Algebra-Number Theory Archives for Academic Year 2016

Organizational meeting

When: Wed, August 31, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Organizational meeting (ANT and Lie Theory - Representation Theory) -

Arithmetic descent

When: Mon, September 12, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Hilaf Hasson (UMD) -
Abstract: I will be describing a joint work with Ryan Eberhart. It is a well known fact that for every finite group G, there exists a G-Galois branched cover of P^1_K for some number field K. Given such a cover, Hilbert Irreducibility tells us that there are infinitely many G-Galois field extensions of K that arise as specializations. We will be asking variants of the question: under what conditions is one of these specializations isomorphic to the compositum of K with a G-Galois field extension of Q?"

Duality for real elliptic curves and motivation from physics

When: Wed, September 14, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jonathan Rosenberg (UMD) -
Abstract: I will explain a problem about duality for real elliptic curves and how it was motivated by physics, and how the result fits with work of Caldararu and Antieau.

Intersection numbers on the Hilbert scheme of points on surfaces

When: Mon, September 19, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Amin Gholampour (UMD) -
Abstract: Carlsson-Okounkov expressed the Chern classes of certain natural K-theory classes over the product of two Hilbert scheme of points on nonsingular surface in terms of Nakajima operators. As an application, taking the trace, they obtain a closed formula for the Euler class of the twisted tangent bundle of the Hilbert scheme generalizing Gottsche's formula.
We study certain top intersection products on the Hilbert scheme of points on a nonsingular surface relative to an effective smooth divisor. We find a formula relating these numbers to the corresponding intersection numbers on the absolute Hilbert schemes. In particular, we obtain a relative version of the formula found by Carlsson and Okounkov. If time permits, we discuss the relation of this to some virtual integrations over the nested Hilbert scheme of points on nonsingular surfaces as well as to some Donaldson-Thomas invariants of threefolds.
This is a joint work with Artan Sheshmani.

A new approach to etale homotopy theory

When: Mon, September 26, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: David Carchedi (George Mason) -
Abstract: Etale homotopy theory, as originally introduced by Artin and Mazur in the late 60s, is a way of associating to a suitably nice scheme a pro-object in the homotopy category of spaces, and can be used as a tool to extract topological invariants of the scheme in question. It is a celebrated theorem of theirs that, after profinite completion, the etale homotopy type of an algebraic variety of finite type over the complex numbers agrees with the homotopy type of its underlying topological space equipped with the analytic topology. We will present work of ours which offers a refinement of this construction which produces a pro-object in the infinity-category of spaces (rather than its homotopy category) and applies to a much broader class of objects, including all algebraic stacks. We will also present a generalization of the previously mentioned theorem of Artin-Mazur, which holds in much greater generality than the original result.


When: Mon, October 3, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Amin Gholampour (UMD) -
Abstract: We construct natural virtual fundamental classes for nested Hilbert schemes (of points and curves) of a nonsingular projective surface S. This allows us to define new invariants of S that recover some of the known important cases such as Poincare invariants (algebraic Seiberg-Witten invariants) of Durr-Kabanov-Okonek and the stable pair invariants of Kool-Thomas. In the case of the nested Hilbert scheme of points, we can express some of our invariants in terms of integrals over the products of Hilbert scheme of points on S, and relate them to the vertex operator formulas found by Carlsson-Okounkov. When the canonical line bundle K of S is positive, in combination with Mochizuki's formulas, we are able to express certain equivariant Donaldson-Thomas invariants of stable 2-dimensional sheaves on the total space of K in terms of our invariants of the nested Hilbert schemes, Seiberg-Witten invariants of S, and the integrals over the products of Hilbert scheme of points on S. This is a joint work with Artan Sheshmani and Shing-Tung Yau.

TBA (Achter)

When: Wed, November 9, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jeff Achter (Colorado State University) -

On the Medvedev-Scanlon Conjecture in dimension 3

When: Mon, November 14, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Matthew Sattriano (Waterloo) -
Abstract: Let X be a variety defined over an algebraically closed field of characteristic 0 and let \phi\colon X\to X be a birational automorphism. The Medvedev-Scanlon conjecture predicts when there is a rational point of X with dense orbit under \phi. We prove their conjecture in positive Kodaira dimension and then, contingent on conjectures in the Minimal Model Program, prove the conjecture for certain minimal threefolds of Kodaira dimension 0. This is joint work with Jason Bell, Dragos Ghioca, and Zinovy Reichstein.

Torsion orders of complete intersections

When: Tue, November 22, 2016 - 1:00pm
Where: Kirwan Hall 1311
Speaker: Andre Chatzistamatiou (MPIM Bonn) -
Abstract: (Joint with M. Levine)
By a classical result of Roitman, a complete intersection $X$ of sufficiently small degree admits a rational decomposition of the diagonal. This means that some multiple of the diagonal by a positive integer $N$, when viewed as a cycle in the Chow group, has support in $X\times D\cup F\times X$, for some divisor $D$ and a finite set of closed points $F$. The minimal such $N$ is called the torsion order. We study lower bounds for the torsion order following the specialization method of Voisin, Colliot-Th\'el\`ene and Pirutka. We give a lower bound for the generic complete intersection with and without point. Moreover, we use methods of Koll\'ar and Totaro to show lower bounds for the very general complete intersection.

Perverse obstructions to flat regular compactifications.

When: Mon, February 13, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Patrick Brosnan (University of Maryland) -
Abstract: Suppose S is a smooth, complex variety containing a dense Zariski
open subset U, and suppose W is a smooth projective family of varieties over U.
It seems natural to ask when W admits a regular flat compactification over S.
In other words, when does there exist a smooth variety X flat and proper over S containing W as a Zariski open subset? Using resolution of singularities, it is not hard to see that it is always possible to find a regular flat compactification when S is a curve.
My main goal is to point out that, when dim S > 1, there are obstructions coming from local intersection cohomology. My main motivation is the recent preprint of Laza, Sacca and Voisin (LSV) who construct a regular flat compactification in the case that W is a certain family of abelian 5-folds over an open subset of 5 dimensional projective space. On the one hand, I'll explain how to compute the intersection cohomology in certain related examples and show that these are obstructed. On the other hand, I'll use the vanishing of the intersection cohomology obstructions implied by the LSV theorem to deduce a theorem on the palindromicity of
the cohomology of certain singular cubic 3-folds.

On rationality of critical L-values for U(3)

When: Mon, March 27, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Yoshihiro Ishikawa (Okayama University, JAPAN) -

Title: On rationality of critical $L$-values for $U(3)$

We introduce Harder type periods as the difference of two rational
structures attached to the zeta integral of Gelbart-Piatetski-Shapiro.
One is obtained from Whittaker model of our generic cohomological cuspidal
representation $\pi$. The other comes from the cohomological interpretation
of the integral, by using Mahnkopf cycles on Picard modular surface. We show
the cuspidality preservation of $Aut(\C)$-action on $\pi$, looking at the
structure of the automorphic spectrum on U(3). The non vanishing problem of
archimedean integral is cleared by my Whittaker new vectors. So we get our
rationality of the critical values of $L$-function for quasi-split $U(3)$.

A Brill-Noether theorem for curves of a fixed gonality

When: Mon, April 3, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Dhruv Ranganathan (Massachusetts Institute of Technology) -
Abstract: The Brill-Noether varieties of a curve C parametrize embeddings of C of prescribed degree into a projective space of prescribed dimension. When C is general in moduli, these varieties are well understood: they are smooth, irreducible, and have the ``expected" dimension. As one ventures deeper into the moduli space of curves, past the locus of general curves, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. I will present a generalization of the Brill--Noether theorem, which determines the dimensions of the Brill--Noether varieties on a general curve of fixed gonality, i.e. "general inside a chosen special locus”. The proof blends recent advances in tropical linear series theory and Berkovich geometry with input from logarithmic Gromov-Witten theory. This is joint work with Dave Jensen.

Thesis defense: exponentiation of motivic zeta functions

When: Fri, April 7, 2017 - 2:00pm
Where: Kirwan Hall 1310
Speaker: Jonathan Huang (UMD)

Degenerations of hyperkähler manifolds

When: Wed, April 12, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Guilia Sacca (Stony Brook University) -
Abstract: The problem of understanding semistable degenerations of K3 surfaces
has been greatly studied and is completely understood. The aim of this
talk is to present joint work in progress with J. Kollár, R. Laza,
and C. Voisin giving partial generalizations to higher dimensional
hyperkähler (HK) manifolds. I will also present some applications,
including a generalization of theorem of Huybrechts to possibly
singular symplectic varieties and shortcuts to showing that certain HK
manifolds are of a given deformation type.

The Hilbert schemes of points on singular surfaces

When: Wed, April 19, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Xudong Zheng (Johns Hopkins University) -
Abstract: The Hilbert scheme of points on a smooth algebraic surface parametrizes its zero-dimensional subschemes. They have received a great deal of attention from representation theory. When the surface is singular, the geometry of the Hilbert scheme should reflect the singularity of the underlying surface, as well as exhibiting some degeneration phenomenon within a family of surfaces. 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 irreducibility.

Wall-crossing for genus zero K-theoretic quasimap theory

When: Mon, April 24, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Fenglong You (Ohio State University) -
Abstract: For each positive rational number epsilon, we define K-theoretic epsilon-stable quasimaps to certain GIT quotients W//G. For epsilon>1, this recovers the K-theoretic Gromov-Witten theory of W//G introduced in more general context by Givental and Y.-P. Lee.

For arbitrary \epsilon_1 and \epsilon_2 in different stability chambers, these K-theoretic quasimap 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.

Tate Days V: Tate's thesis

When: Sat, May 6, 2017 - 9:30am
Where: Kirwan Hall 1311
Speaker: D. Bekkerman, S. Gilles, D. Kaufman, D. Zollers, R. Cowan, N. Dykas (UMD) -
Abstract: "Tate Days" is a one-day graduate student seminar centered around a paper of John Tate. This seminar, the fifth in the series, is centered around the Ph.D. thesis of Tate (Princeton, 1950).

The talks will be an exposition of Tate's thesis which used harmonic analysis and Fourier transforms to prove results in number theory, namely, the proof of the meromorphic continuation and functional equation of the zeta function of number fields; these properties for the Riemann zeta function were proved by Riemann.

The talks will be mostly self-contained, beginning with the basics of Fourier analysis on locally compact groups (especially p-adic numbers, adeles, ideles) and then Poisson summation, Riemann-Roch theorems and ending with the functional equation and meromorphic continuation of the zeta function of number fields.

There will be a break for lunch and coffee. Speakers are our own graduate students: D. Bekkerman, S. Gilles, D. Kaufman, D. Zollers, R. Cowan, N. Dykas

The Igusa cusp form conjecture

When: Mon, May 8, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Georg Oberdieck (Massachusetts Institute of Technology) -
Abstract: Curve counting invariants of Calabi-Yau threefolds are conjecturally related to modular forms.
A beautiful example arises for the product of a K3 surface and an elliptic curve. Here the
invariants are conjectured to be given by the reciprocal of the Igusa cusp form, a Siegel
modular form. In this talk I will explain recent work with Aaron Pixton and Junliang Shen
which yields a proof of this conjecture. The proof uses both sheaf theoretic methods
(via derived auto-equivalences, with J. Shen) and new holomorphic anomaly
equations in Gromov-Witten theory (with A. Pixton).