Abstract: The Grothendieck ring of complex algebraic varieties is defined as the space of
formal sums $\sum_i a_i [X_i]$ of algebraic varieties with integer
coefficients, subject to the relations $[X]=[X-Z]+[Z]$ for closed
subvarieties $Z$ of $X$. I will talk about relatively recent
developments that show that the class of the affine line is a zero
divisor in the Grothendieck ring.
Olivier Debarre (École normale supérieure/Duke): Unexpected isomorphisms between hyperkähler fourfoldsAbstract: In 1985, Beauville and Donagi showed by an explicit geometric construction that the variety of lines contained in a Pfaffian cubic hypersurface in $P^5$ is isomorphic to a canonical desingularization of the symmetric self-product of a K3 surface (called its Hilbert square). Both of these projective fourfolds are hyperkähler (or symplectic): they carry a symplectic 2-form. In 1998, Hassett showed by a deformation argument that this phenomenon occurs for countably many families of cubic hypersurfaces in $P^5$. Using the Verbitsky-Markman Torelli theorem and results of Bayer-Macri, we show these unexpected isomorphisms (or automorphisms) occur for many other families of hyperkähler fourfolds. This involves playing around with Pell-type diophantine equations. This is joint work with Emanuele Macrì. Hiraku Nakajima (Kyoto University): Symplectic varieties and their quantizationAbstract: Symplectic varieties are algebraic varieties with symplectic forms on their regular loci. In representation theory people are interested in symplectic varieties as their quantization are often noncommutative algebras with rich representation theory. Quiver varieties, which I introduced more than 20 years ago, are examples ofsymplectic varieties. Their quantization are studied by various people. If time permits, I explain a new construction, called Coulomb branches, motivated by theoretical physics.Raman Parimala (Emory University): Isotropy of quadratic forms over function fieldsAbstract: A
consequence of Hasse-Minkowski theorem is that every quadratic form in
at least five variables over a totally imaginary number field is
isotropic (i.e., admits a nontrivial zero). It is an open question
whether quadratic forms in sufficiently many variables over function
fields of curves over totally imaginary number fields is isotropic. We
discuss recent progress towards understanding this question.Mihai Paun (University of Illinois at Chicago): Singular Hermitian metrics and positivity of direct images of pluricanonical bundlesAbstract: The positivity properties of the direct images of relative pluricanonical bundles were extensively studied in algebraic geometry, starting with the work of T. Fujita, Y. Kawamata, J. Kollár, E. Viehweg among many others. In our talk we will explore some of the metric properties of these sheaves. We will first recall the main notions involved. Then we will explain some of the results obtained via analytic methods, such as:(1) The metric counterpart of the "weak semi-positivity" type results in algebraic geometry (2) A few applications (a flatness criteria and some particular cases of the Iitaka conjecture). Karl Schwede (University of Utah): Perfectoid multiplier/test ideals and applicationsAbstract: Building on the ideas of recent proofs of the direct summand conjecture of Andre and Bhatt, we introduce a mixed characteristic analog of the multiplier ideal from birational geometric and the test ideal from characteristic p commutative algebra. We prove basic properties of this perfectoid multiplier/test ideal and as an application, we deduce a uniform bound on the growth of symbolic powers in regular rings (similar to previous results of Ein-Lazarsfeld-Smith and Hochster0Huneke in the equal characteristics setting). This is joint work with Linquan Ma.Behrouz Taji (Northwestern University/Freiburg): Orbifold stability and Miyaoka-Yau inequality for minimal pairsAbstract: It
was proved by Yau that complex projective varieties with ample
canonical bundle verify a certain Chern class inequality that is
stronger than Bogomolov-Gieseker inequality for semistable holomorphic
bundles. Through sophisticated characteristic-p methods, Miyaoka
established a different -- and in some sense weaker -- inequality for
higher dimensional minimal models; an inequality that was crucial in the
resolution of the Abundance Conjecture for threefolds. In a joint work
with H. Guenancia we use conical Kahler-Einstein theory to bridge the
gap between the two inequalities of Miyaoka and Yau and generalize their
results to the context of log-minimal models. Julianna Tymoczko (Smith College): Representation theory associated to Hessenberg varietiesAbstract:Hessenberg
varieties are a family of varieties that generalize Springer fibers,
which are the central objects in one of first constructions of a
geometric representation. We will describe symmetric group
representations associated to Hessenberg varieties, including recent
work on a longstanding conjecture of Stanley and Stembridge by Brosnan
and Chow and, independently, Guay-Pacquet. Time permitting, we will also
discuss open and related questions. |

Stony Brook 2017 >