Schedule of talks:

Abstracts

(Cueto): Tropical geometry is a piecewise-linear shadow of algebraic geometry that preserves important geometric invariants.  Often, we can derive classical statements from these (easier) combinatorial objects.  One general difficulty in this approach is that tropicalization strongly depends on the embedding of the algebraic variety.  Thus, the task of funding a suitable embedding or of repairing a given "bad" embedding to obtain a nicer tropicalization that better reflects the geometry of the input object becomes essential for many applications.  In this talk, I will show how to use linear tropical modifications and Berkovich skeleta to achieve such goal in the curve case.  Our motivating example will be plane elliptic cubics defined over a non-Archimedean valued field.  This is joint work with Hannah Markwig.

(Fusi): Let X be a complex projective variety.  Properties of special families of rational curves capture the geometry of X.  In 2006 Ionescu and Russo gave a criterion that describes rationality in terms of suitable families of rational curves through a point of X.  Inspired by their result we prove a stronger version of the criterion, namely we give necessary and sufficient conditions for the local ring of X at a point x to be C-isomorphic to the local ring of a point of the projective space.  As an application we define a degree of rationality in a natural way and we provide a classification result for small values.

(Küronya): The main theme of the talk is positivity of line bundles at a given point on a projective variety.  After discussing the definition of local positivity and how to measure it, we explain a way to describe local positivity in terms of Newton-Okounkov bodies.  If time permits, we present some applications to abelian surfaces.

(Kaveh): The purpose of this talk is to discuss some recent general results about symplectic geometry of smooth projective varieties using toric degenerations (motivated by commutative algebra and specifically Dave Anderson's toric degenerations).  The main result is the following: Let X be a smooth n-dimensional complex projective variety embedded in a projective space and equipped with a Kahler structure induced from a Fubini-Study Kahler form.  We show that for any epsilon>0, the manifold X has an open subset U (in the usual topology) such that vol(X - U) < epsilon, and moreover U is symplectomorphic to the algebraic torus (C^*)ⁿ equipped with a toric Kahler form.  The proof is based on construction of a toric degeneration of X.  As applications we obtain lower bounds on the Gromov width of X.  Moreover, we show that X has a full symplectic ball packing by d balls of capacity 1 where d is the degree of X.

(Vogiannou): A tropical compactification of a subvariety Y of a torus T is a compactification Y^* in a toric variety X such that (i) Y^* is proper, and (ii) the structure morphism from T times Y^* to X is faithfully flat.  The support of the fan associated to X coincides with the tropicalization Trop Y, which can be computed by other means; this suggests a way to compute Y^*.  These compactifications possess a set of nice properties.  For instance if X is smooth, any set of n boundary divisors intersect in codimension n (combinatorial normal crossings condition).  Tropical compactifications were introduced and their existence was shown by Tevelev in 2007.  We extend this idea to compactifications of subvarieties of spherical homogeneous spaces G/H for a (non-abelian) reductive group G.  We show their existence and provide some examples.

(Zhou): In this talk I will introduce the generalization of relative Donaldson-Thomas theory to 3-dimensional smooth Deligne-Mumford stacks.  We adopt Jun Li's construction of expanded pairs and degenerations and prove an orbifold DT degeneration formula.  I'll also talk about the application in the case of local gerby curves, and its relationship to the work of Okounkov-Pandharipande and Maulik-Oblomkov.

(Escobar): Bott-Samelson varieties are twisted products of CP¹'s with a map into G/B.  This talk will be about brick varieties which are the general fibers of Bott-Samelson maps.  I will describe the moment polytopes of the brick varieties and describe some instances in which these varieties are toric.  In particular, I will give a description of the toric variety of the associahedron.

(Ranganathan): Non-archimedean analytic techniques provide precise geometric connections between tropical enumerative geometry and logarithmic Gromov-Witten theory.  This framework leads to a simple new description of the algebraic moduli space of rational curves in toric varieties with prescribed contact orders.  Applications include a new proof of Nishinou and Siebert's classical/tropical correspondence theorem and a new result on the structure of the tropical rational double Hurwitz cycle.  Time permitting, I will explain how these ideas lead to a new perspective on the realization problem for embedded tropical curves in higher genus.

(Cartwright): The dual complex of a degeneration records how the components of the special fiber intersect.  It appears in tropical geometry as a parameterizing object for the tropicalization.  While any graph appears as the dual complex of a degeneration of curves, the same is not true in higher dimensions.  I will discuss an obstruction to realizing dual complexes homeomorphic to topological surfaces with positive Euler characteristic.

(Jabbusch): In this talk I will describe how to relate a collection of rational convex polytopes to a toric vector bundle on a smooth complete toric variety.  We'll see that this parliament of polytopes encodes much information about the toric vector bundle, for example the lattice points in the polytopes will correspond to generators of the global sections, and the edge lengths can be related to jets.  Further notions of positivity can also be related to these polytopes, and we'll see examples of ample toric vector bundles that are not globally generated, and ample, globally generated toric vector bundles that are not very ample.  This is joint work with Sandra Di Rocco and Greg Smith.

(Deopurkar): One of the easiest ways of writing down an algebraic curve is as the zero locus of a polynomial function on the plane.  For generic values of the coefficients of the polynomial, the zero locus will be a smooth curve, representing a point in the moduli space of curves.  But what happens as the coefficients specialize?  Can we describe the limits of smooth plane curves in the Deligne-Mumford compactification of the moduli space of all curves?  I will describe an explicit and complete answer to this old question in the first non-trivial case: plane quintics.  The solution will use stacky curves and will also explain a web of inter-relations between special 4-gonal and 3-gonal curves described by Vakil.

(Liu): The Norbury-Scott conjecture relates stationary Gromov-Witten invariants of the projective line to Eynard-Orantin invariants of the spectral curve x= Y+ 1/Y, y=logY.  In this talk, I will describe a correspondence between equivariant Gromov-Witten invariants of the projective line and Eynard-Orantin invariants of the spectral curve associated to the equivariant Landau-Ginzburg mirror of the projective line.  The correspondence specializes to the Norbury-Scott conjecture in the non-equivariant limit, and specializes to the Bouchard-Marino conjecture on simple Hurwitz numbers in the large radius limit.  This talk is based on joint work with Bohan Fang and Zhengyu Zong.

(Schaug): Mirror symmetry is a fundamental duality of Calabi-Yau threefolds, critical to string theory. Borcea-Voisin manifolds provided some of the earliest pairs of Calabi-Yau threefolds to exhibit a simple topological version of mirror symmetry, but their deeper mirror symmetry at the level of enumerative geometry has been poorly studied. This should relate the Gromov-Witten invariants of a Borcea-Voisin threefold to the periods of its mirror partner. We shall discuss results in this regard after introducing some of the concepts involved, and explore other aspects of the Gromov-Witten theory of Borcea-Voisin manifolds, which forms part of a richer structure.

(You): For each positive rational number ε, we define K-theoretic ε-stable quasimaps to certain GIT quotients W//G. In particular, for ε>1, we obtain the K-theoretic Gromov-Witten theory of W//G introduced by A. Givental and Y.P. Lee. For arbitrary ε₁ and ε₂ in different stability chambers, the numerical invariants are expected to be related by wall-crossing formulas. We prove wall-crossing formulas for genus zero K-theoretic quasimap theory when the target W//G admits a torus action with isolated fixed points and isolated one-dimensional orbits. This is joint work with Hsian-Hua Tseng.

(Bapat): To a singularity of an algebraic hypersurface, one can associate an invariant called the Bernstein-Sato polynomial or the b-function. Although the b-function is important and interesting, it is usually difficult to compute. It is conjectured (Strong Monodromy Conjecture or SMC) that some roots of the b-function can be obtained from the poles of another singularity invariant, the topological zeta function. I will sketch the proof of the SMC for the case of Weyl hyperplane arrangements, via the "n/d conjecture" of Budur, Mustaţă, and Teitler. I will also describe some results towards computing the b-function of these arrangements, focusing on a special case (the Vandermonde determinant). This is joint work with Robin Walters.

(Kass): A celebrated result of Eisenbud-Kimshaishvili-Levine computes the local degree of a smooth function f: Rⁿ → Rⁿ with an isolated zero at the origin as the signature of the degree quadratic form. We prove a parallel result computing the A1-local degree of a polynomial function with an isolated zero at the origin, answering a question posed by David Eisenbud in 1978. This talk will present this result and then discuss applications to the study of singularities if time permits. This is joint work with Kirsten Wickelgren.

(Raicu): Bernstein-Sato polynomials are important invariants in the study of singularities: they are associated classically to hypersurfaces, and in work of Budur- Mustaţă-Saito to more general algebraic varieties. Despite their simple definition, Bernstein-Sato polynomials are notoriously difficult to compute, and they have been described explicitly in only a few cases. In my talk I will explain how basic results on local cohomology modules and invariant differential operators can be used to describe the Bernstein-Sato polynomials for the varieties of general (resp. skew-symmetric) matrices of sub-maximal rank. This can then be used to verify the Strong Monodromy Conjecture for the said varieties. Joint work with Andrá Lőrincz, Uli Walther and Jerzy Weyman.

(Can): We are going to present two general methods for computing equivariant K-theory of a smooth spherical variety. If time permits we are going to apply our theorems to the following spherical varieties: the wonderful compactifications of PSL(2n)/PSp(n), PSL(n), and of PSL(n)/PSO(n). This talk is based on our joint work with Soumya Banerjee and Michael Joyce.

(Lewis): This talk serves as a precursor to a 24 lecture mini course delivered at the USTC in Hefei, China, June 23 - July 12, 2014. The 24 lectures will be published in the Communications in Mathematics and Statistics, and this talk will be published in Yau's ICCM Notices.

(Fusi): Given a property, it is natural to ask how it behaves in families. A family of projective varieties is a projective morphism f: X → T of schemes, the fibers of f are the members of the family and T is the parameter space. Little is known about how the properties of being rational, unirational, and stably rational behave in families. In the talk we will present recent results in this direction.

(Z. Lin): Hall algebra was first defined to capture properties of module extensions over certain PID. Ringel's application of Hall algebra to give a new construction of part of quantum enveloping algebras opened up a new frontier to construct certain algebraic objects which encode geometric properties. Roughly speaking, Hall algebras are convolution algebras over certain systems of geometric objects. The algebraic formulation provides ways to compute geometric invariants. This approach has been used widely in many directions of mathematics. This seminar talk will outline some of the different approaches: counting over finite fields, cohomological, K-theoretic, as well as motivic setting. All of these are closely related to representation theory.

(B. Lin): The linear system |D| of a divisor D on a metric graph has the structure of a cell complex. We introduce the anchor divisors and anchor cells in it - they serve as the landmarks for us to compute the f-vector of the complex and find all cells in the complex. A linear system can also be identified as a tropical convex hull of rational functions. We compute its extremal generators using the landmarks. We apply these methods to some examples - namely the canonical linear systems on K₄ and K_3,3.

(González Villa): We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (assuming certain finiteness conditions are satisfied) a motivic zeta function and a motivic infinite cyclic cover and show its birational invariance. Our constructions generalize the notions of motivic zeta function and motivic Milnor fibre of a complex hypersurface singularity germ due to Denef and Loeser. This is joint work with Anatoly Libgober and Laurentiu Maxim.

(A. Zahariuc): The idea to analyze the geometric or enumerative properties of algebraic curves on Calabi-Yau threefolds by degenerating the underlying threefold has been around for a long time. The first step in applying this method is to understand the space of stable maps to the degenerate fiber. In this talk, I will show an elementary construction of a degeneration of quintic threefolds, for which this step can be carried out in a satisfactory way in the genus zero case. An application to higher genus and some questions relating to Clemens' Conjecture will also be discussed.

(Park): Faltings' theorem states that curves of genus g > 1 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry to show that we can also give an effective bound on the number of rational points outside of the special set of the d-th symmetric power of X, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d.

Ohio State University Algebraic Geometry Seminar-Year 2014-2015

Ohio State University Algebraic Geometry Seminar-Year 2013-2014

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