Algebra-Number Theory Archives for Academic Year 2014

Jumps in the Archimedean height

When: Mon, September 9, 2013 - 2:00pm
Where: Math 1311
Speaker: Gregory Pearlstein (Texas A&M) -
Abstract: We will discuss recent with P. Brosnan
towards the reformulation of the Hodge conjecture
in terms of the asymptotic behavior of the metric
on an associated biextension line bundle.

Integral models of Siegel modular varieties with $Gamma_1(p)$-type level-structure

When: Mon, September 23, 2013 - 2:00pm
Where: Math 1311
Speaker: Richard Shadrach (MSU) -
Abstract: The Siegel modular varieties are moduli spaces for abelian
schemes with certain additional structures. Integral models of these
varieties can be defined by posing a moduli problem over the p-adic
integers. In the case of Gamma_1(p)-type level structure, we consider
moduli problems that use ``Oort-Tate generators" for certain group
schemes. In this case I will construct explicit local models, i.e.
simpler schemes which can be used to study local properties of the
integral models. I will then use the local model for the Siegel
modular variety of genus 2 to construct a resolution of the integral
model which is regular with special fiber a divisor of nonreduced
normal crossings.

Local-Global Principles in Orbits

When: Mon, September 30, 2013 - 2:00pm
Where: 1311
Speaker: Alex Kontorovich (Institute for Advanced Study) -
Abstract: We will discuss some natural problems in arithmetic which can be (re)formulated as local-global principles for orbits of certain "thin" semigroups of integer matrix groups. Applications include partial progress towards Zaremba's conjecture and McMullen's "Arithmetic Chaos" Conjecture on the ubiquity of "low-lying" closed geodesics on the modular surface defined over a given number field. The main tools are expander graphs, bilinear forms, and bounds for exponential sums. This is joint work with Jean Bourgain.

Conformal blocks and cohomology in genus 0

When: Wed, October 2, 2013 - 2:00pm
Where: Math 1311
Speaker: Swarnava Mukhopadhyay (UMCP) -
Abstract: In this joint work with Prakash Belkale, we give a characterization of conformal blocks in terms of the singular cohomology of suitable smooth projective varieties, in genus 0 for classical Lie algebras and G2.

Three Hopf algebras and their common categorical background

When: Mon, October 7, 2013 - 2:00pm
Where: Math 1311
Speaker: Ralph Kaufmann (Institute for Advanced Study) -
Abstract: We discuss the renormalization Hopf algebra of Connes and Kreimer, Goncharov's Hopf algebra for multi-zeta values and the Hopf algebra appearing in Baues' double cobar construction. We show that these are an example of a common algebraic framework. Moreover this framework is a manifestation of one of the properties of Feynman categories, which we briefly define and discuss at the end.

Stability condition in the derived category of coherent sheaves and counting invariants

When: Mon, November 4, 2013 - 2:00pm
Where: Math 1311
Speaker: Amin Gholampour (University of Maryland) -
Abstract: I will give an introduction to and a review of some of the known results on the stability conditions in the derived category of coherent sheaves, Donladson-Thomas type invariants, and wall-crossing techniques. This is a preliminary talk for my second lecture, scheduled one week later.

Stable pair theory of K3 fibrations

When: Mon, November 11, 2013 - 2:00pm
Where: Math 1311
Speaker: Amin Gholampour (University of Maryland) -
Abstract: I will talk about my joint work with A. Sheshmani and Y. Toda. We study the stable pair theory of K3 fibrations over curves with possibly nodal fibers. We express the stable pair invariants of the fiberwise irreducible classes in terms of the famous Kawai-Yoshioka formula for the Euler characteristics of moduli space of stable pairs on K3 surfaces and Noether-Lefschetz numbers of the fibration. In the case that the K3 fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants of the fiberwise curve classes in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.

Explicit high ranks in higher genus

When: Wed, November 13, 2013 - 2:00pm
Where: Math 1311
Speaker: Douglas Ulmer (Georgia Institute of Technology) -
Abstract: For every genus g>0 and most primes p, we write down a curve over Fp(t) with Jacobian of large rank and with explicit divisors to fill out a finite index subgroup of the Mordell-Weil group. The height pairings among the explicit points have a very nice group-theoretic description, and we have an analytic class number formula of the type |sha|=square of index of the explicit points in the full Mordell-Weil group.

Twisted K-theory and division algebras.

When: Fri, November 15, 2013 - 2:00pm
Where: Math 3206
Speaker: Benjamin Antieau (University of Washington) -
Abstract: I will discuss topological twisted K-theory and its role in gaining insight about the period-index conjecture in algebraic geometry. This conjecture is about division algebras, but has turned out to be very difficult in complex dimension at least 3. On the other hand, using twisted K-theory, I solved with Ben Williams the analogous problem for topological spaces, obtaining a result that strongly suggests that the original, algebraic conjecture is false.

Regularity of general rational curves on Fano hypersurfaces

When: Mon, November 18, 2013 - 2:00pm
Where: Math 1311
Speaker: Sara Gharahbeigi (University of Missouri) -

Upper bounds on class numbers and applications to cyclotomic fields

When: Fri, November 22, 2013 - 2:00pm
Where: Math 1311
Speaker: John Miller (Rutgers University) -
Abstract: In number theory, "explicit formulas" relate sums over primes to sums
over zeta zeros. Explicit formulas have been used to find upper bounds
on class numbers for number fields of small discriminant. By finding
lower bounds for sums over prime ideals, we can extend this approach to
fields of larger discriminant. We apply these results to calculate the
class numbers of cyclotomic fields which have not been treatable by
other methods. This talk will be accessible to graduate students and

L^p bounds for eigenfunctions on locally symmetric spaces

When: Fri, December 13, 2013 - 2:00pm
Where: Math 3206
Speaker: Simon Marshall (Northwestern University) -
Abstract: Let M be a compact Riemannian manifold, and f an L^2-normalised Laplace
eigenfunction on M. If p > 2, a theorem of Sogge tells us how large the
L^p norm of f can be in terms of its Laplace eigenvalue. For instance,
when p is infinity this is asking how large the peaks of f can be. I will
present an analogue of Sogge's theorem for eigenfunctions of the full ring
of invariant differential operators on a locally symmetric space, and
discuss some links between this result and number theory.

Special Values of Automorphic L-functions

When: Mon, January 13, 2014 - 2:00pm
Where: Math 1310
Speaker: A. Raghuram (IISER (Pune), India) -
Abstract: One approach to studying the special values of automorphic L-functions is to interpret a given analytic theory of L-functions in terms of maps in the cohomology of arithmetic groups. In this talk, I will illustrate how one can view the classical theorem of Langlands on L-functions appearing in the constant terms of Eisenstein series in terms of restriction maps from the cohomology of the Borel-Serre compactification of a locally symmetric space to the cohomology of the boundary of this compactification. This is a report on an ongoing collaboration with G√ľnter Harder. I will begin my talk with some easy examples involving L-functions of modular forms.

Eisenstein cocycle and a conjecture of Gross

When: Mon, February 24, 2014 - 2:00pm
Where: Math 1311
Speaker: Michael Spiess (IAS, Princeton/Bielefeld) -
Abstract: I will present a construction of the Eisenstein cocycle
and discuss its relation to values of partial Zeta-functions of
totally real fields at non-positive integers. I will also discuss
relations to conjectures of Gross and Dasgupta (this is joint
work with Samit Dasgupta).

Two kinds of Steenrod operations on motivic cohomology

When: Mon, March 24, 2014 - 2:00pm
Where: MATH 1311
Speaker: Patrick Brosnan (UMD) -
Abstract: In topology, Steenrod squares are operations on the mod 2 cohomology of a space. The i-th Steenrod square sends mod 2 cohomology in degree n to mod 2 cohomology in degree n+i.
For each prime odd p there are analogues of the Steenrod squares
called the p-th operations on mod p cohomology.

Motivic cohomology seems to be the best analogue in algebraic geometry for the topologists' cohomology groups. Unlike cohomology groups which are graded by degree, motivic cohomology groups are bigraded. The first degree is called the motivic degree and the second is called the simplicial degree. There are also two kinds of cohomology operations. The first are the simplicial operations constructed by Kriz and May. The second are the motivic ones constructed by Voevodsky.

In my talk, I will explain all of this along with a theorem I proved with
Roy Joshua comparing the two types of operations. (I will also sketch an independent proof of the result due to Guillou and Weibel.)

Special values of zeta functions and the de Rham Witt complex

When: Mon, April 7, 2014 - 2:00pm
Where: MATH 1311
Speaker: Niranjan Ramachandran (UMD) -
Abstract: I will report on the recent results obtained with J. Milne which provide a conjectural formula for the p-part of the special values of the L-functions of motives over finite fields. I will review some of the history (Artin-Tate conjecture) of this subject as well as the classical results of Illusie-Raynaud-Ekedahl on the de Rham Witt complex.

Special values of zeta functions and the de Rham Witt complex II

When: Mon, April 14, 2014 - 2:00pm
Where: MATH 1311
Speaker: Niranjan Ramachandran (UMD) -
Abstract: Last week's talk continued, focusing on the de Rham Witt complex.