Algebra-Number Theory Archives for Academic Year 2019

Theta polynomials in geometry, Lie theory, and combinatorics

When: Wed, September 12, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Harry Tamvakis (University of Maryland) -
Abstract: The classical Schur polynomials form a natural basis for the ring
of symmetric polynomials, and have geometric significance since Giambelli
showed that they represent the Schubert classes in the cohomology ring of
Grassmannians. Moreover, these polynomials enjoy rich combinatorial properties. In the last decade, an exact analogue of this picture has emerged in the symplectic and orthogonal Lie types, with the Schur polynomials replaced by the theta and eta polynomials of Buch, Kresch, and the speaker. I will discuss this correspondence in the case of the symplectic group and theta polynomials.

The P-adic groups and their Weyl groups

When: Mon, October 15, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Xuhua He (University of Maryland)
Abstract: In a 1957 paper, Tits explained the analogy between the symmetric group $S_n$ and the general linear group over a finite field $\mathbb F_q$ and indicated that $S_n$ should be regarded as the general linear group over $\mathbb F_1$, the field of one element.

Following Tits' philosophy, we would like to regard the affine Weyl groups as the reductive group over $\mathbb Q_1$, the $1$-adic field. Although it is still premature to develop the theory of $1$-adic field at the current stage, we do have a fairly good understanding on the conjugacy classes of the affine Weyl groups, together with the length function on it, and such knowledge allows us to reveal a great part of the structure of the conjugacy classes of $p$-adic groups. I will explain how this idea may be used in representation theory and in arithmetic geometry.

Height pairings and torsion points

When: Mon, October 22, 2018 - 2:00pm
Where: Kirwan Hall 3206

Speaker: Laura DeMarco (Northwestern University) -

Abstract: I will discuss uniformity questions surrounding Unlikely Intersection problems -- the most famous of which is the Manin-Mumford Conjecture (proved by Raynaud, 1983) -- and a new result about elliptic curves and the geometry of their torsion points, joint with Holly Krieger and Hexi Ye. Our results hold over C, the field of complex numbers, but the proofs are carried out first over the field of algebraic numbers (and involve an analysis of certain height functions on P^1).

Degeneracy loci and nested Hilbert schemes II

When: Mon, October 29, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Amin Gholampour (University of Maryland) -
Abstract: Last year, I talked about a natural perfect obstruction theory on the "deepest" degeneracy locus of a map of vector bundles with applications to the nested Hilbert schemes of points on a projective surface S. In this talk, I will discuss other degeneracy loci and show that a certain "virtual resolution" of each of these loci is equipped with a natural perfect obstruction theory. This is used to construct and study a virtual fundamental class on the nested Hilbert schemes of points and curves on S. This virtual cycle arises naturally from Vafa-Witten/reduced local Donaldson-Thomas theory of S. This is a joint work with Richard Thomas.

Irreducible components of affine Deligne-Lusztig varieties, and orbital integrals

When: Thu, November 1, 2018 - 2:00pm
Where: Kirwan 1313

Speaker: Yihang Zhu (Columbia University)
Abstract: Affine Deligne-Lusztig varieties (ADLV) naturally arise from the study of Shimura varieties and Rapoport-Zink spaces. Their irreducible components provide interesting algebraic cycles on special fibers of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which equates the number of irreducible components of an ADLV (modulo a symmetry group) with a weight multiplicity of the Langlands dual group. Our method is to count the number of F_q points on the ADLV, and study the growth rate of the counting as q grows. This allows us to apply tools from local harmonic analysis, including twisted orbital integrals and the Base Change Fundamental Lemma. We reduce the problem to a problem about q-analogues of Kostant's partition functions. After estimating these partition functions, we reduce the conjecture to the previously known cases proved by Hamacher-Viehmann and Nie. Along the way we also use an independent-of-p argument, which allows us to "set p=0". This is joint work with Rong Zhou.

On the Beilinson-Bloch-Kato conjecture in the unitary Gan-Gross-Prasad paradigm

When: Mon, November 5, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Liang Xiao (University of Connecticut) -

Abstract: The Birch and Swinnerton-Dyer conjecture is known in the case of rank 0 and 1 thanks to the foundational work of Kolyvagin and Gross-Zagier. In this talk, I will report on a joint work in progress with Yifeng Liu, Yichao Tian, Wei Zhang, and Xinwen Zhu, which studies the analogue and generalizations of Kolyvagin's result to the unitary Gan-Gross-Prasad paradigm. More precisely, our ultimate goal is to show that, under some technical conditions, if the central value of the Rankin-Selberg L-function of an automorphic representation of U(n)*U(n+1) is nonzero, then the associated Selmer group is trivial; Analogously, if the Selmer class of certain cycle for the U(n)*U(n+1)-Shimura variety is nontrivial, then the dimension of the corresponding Selmer group is one.

Torsors on semistable curves and the problem of degenerations

When: Mon, November 12, 2018 - 2:00pm
Where: Kirwan Hall 1311

Speaker: V Balaji (Chennai Math Institute) -

Monodromy representations and arithmetic

When: Wed, December 5, 2018 - 2:00pm
Where: Kirwan Hall 3206

Speaker: Daniel Litt (IAS, Princeton) -

Abstract: Let $X$ be a smooth curve over a finitely generated field $k$, and let $\ell$ be a prime different from the characteristic of $k$. We analyze the dynamics of the Galois action on the deformation rings of mod $\ell$ representations of the geometric fundamental group of $X$. Using this analysis, we prove analogues of the Shafarevich and Fontaine-Mazur finiteness conjectures for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey-Mazur conjecture for function fields in characteristic zero.
For example, we show that if $X$ is a normal, connected variety over $\mathbb{C}$, the (typically infinite) set of representations of $\pi_1(X^{\text{an}})$ into $GL_n(\overline{\mathbb{Q}_\ell})$, which come from geometry, has no limit points. As a corollary, we deduce that if $L$ is a finite extension of $\mathbb{Q}_\ell$, then the set of representations of $\pi_1(X^{\text{an}})$ into $GL_n(L)$, which arise from geometry, is finite.

Openness of uniform K-stability in families of Q-Fano varieties

When: Mon, December 10, 2018 - 2:00pm
Where: Kirwan Hall 3206

Speaker: Yuchen Liu (Yale University) -

Abstract: K-stability is the algebraic notion which is supposed to characterize whether a Fano variety admits a K\"ahler-Einstein metric. One important feature of the notion of K-stability is that it is supposed to give a nicely behaved moduli space. To construct the K-moduli space of Q-Fano varieties as an algebraic space, one important step is to prove the openness of K-(semi)stable locus in families. In this talk, I will explain the proof of openness of uniform K-stability in families of Q-Fano varieties. This is achieved via showing the lower semi-continuity of delta-invariant, an interesting invariant introduced by Fujita and Odaka as a variant of Tian's alpha-invariant. This is a joint work with Harold Blum.

Transgression of the Euler class and arithmetic applications

When: Tue, January 15, 2019 - 2:00pm
Where: Kirwan Hall 3206

Speaker: Luis Garcia (University of Toronto, Canada) -

Abstract: I will survey recent work (joint with N. Bergeron and P.
Charollois) giving a new construction of certain cohomology classes of
SL_N(\bbZ) that were first defined by Nori and Szcech. To motivate our
approach, I will start by discussing the problem of how to compute linking
numbers in certain three-manifolds that fiber over the circle, e.g in the
complement of the trefoil knot in the 3-sphere. We will see that these
linking numbers are special values of L-functions, which implies that the
latter are rational numbers. Then I will explain some generalizations that
relate the topology of real locally symmetric spaces with the arithmetic
world of modular forms.

Gromov-Witten theory of CP^1-bundles over ruled surfaces

When: Mon, February 25, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Eric Lownes (University of Maryland) -
Abstract: We will discuss the techniques of virtual localization and degeneration in Gromov-Witten theory and the framework of topological quantum field theory. We will discuss a new result obtained by applying these techniques to families of non-toric threefolds given by CP^1-bundles over ruled surfaces over any non-singular projective curve. We compute the equivariant Gromov-Witten partition functions for all "section classes".

The Breuil-Mezard conjecture for potentially crystalline deformation rings

When: Fri, March 8, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Brandon Levin (U. Arizona) -
Abstract: The Breuil-Mezard conjecture predicts the geometry of local Galois deformation rings with p-adic Hodge theory condition in terms of modular representation theory. I will discuss joint work in progress with Daniel Le, Bao V. Le Hung, and Stefano Morra where we prove the conjecture in generic situations for a class of potentially crystalline deformation rings. The key ingredient is the construction of a local model which models the singularities of these Galois deformation rings.

The intersection motive of the moduli stack of shtukas

When: Mon, March 11, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Timo Richarz (U. Darmstadt) -
Abstract: Moduli stacks of shtukas are regarded as the function field analogues of Shimura varieties, and their étale cohomology is known to realize the Langlands correspondence for these fields. For the general linear group such a correspondence was established by L. Lafforgue in the 90’s building upon earlier work of Drinfeld in the two dimensional case. In a recent breakthrough V. Lafforgue constructs the automorphic to Galois direction of the correspondence for general reductive groups G over function fields. His completely new method makes it possible to systematically analyze the requirements of the cohomology theory in order to establish such a correspondence.

In the talk I report on joint work with J. Scholbach which aims at applying the theory of motives as developed by Voevodsky, Levine, Hanamura, Ayoub and Cisinski-Déglise to the constructions in the work of V. Lafforgue. We show that the intersection (cohomology) motive of the moduli stack of iterated G-shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic t-structures on triangulated categories of motives, and establish the analogue of the Geometric Satake Isomorphism of Lusztig, Ginzburg and Mirkovic-Vilonen in this set-up. This is in accordance with general expectations on the independence of \ell in the Langlands correspondence for function fields.

K-moduli conjecture and the recent progress

When: Wed, March 13, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Chenyang Xu (MIT)
Abstract: K-stability of Fano varieties was originally defined to capture the existence of a Kahler-Einstein metric. One of the most deep applications in algebraic geometry is that it conjecturally provides a good behaved moduli, which we call the K-moduli conjecture. I will discuss some recent progress toward the conjecture.

A Tannakian theory of displays and Rapoport-Zink spaces

When: Mon, March 25, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Patrick Daniels (UMCP) -
Abstract: We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by Bueltel and Pappas, and further studied by Lau. We use this framework to generalize the purely group-theoretic definition of Rapoport-Zink spaces given by Bueltel and Pappas, and to show that this definition coincides with the classical one in the case of unramified EL-type local Shimura data.

Codimension two cycles in Iwasawa theory

When: Wed, March 27, 2019 - 2:00pm
Where: Kirwan Hall 3206

Speaker: Bharathwaj Palvannan (UPenn) -
Abstract: Classical Iwasawa theory studies a relationship, called the Iwasawa main conjecture, between a $p$-adic $L$-function and a Selmer group. This relationship involves codimension one cycles of an Iwasawa algebra. This talk will discuss results on the topic of higher codimension Iwasawa theory. We will consider the restriction to an imaginary quadratic field of an elliptic curve defined over the rational numbers with good supersingular reduction at an odd prime. We shall also consider the tensor product of Hida families. This is joint work with Antonio Lei.

Sharpness of Tian's criterion of K-stability

When: Mon, April 22, 2019 - 2:00pm
Where: Kirwan 3206

Speaker: Ziquan Zhuang (Princeton University)

Abstract: K-stability is an algebraic notion that captures theexistence of K\"ahler-Einstein metric on Fano varieties. A famous
criterion of Tian states that a Fano variety of dimension n whose
alpha invariant is greater than n/(n+1) is K-stable. In this talk, I
will discuss some recent refinement of this criterion by K. Fujita and
present the construction of some (singular) Fano varieties with alpha
invariant n/(n+1) that is not K-stable to show that Tian's criterion
is sharp. This is joint work with Yuchen Liu.

Steenrod operations and the Artin-Tate pairing

When: Wed, April 24, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Tony Feng (Stanford University)

Abstract: In 1966 Artin-Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. I will present a resolution to this conjecture, based on a new connection to Steenrod operations and other ideas originating in algebraic topology.

Automorphisms of affine cubic surfaces

When: Wed, May 1, 2019 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Bill Goldman (UMCP) -
Abstract: In the 19th century, Vogt (and later Fricke and Klein) showed that the GIT quotient S of
SL(2) x SL(2,C) by Inn(SL(2,C) ) is an affine space C^3. Fricke and Klein were motivated by
uniformizing Riemann surfaces by hyperbolic non-Euclidean geometry. Viewing SL(2) x SL(2) as Hom(F2, SL(2))
(where F2 is the two-generator free group) leads to an algebraic action of Out(F2) (which is isomorphic
to the modular group GL(2,Z)) on X. This action has interesting dynamical properties. In particular this action
preserves the trace of the commutator of the free generators of F2, which is the cubic polynomial k = x^2 + y^2 + z^2 - xyz - 2
For simplicity we consider only the set of R-points, although the C-points are extraordinarily rich and complicated.
The topology bifurcates at the level sets of the critical values of k, which are +2 and -2 --- these are already
interesting and classically studied: integer points on the level set k=-2 are the Markoff triples,
and the level set k=-2 is the Cayley cubic characterized by having 4 nodes and a rational parametrization of its
affine patch. The dynamics bifurcates at the level k =18, which turns out to be an affine patch on the famous Clebsch diagonal surface.
We relate the geometry and dynamics of these affine cubics to their classical projective geometry,
also developed in the 19th century by Cayley, Salmon, Schlafli, Sylvester, and Cremona.

Tame derivatives and the Eisenstein ideal

When: Mon, May 6, 2019 - 2:00pm
Where: Kirwan Hall 3206

Speaker: Preston Wake (IAS, Princeton) -
Abstract: As was made famous by Mazur, the mod-5 Galois representation associated to the elliptic curve X_0(11) is reducible. In fact, the mod-25 Galois representation is also reducible. We’ll talk about this kind of extra reducibility phenomenon, and how its related, on one hand, to a ‘tame deriviative’ of an L-function, and on the other hand, to an algebraic invariant. This type of relation is predicted by the Bloch-Kato conjecture. This is a joint work with Akshay Venkatesh.

Do sums of squares dream of free resolutions?

When: Wed, May 8, 2019 - 2:00pm
Where: Kirwan 3206
Speaker: Greg Blekherman (Georgia Tech),
Abstract: A polynomial with real coefficients is called nonnegative if it takes only nonnegative values. For example, any sum of squares of polynomials is obviously nonnegative. The study of the relationship between nonnegative polynomials and sums of squares is a classical area in real algebraic geometry. I will review the rich history of this area and discuss some modern applications. Then I will discuss a recent line of work which shows how topic is inextricably linked to classical topics in algebraic geometry and commutative algebra, such as properties of minimal free resolutions.

Isolated Abelian Varieties and Cryptography

When: Thu, May 30, 2019 - 2:00pm
Where: Kirwan 3206

Speaker: Travis Scholl (UC Irvine)
Abstract: We call an elliptic curve over a finite field super-isolated if it admits no (rational) isogenies to other curves. In this talk, we will discuss the cryptographic motivations for super-isolated curves and give several examples. This concept can also be applied to general abelian varieties. Our main result is that for any g >= 3, there are finitely many super-isolated simple ordinary abelian varieties of dimension g.