Algebraic geometry seminar

from Friday
June 01, 2018 to Monday
December 31, 2018
Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars

Wednesday
September 05, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Gabriele Di Cerbo, Princeton
Birational boundedness of low-dimensional elliptic Calabi-Yau varieties

I will discuss new results towards the birational boundedness of low-dimensional elliptic Calabi-Yau varieties, a joint project with Roberto Svaldi. Recent work in the minimal model program suggests that pairs with trivial log canonical class should satisfy certain boundedness properties. I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are indeed log birationally bounded. This implies birational boundedness of elliptically fibered Calabi-Yau manifolds with a section, in dimension up to 5. I will also explain how one could adapt our strategy to try and generalize the results in higher dimension, partly joint with W. Chen, J. Han and, C. Jiang and R. Svaldi.


Thursday
September 13, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Andrei Okounkov, Columbia University
Enumerative symplectic duality

Please note special day: Thursday instead of Wednesday.

Inspired by theoretical physicists, enumerative geometers study various highly nonobvious "dual" ways to describe curve counts in algebraic varieties (the traditional mirror symmetry being perhaps the best known example). The notion of symplectic duality, or 3-dimensional mirror symmetry originated in the study of 3-dimensional supersymmetric field theories and interchanges the degree-counting variables in generating functions with torus variables for equivariant counts. My goal in the talk is to explain the basic features of this phenomenon and indicate how it can be proven under a restricted definition of a symplectically dual pair (following ongoing joint work with Mina Aganigic).


Wednesday
September 19, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Chuanhao Wei, Stony Brook University
Log-Kodaira dimension and zeros of holomorphic log-one-forms

I will introduce the result about the relation between the zeros of holomorphic log-one-forms and the Log-Kodaira dimension, which is a natural generalization of Popa and Schnell's result on zeros of one-forms. Some geometric corollaries will be stated, e.g. algebraic Hyperbolicity of log-smooth family of log-general type. I will also briefly introduce the idea that a log-D module underlies a Mixed Hodge module which is a natural generalization of Deligne's canonical extension of variation of Hodge structures. All are welcome!


Wednesday
September 26, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Aaron Bertram, University of Utah
Stability Conditions on Projective Space.

Gieseker stability uses the Hilbert polynomial of a coherent sheaf divided by its leading coefficient as an "asymptotic" slope function. We propose a family of stability conditions on Castelnuovo-Mumford regular sheaves that use the Hilbert polynomial divided by its derivative as "exact" slopes. We conjecture that this converges to Gieseker stability (it's true in dimensions 1,2,3). This is joint work with Matteo Altavilla and Marin Petkovic, and an application to the classification of Gorenstein rings is joint work with Brooke Ullery.


Wednesday
October 03, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Angela Gibney, Rutgers University
Basepoint free classes on the moduli space of stable n-pointed curves of genus zero

In this talk I will discuss basepoint free classes on the moduli space of stable pointed rational curves that arise as Chern classes of Verlinde bundles, constructed from integrable modules over affine Lie algebras, and the Gromov-Witten loci of smooth homogeneous varieties. We'll see that in the simplest cases these classes are equivalent. Examples and open problems will be discussed.


Wednesday
October 17, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Matt Kerr, Washington University in St Louis
Hodge theory of degenerations

The asymptotics and monodromy of periods in degenerating families of algebraic varieties are encountered in many settings -- for example, in comparing (GIT, KSBA, Hodge-theoretic) compactifications of moduli, in computing limits of geometric normal functions, and in topological string theory. In this talk, based on work with Radu Laza, we shall describe several tools (beginning with classical ones) for comparing the Hodge theory of singular fibers to that of their nearby fibers, and touch on some relations to birational geometry.


Wednesday
October 24, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Will Sawin, Columbia University
What circles can do for you

In joint work with Tim Browning, we study the moduli spaces of rational curves on smooth hypersurfaces of very low degree (say, a degree $d$ hypersurface in $n$ variables in $n > 3 (d-1)2^{d-1}$). We show these moduli spaces are integral locally complete intersections and that they are smooth outside a set of high codimension. We get stronger results, with better codimensions, as the degrees of the rational curves grow. These results rely on the circle method from analytic number theory. I will explain how this application works, and how the same technique should apply to recent conjectures of Peyre about rational points on these hypersurfaces.


Monday
November 05, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Wei Ho, U. Michigan Ann Arbor and Columbia U.
Splitting Brauer classes with the universal Albanese

We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the class also splits the class. This can fail when the index is congruent to 2 modulo 4, but adding a single genus 1 factor to the Albanese suffices to split the class. This is joint work with Max Lieblich.


Wednesday
November 07, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Melody Chan, Brown University
Tropical curves, graph homology, and cohomology of M_g

Joint with Søren Galatius and Sam Payne. The cohomology ring
of the moduli space of curves of genus g is not fully understood, even for g small. For example, in the 1980s Harer-Zagier showed that the Euler characteristic (up to sign) grows super-exponentially with g---yet most of this cohomology is not explicitly known. I will explain how we obtained new results on the rational cohomology of moduli spaces of curves of genus g, via Kontsevich's graph complexes
and the moduli space of tropical curves.


Wednesday
November 14, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Jakub Witaszek, IAS
Frobenius liftability of projective varieties

The celebrated proof of the Hartshorne conjecture by Shigefumi Mori allowed for the study of the geometry of higher dimensional varieties through the analysis of deformations of rational curves. One of the many applications of Mori's results was Lazarsfeld's positive answer to the conjecture of Remmert and Van de Ven which states that the only smooth variety that the projective space can map surjectively onto, is the projective space itself. Motivated by this result, a similar problem has been considered for other kinds of manifolds such as abelian varieties (Demailly-Hwang-Mok-Peternell) or toric varieties (Occhetta-Wiśniewski). In my talk, I would like to present a completely new perspective on the problem coming from the study of Frobenius lifts in positive characteristic. Furthermore, I will provide applications of the theory of Frobenius lifts to varieties with trivial logarithmic cotangent bundle. This is based on a joint project with Piotr Achinger and Maciej Zdanowicz.


Wednesday
November 28, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Jason Starr, Stony Brook University
Symplectic Invariance of Rational Surfaces on Kaehler Manifolds

Gromov-Witten invariants are manifestly symplectically invariant and count holomorphic curves of given genus and homology class satisfying specified incidence conditions. The corresponding differential equations for holomorphic *surfaces* are not well-behaved and do not give invariants. Nonetheless, I will explain how the symplectically invariant Gromov-Witten theory can produce covering families of rational surfaces in Kaehler manifolds, e.g., every Kaehler manifold symplectically deformation invariant to a projective homogeneous space has a covering family of rational surfaces. The key input is a positive curvature result for spaces of stable maps proved jointly with de Jong.


Wednesday
December 05, 2018

4:00 PM - 5:00 PM
Math Tower P-131
Shizang Li, Columbia Universtiy
\ An example of liftings with different Hodge numbers

Does a smooth proper variety in positive characteristic know the
Hodge number of its liftings? The answer is ”of course not”. However, it’s not
that easy to come up with a counter-example. In this talk, I will first introduce
the background of this problem. Then I shall discuss some obvious constraints
of constructing a counter-example. Lastly I will present such a counter-example
and state a few questions of similar flavor for which I do not know an answer.


Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars