Wednesday September 06, 2017 4:00 PM  5:30 PM Math Tower P131
 Rob Silversmith, SCGP
GromovWitten invariants of symmetric products of projective spaceThrough 3 general points and 6 general lines in $P^3$, there are exactly 190 twisted cubics; 190 is a (genuszero) GromovWitten invariant of $P^3$. I will introduce GromovWitten invariants of a smooth complex projective variety X, and discuss how a torus action on X can help us compute its GromovWitten invariants. In the case when X is a toric variety, Kontsevich used this method to compute any GromovWitten invariant of X. Givental and LianLiuYau used Kontsevich’s computation to prove a mirror theorem, which states that genuszero GromovWitten invariants of X have an interesting rigid structure, which had been previously predicted by physicists. I will discuss the difficulties that arise when X is not toric. In particular, I will talk about the nontoric orbifold $X=Sym^d(P^r)$, the symmetric product of projective space. By studying the equivariant geometry of $Sym^d(P^r)$, I extended the strategies of Givental/LianLiuYau to prove a mirror theorem for $Sym^d(P^r)$.

Wednesday September 13, 2017 4:00 PM  5:30 PM Math Tower P131
 Francois Greer, Stony Brook University
Elliptic Fibrations and NoetherLefschetz TheoryThe mirror symmetry philosophy suggests that the GromovWitten series of an elliptic fibration should be a modular form. Using the theta correspondence from automorphic forms and the Hodge theory of surfaces, we give a proof of this statement for genus 0 and base degree 1, and compute the series explicitly in many examples. Time permitting, we will discuss some ideas on how to generalize to all genera and base degrees.

Wednesday September 20, 2017 4:00 PM  5:30 PM Math Tower P131
 Philip Engel, Harvard
Algebraic/symplectic K3 dictionarySingular integralaffine structures on the sphere arise in connection to K3 surface in two distinct ways: Algebraically, as the dual complex of the central fiber of a degenerating family, and symplectically, as the base of a Lagrangian torus fibration. The Teichmuller space of such structures thus provides a dictionary between algebraic and symplectic moduli. Joint work with Simion Filip proves various structure results on the Teichmuller space of integralaffine structures on S^2it is itself a separated 44dimensional integralaffine manifold, with a natural 24 dimensional foliation, whose leaf space maps to the positive cone in R^{2,18}. Heuristically, any extension of the universal family of polarized K3 surfaces to a toroidal compactification would necessarily correspond to a section of the foliation over a certain hyperplane in R^{2,18}. Thus, the existence of such a section constructed by hyperKahler rotation evinces a program to extend the universal family.

Wednesday October 04, 2017 4:00 PM  5:30 PM Math Tower P131
 Luca Schaffler, University of Massachusetts, Amherst
The KSBA compactification of the moduli space of $D_{1,6}$polarized
Enriques surfacesIn this talk we describe the moduli compactification by stable pairs (also known as KSBA compactification) of a 4dimensional family of Enriques surfaces, which arise as the bidouble covers of the blow up of the projective plane at three general points branched along a configuration of three pairs of lines. The chosen divisor is an appropriate multiple of the ramification locus. Using the theory of stable toric pairs we are able to study the degenerations parametrized by the boundary and its stratification. We relate this compactification to the BailyBorel compactification of the same family of Enriques surfaces. Part of the boundary of this stable pairs compactification has a toroidal behavior, another part is isomorphic to the BailyBorel compactification, and what remains is a mixture of these two. To conclude, we construct an explicit Looijenga semitoric compactification of this 4dimensional family which we conjecture is isomorphic to the KSBA compactification studied.

Wednesday October 11, 2017 4:00 PM  5:30 PM Math Tower P131
 Xudong Zheng, J. Hopkins University
The Hilbert scheme of points on singular surfacesThe Hilbert scheme of points on a quasiprojective variety parametrizes its zerodimensional subschemes. When the variety is a singular surface, the geometry of the Hilbert scheme should reflect the singularity of the underlying surface. I will present a sufficient condition for the Hilbert scheme to be irreducible in terms of the singularity of the surface, namely, the surface has only Kleinian singularities, via a purely algebraic approach. I will also report work in progress on some geometric consequences following their irreducibility.

Monday October 16, 2017 4:00 PM  5:00 PM Math Tower P131
 Daniele Agostini, Visiting Stony Brook University
Asymptotic syzygies and higher order embeddings.The syzygies of an algebraic variety are the algebraic relations between its equations, and they often encode surprising geometric informations about the variety. For example, recently Ein and Lazarsfeld proved that one can read
the gonality of a curve off the syzygies of an embedding of very high degree. I will present a partial extension of this result in higher dimensions, especially in the case
of surfaces.

Wednesday October 18, 2017 4:00 PM  5:00 PM Math Tower P131
 Harold Blum, U. Michigan, Ann Arbor
Valuations, Singularities, and KstabilityLet L be a line bundle on a projective variety X. We use valuations to measure the singularities of the linear system mL as m goes to infinity.
Specifically, we consider the global log canonical threshold, also known as Tian’s alpha invariant, and the stability thresholds of L. The stability threshold generalizes an invariant recently introduced by Kento Fujita and Yuji Odaka. When X is a Fano variety, we show that the stability threshold detects the K(semi)stability of X. This talk is based on joint work with Mattias Jonsson.

Wednesday October 25, 2017 4:00 PM  5:30 PM Math Tower P131
 Sebastian CasalainaMartin, U. of Colorado,. Boulder
Algebraic representatives and intermediate Jacobians over perfect fieldsIntermediate Jacobians and AbelJacobi maps provide a powerful tool for the study of complex projective manifolds. In positive characteristic, over algebraically closed fields, algebraic representatives and regular homomorphisms provide a replacement for the intermediate Jacobian and AbelJacobi map. I will discuss recent progress, with Jeff Achter and Charles Vial, extending this theory to the case of perfect fields, as well as some applications to a question of Barry Mazur on weight one Galois representations arising from geometry.

Wednesday November 01, 2017 4:00 PM  5:30 PM Math Tower P131
 Inna Zakharevich, Cornell
Constructing derived motivic measuresMotivic measures can be thought of as homomorphisms out of the Grothendieck ring of varieties. Two wellknown such measures are the LarsenLunts measure (over $\mathbf{C}$) and the HasseWeil zeta function (over a finite field). In this talk we will show how to lift the HasseWeil zeta function to a map of $K$theory spectra which restricts to the usual zeta function on $K_0$. As an application we will show that the Grothendieck spectrum contains nontrivial elements in the higher homotopy groups.

Wednesday November 08, 2017 4:00 PM  5:30 PM Math Tower P131
 Fedor Bogomolov, NYU
$PGL(2)$invariants of collections of torsion points of elliptic curvesThe main object of the talk is a (complex) elliptic curve $E$ with a standard degree $2$ projection $π$ on $P^1$. Assuming that we fix one of the ramification points as a zero we obtain a subset $PE_{tors}$ of the images of torsion points on $E$ inside $P^1$.
This sets are different and we have shown jointly with Yuri Tschinkel that these sets are very different for different elliptic curves  they have finite intersection for any two nonisomorphic elliptic curves. However some subsets of the above $PE_{tors}$ are $PGL(2)$ equivalent. This holds for the images of points of order $3$ and order $4$.
In this talk I am going to discuss general problem of the behavior of $PGL(2)$invariants of the $k$tuples of the images of torsion points of different order.
For every subset of different $k$ points in $P^1$ we can define it's image in the moduli $M_{0,k}$ of $k$tuples of points which is essentially a quotient of projective space $S^kP^1= P^k$ by the action of $PGL(2)$. Thus $M_{0,k}$ is a rational variety of dimension $k3$. If we consider the images of points of finite order in different elliptic curves under natural projections then we obtain an( infinite) system of modular type curves with maps into $M_{0,k}$.
I will formulate three conjectures (semi theorems) about properties of such maps which provide a possiblity of realistic universal estimate for intersections between subset of $PE^i_{tors},i=1,2$ for different elliptic curves $E^i$.
These conjectures are formulated in our joint article with Yuri Tschinkel and Hang Fu. Note that there are pairs of curves $E^i$ with big intersection $≥ 22$ of $PE^i_{tors},i=1,2$ as it was shown in our joint article.

Wednesday November 15, 2017 4:00 PM  5:30 PM Math Tower P131
 Nicolas Addington, U. Oregon
Special cubic fourfolds and apolarityIn the moduli space of cubic fourfolds, Hassett's NoetherLefschetz divisors, which parametrize cubics containing special surfaces, are geometrically rich and heavily studied. Recently, Ranestad and Voisin considered some divisors parametrizing cubics "apolar" to special surfaces, and showed that one of them is _not_ a NoetherLefschetz divisor. I will explain why this is surprising, and present a new, more direct proof that three of their divisors are not NoetherLefschetz, using pointcounting methods over finite fields. Joint with Asher Auel.

Wednesday November 29, 2017 4:00 PM  5:30 PM Math Tower P131
 Christian Schnell, Stony Brook University
Extension theorems for differential forms on singular spacesThe talk is about the following problem: Suppose we have an algebraic (or holomorphic) differential form, defined on the smooth locus of an algebraic variety (or analytic space). Under what conditions does it extend to an algebraic (or holomorphic) differential form on a resolution of singularities of X?

Wednesday December 06, 2017 4:00 PM  5:30 PM Math Tower P131
 Davesh Maulik, MIT
GopakumarVafa invariantsIn this talk I want to explain a conjectural picture (joint with Toda) relating curvecounting invariants on CalabiYau threefolds with certain perverse sheaves on the Chow variety. If time permits, I'll try to discuss some more recent work connecting it with conjectural formula on the cohomology of moduli spaces of Higgs bundles.

