complete classification of all smooth Fano intrinsic Grassmannians with Picard number two and study their geometry.

Dmitry Zakharov: Vector bundles on metric graphs

Abstract: Reductive groups over an algebraically closed field are classified by their root systems. Starting with the Weyl group of a root system, we construct a tropical analogue of the corresponding reductive group. For the systems A_n, B_n, and D_n, the groups we obtain natural descriptions in terms of tropical linear algebra. 

We then define a tropical vector bundle with structure group G on a metric graph X (more generally, on a rational polyhedral space) as a Cech cocycle in the sheaf of G-valued harmonic piecewise-linear functions on X. When G is the tropical general linear group, we prove analogues of a number of algebraic results, such as Grothendieck’s classification of vector bundles on the projective line, Atiyah’s classification of vector bundles on elliptic curves, the Weil—Riemann—Roch theorem, and the Narasimhan—Seshadri correspondence. 

Joint work with Andreas Gross, Arne Kuhrs, and Martin Ulirsch.