Schedule for: 17w5046 - Newton-Okounkov Bodies, Test Configurations, and Diophantine Geometry

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I will introduce the notion of height, and how Diophantine inequalities help in Diophantine geometry. Specifically, I'll discuss the theorem of Faltings-Wuestholz and how it uses filtrations to derive Diophantine inequalities.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will present some surprising relations between Baker-Norine theory of linear systems on graphs and Newton-Okounkov bodies over discrete valuation rings. This can be thought as a step further connecting Tropical Geometry and Newton-Okounkov bodies. This is joint work with Eric Katz.

In this talk, we will discuss our construction of compact Hausdorff Moishezon moduli spaces parametrizing smoothable K-stable Fano varieties. The solution relies on the recent solution of the Yau-Tian-Donaldson conjecture by Chen-Donaldson-Sun and Tian. In particular, we prove the uniqueness of the degeneration of Fano Kähler-Einstein manifolds and more algebraic properties that are needed to construct a good algebraic moduli space. (This is a joint work with Chi Li and Chenyang Xu)

The goal of this talk is to explain how to construct a large family of Newton-Okounkov bodies for partial flag varieties, by means of a notion called birational sequence. This construction will be then applied to unify the known toric degenerations of partial flag varieties, as well as to determine Gromov widths of coadjoint orbits.

The theory of toric varieties is a well established tool to study algebro-geometric objects by combinatorial methods (and vice versa). It translates varieties with a (full) torus action into cones, polyhedra, and fans. Moreover, many of the properties of the varieties have a combinatorial counterpart via this correspondence. However, in many situations, varieties with a lower-⁠dimensional torus action arise in a natural way: The striking example is the total spaces of deformations of toric varieties. For those situations we show how to adapt the toric language to keep a description involving as much combinatorics possible. The associated notion is that of combinatorial divisors and polyhedral fans developed together with Juergen Hausen and Hendrik Suess. A special example of this construction is the representation of the Cox ring of Del Pezzo surfaces (under the action of the Picard torus) as a polyhedral divisor. This turns out to be strongly related to the Zariski decomposition of divisors. This is joint work with Jarek Wisniewski.

This is a joint work with Huayi Chen. We can easily recognize that the core part of the proof of the Dirichlet unit theorem is nothing but the existence of a global section of a pseudo-effective arithmetic divisor on an arithmetic curve. In this sense, the existence of a global section is often called the Dirichlet property. In this talk, I would like to discuss with the Dirichlet property on an arithmetic dynamical system. The affirmative answer of the Dirichlet property on a toric variety is closely related to the integral formula of the arithmetic volume on a Newton-Okounkov body.

Manin's conjecture predicts the growth rate of points of bounded height on a Fano variety after removing an exceptional set. We study the geometry of the exceptional set using techniques from the minimal model program. This is joint work with Sho Tanimoto.

Given a projective manifold, it is very difficult to detect in the Kähler cone which classes admit a Kähler metric with constant scalar curvature. Fix a Kahler class $\alpha$ with such a special Kähler metric, one can ask if it is easier to describe an explicit neighborhood V of $\alpha$, such that for all $\beta$ in V, the Kähler class $\beta$ admits a Kähler metric with constant scalar curvature, and of course to find the largest possible neighborhood V. Under certain conditions, a partial answer to these questions can be given by studying Donaldson's J-flow. We will present some analytic and algebraic results about this geometric flow. In particular, we will discuss a G.I.T framework associated to this flow and its relationship with the Yau-Tian-Donaldson conjecture and (uniform) K-stability. This is joint works with R. Dervan and Y. Hashimoto.

Using the infinitesimal Newton-Okounkov bodies, Küronya-Lozovanu show a Reider-type theorem for higher syzygies of ample line bundles on abelian surfaces. In this talk, I will explain a slight improvement of their result.

Adapting the concept of symplectic packings to the Kähler setting we introduce Kähler packings and show how on complex surfaces, their properties are related to (multi-point) Seshadri constants. We explicitly construct Kähler packings on toric surfaces and connect them to the toric moment maps and their images, the moment polytopes. Finally we discuss some examples generalizing these results to non-toric situations and Newton-Okounkov bodies.

I will talk about a recent work with Chris Manon giving a direct relation between the theory of Newton-Okounkov bodies (which is concerned with full rank valuations on a graded algebra) and tropical geometry (which is concerned with rank 1 valuations). More specifically, we show that an algebra has a full rank valuation with a finitely generated value semigroup if and only if there is a "prime cone" in its tropical variety for some presentation of this algebra as a quotient of a polynomial ring. A key concept in our theory is that of a Khovanskii basis. Roughly speaking, it is an algebra analogue of a Grobner basis for an ideal. This approach unites "toric degenerations" arising in the context of Newton-Okounkov bodies and the ones coming from tropical geometry. Representation theory provides many interesting and natural examples of this theory, in particular, wonderful compactification makes an appearance.

On the triaxial ellipsoid there are various classical dynamical systems. In the talk we shall report on joint work of Ronald Garcia on curvature lines and geodesics. We are dealing with the question under which conditions these curves are closed or not. In the case of geodesics we give a complete answer if the ellipsoid is defined over a number field. In the case of curvature lines we show that there is a countable set of curvature lines which are not closed. These are classical problems in the theory of dynamical systems and there were almost no results in this direction. We also discuss the cases of ellipsoids of revolution and ellipsoids in Minkowski space. The proofs make basic use of the analytic subspace theorem in the case of elliptic periods and periods on abelian surfaces coming up naturally. In the elliptic case we solve an extended problem of Th. Schneider dealing with periods of differentials of the third kind.

Introduced by Postnikov in 2006, plabic graphs are certain planar bicolored graphs embedded in a circle which played a central role in Rietsch and Williams' recent work on mirror symmetry for Grassmannians. Inspired by their construction, we use plabic graphs to define weight vectors and show that in the cases of Gr(2,n) and Gr(3,6) these vectors lie in the tropical Grassmannian. Moreover, we provide an explicit bijection between toric degenerations of Gr(2,n) arising from maximal cones in the tropical Grassmannian and the ones coming from plabic graphs. This is joint work with L. Bossinger, X. Fang, G.Fourier, and M. Hering.

I will define semistability for pairs of repesentations of any linear reductive algebraic group and provide the link with metrics of constant scalar curvature in any Hodge class. The emphasis will be on examples and the transparent connection to Hilbert and Mumford's GIT.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.