Schedule of talks:

Abstracts

(Castravet): The Mori cone of curves of the Grothendieck-Knudsen moduli space \bar{M}_{0,n} of stable rational curves with n markings, is conjecturally generated by the one-dimensional strata (the so-called F-curves). A result of Keel and McKernan states that a hypothetical counterexample must come from rigid curves that intersect the interior. In this talk I will show several ways of constructing rigid curves. In all the examples a reduction mod p argument shows that the classes of the rigid curves that we construct can be decomposed as sums of F-curves. This is joint work with Jenia Tevelev.

(Clemens): Click here for abstract.

(Marsano): The purpose of this talk is to discuss the construction of a particular Calabi-Yau fourfold and computations of some cohomology groups attached to it. The interpretation of this explicit geometry in stringy theory and quantum physics will also be explained.

(Tseng): For a compact semi-Fano toric manifold X, the toric mirror theorem of Givental and Lian-Liu-Yau says that a generating function of 1-point genus 0 descendant Gromov-Witten invariants, the J-function of X, coincides up to a mirror map with a function I_X which is written using the combinatorics of X. The procedure of obtaining the mirror map, which involves expanding I_X as a suitable power series, is somewhat mysterious. In this talk we'll describe some attempts at understanding the mirror maps more geometrically.

(Pixton): I'll discuss various conjectures about divisors on $\overline{M}_{0,n}$ and describe a counterexample to one of them: it is not true that every nef divisor is numerically equivalent to an effective sum of boundary divisors. The counterexample is combinatorial in nature and is closely related to the (11, 5, 2) biplane.

(Joshua): We explore several variations of the notion of |purity for the action of Frobenius on schemes defined over finite fields. In particular, we study how these notions are preserved under certain natural operations like quotients for principal bundles and also geometric quotients for reductive group actions. We then apply these results to study the cohomology of quiver moduli. We prove that a natural stratification of the space of representations of a quiver with a fixed dimension vector is equivariantly perfect and from it deduce that each of the l--adic cohomology groups of the quiver moduli space is strongly pure.

(Satriano): In this talk we explore the following local-global question: if X is locally a quotient of a smooth variety by a finite group, then is it globally of this form? We show that the answer is "yes" whenever X is quasi-projective and already known to be a quotient by a torus. In particular, this applies to all quasi-projective simplicial toric varieties. We discuss the proof and show how it can be made explicit in the case of toric varieties. This is joint work with Anton Geraschenko.

(Păun): Cancelled.

(Liu): We prove a mirror theorem on counting holomorphic orbidisks in toric Calabi-Yau 3-orbifolds. This is based on joint work in progress with Bohan Fang and Hsian-Hua Tseng.

(Kuznetsov): A categorical resolution of singularities of an algebraic variety Y is a triangulated category T with an adjoint pair of functors between D(Y), (the derived category of quasicoherent sheaves on Y) and T, such that the composition is an identity endofunctor of D(Y). If X \to Y is a usual resolution, the derived category D(X) with pullback and pushforward functors is a categorical resolution ONLY if Y has rational singularities. However, I will explain that even if Y has nonrational singularities, still one can construct a categorical resolution of D(Y) by gluing derived categories of appropriate smooth algebraic varieties. This is a work in progress, joint with Valery Lunts.

(Oprea): The strange duality conjecture concerns spaces of generalized theta functions constructed from moduli spaces of sheaves. I will discuss versions of the strange duality conjecture over abelian surfaces, as well as some evidence and partial results. This is joint work with Alina Marian.

(Prendergast-Smith): I will give an overview of the different versions (due to Morrison, Kawamata, and Totaro) of the Cone Conjecture in birational geometry, and discuss some cases in which the conjecture has been proven.

(Urbinati): Based on the construction of de Fernex and Hacon, we study the behavior of the singularities on non-Q-Gorenstein varieties, in particular for which the canonical ring is not a finitely generated O_X-algebra. We give an example of a non Q-Gorenstein variety whose canonical divisor has an irrational valuation and example of a non Q-Gorenstein variety which is canonical but not klt. We also give an example of an irrational jumping number and we prove that there are no accumulation points for the jumping numbers of normal non-Q-Gorenstein varieties with isolated singularities.We also prove that Canonical singularities are equivalent to the finite generation of the canonical ring.Finally, we are trying to introduce a notion of nefness in this new setting, with studying the behavior of the new feature.

(Witten): Chern-Simons gauge theory in three dimensions gives a natural framework to understand the Jones polynomial of knots and related invariants of knots and three-manifolds. It is also closely related to conformal field theory in two dimensions as well as to quantum groups. In this talk, we will sketch a more recent point of view that relates Chern-Simons theory to mathematical physics in four dimensions.

(Docampo Álvarez): In this talk I will introduce the notion of Jacobian discrepancy, an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. This invariants, very natural from the point of view of jet schemes and arc spaces, lead to a framework in which adjunction and inversion of adjunction hold in full generality. Moreover, they allow us to give explicit formulas measuring the gap between the dualizing sheaf and the Grauert-Riemenschneider canonical sheaf of a normal variety, leading to characterizations of rational and Du-Bois singularities in the normal Cohen-Macaulay case in terms of Jacobian discrepancies.

(Kaveh): This talk is an invitation to the theory of Okounkov bodies, a new connection between algebraic geometry and convex geometry. It generalizes the well-known and extremely rich correspondence between toric varieties and convex integral polytopes. We will discuss a general combinatorial construction to associate convex bodies to algebraic varieties. As examples, it unifies seemingly unrelated constructions such as Newton polytope of a toric variety, moment polytope of a Hamiltonian action and string polytope of a representation. Among the many applications are an elementary proof of the Hodge inequality for intersection numbers, far generalization of the Kushnirenko theorem on the number of solutions of a system of polynomial equations, and construction of (completely) integrable systems on varieties. Building on earlier work of A. Okounkov, in 2008 this theory was developed and applied in joint work of A. G. Khovanskii and the speaker, and independently by Lazarsfeld and Mustata. For the most part, the talk is accessible to anyone with just basic background in algebra and geometry.

(Ilten): In this talk, I will report on recent joint work with Jan Christophersen. Motivated by the search for toric degenerations of Fano varieties, we construct degenerations of rank one index one smooth Fano threefolds to Stanley-Reisner schemes which are smooth points in the corresponding Hilbert schemes. Coupled with deformation-theoretic calculations, this allows us to classify all smoothings of canonical Gorenstein toric Fano varieties of degree at most 12. Generalizing the situation to higher dimension, we show that the Stanley-Reisner scheme coming from the boundary complex of the dual associahedron is unobstructed, and construct a series of more unobstructed Stanley-Reisner schemes via stellar subdivisions. In particular, this gives a technique of finding new toric degenerations of the Grassmannian G(2,n).

(Morrison): The work of Seiberg and Witten in the mid-1990's on a class of supersymmetric quantum field theories was important for both physics and mathematics. In physics, it pointed the way to the study of non-perturbative aspects of a variety of quantum field theories and string theories, and led fairly directly to the so-called ``second superstring revolution'' of 1995. In mathematics, the work had remarkable consequences for topology in four dimensions, leading to the rapid solution of many long-standing problems.

(Krishna): We shall define the algebraic analogue of the equivariant cobordism for schemes with group action. We shall show how the equivariant cobordism is related to the other equivariant cohomology theories. We shall how one can use the equivariant cobordism to study some conjectures of Totaro about the cycle class map for the classifying spaces.

(Ballard): We describe semi-orthogonal decomposiitions between bounded derived categories of coherent sheaves on two different GIT quotients related by a "nice" variation of the linearization. As a warm-up, we will demonstrate how the main theorem encompasses two results, well-known in the study of birational geometry and derived categories: Beilinson's exceptional collection on projective space and Bondal-Orlov's derived equivalence for the Atiyah flop. We will then state the main result and discuss some other applications and related results, including descriptions of derived categories for projective toric DM stacks similar to work of Kawamata and a generalization of Orlov's sigma-model/Landau-Ginzburg semi-orthogonal decomposition. All results are taken from joint work, arXiv:1203.6643, with D. Favero (U. Wien) and L. Katzarkov (U. Miami/Wien).

(Harbourne): Motivated by work in number theory, in the 1970s Waldschmidt defined an asymptotic measure of the least degree of a polynomial in n variables with given order of vanishing on a finite set of points in projective space. In the case of generic points in P2, determining the value of Waldschmidt's constant is equivalent to an open conjecture of Nagata. Recent work has related Waldschmidt's constant to an ideal containment problem: which symbolic powers of the ideal of the points are contained in a given power of the ideal? In joint work with E. Guardo and A. Van Tuyl, we generalize this work from points to lines in projective space and we formulate a version of Nagata's conjecture in this context. Additional joint work with M. Dumnicki, T. Szemberg and H. Tutaj-Gasinska extends this to r-planes in projective N-space.

(Lau): In this talk I report on my joint work with K.W. Chan, N.C. Leung and H.H. Tseng on the correspondence between certain symplectic disk-counting invariants and mirror maps in the case of toric manifolds with nef canonical divisors. It is an open analog of the mirror phenomenon that closed Gromov-Witten invariants are encoded by the complex geometry in the mirror side. This gives a `mirror method' to compute the open invariants, which are in general difficult to deal with when the moduli theory has non-trivial obstructions.

(Soprunov): The Toric Euler-Jacobi theorem, discovered by Khovanskii in 1970s, concerns the global residue, which is the sum of local residues of a rational form over a complete intersection in the algebraic torus. Global residues have numerous applications including GKZ hypergeometric systems and computations on sparse polynomial systems.

(Isaksen): A sums-of-squares formula over a field k is a polynomial identity of the form
\[
\left( x_1^2 + \cdots + x_r^2 \right) \left( y_1^2 + \cdots + y_s^2 \right) = z_1^2 + \cdots z_t^2,
\]
where the z's are bilinear in the x's and y's over k. If a sums-of-squares formula exists over R, then a theorem of Hopf from 1940 gives numerical restrictions on r, s, and t. This result was one of the earliest uses of the cup product in singular cohomology.

(Eisenberg): Connecting models with data to yield predictive results requires a range of parameter estimation, identifiability, and uncertainty quantification techniques. Identifiability analysis addresses the question of whether it is possible to uniquely recover the model parameters from a given set of data. In this talk, I will discuss my recent work developing identifiability methods using tools from computational algebra and systems theory, and present some applications to problems in human disease, including cholera, thyroid hormone regulation, and cancer.

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