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Algebra and Number Theory Seminar

Fall 2007/Spring 2008 Schedule

Mondays 2:00-3:00pm, MATH 1311

Date	Speaker	Title	Abstracts

Spring 2008
4 Feb 2008	Harry Tamvakis UMCP	Giambelli formulas for Grassmannians, I
18 Feb 2008	Harry Tamvakis UMCP	Giambelli formulas for Grassmannians, II


10 March 2008	Yuri Zarhin (PSU)	Superelliptic jacobians and homotheties
24 March 2008	Jacob Lurie, MIT/Harvard/AIM	Two talks: Classical Elliptic cohomology and Derived Algebraic Geometry.	1-1:50pm, 2-2:50pm in MATH 1311
31 March 2008	Serkan Hosten , SFSU	The number of complex critical points of the product of powers of polynomial functions and maximum likelihood estimation	Let p_1, ..., p_n be polynomials in d variables, and u_1, ..., u_n be integers. We call the number of complex critical points of the product of p_i^u_i the maximum likelihood degree (ML degree) of p_1, ..., p_n. In this talk I will show that this (statistics motivated) number is a constant for generic u_1, ..., u_n. Moreover, for generic p_1, ..., p_n, I will give a formula. We will look at the case when p_i's are linear and we will also treat the "toric" case. As applications I will present the ML degrees for various nucleitide substitution models on small phylogenetic trees, the ML degree of Behrens-Fisher problem and also that of the bivariate missing data problem.
7 April	Michael Harris, Paris/NY	Theta correspondences for close unitary groups	(Joint ANT/Lie Theory & Representation theory seminar) The stabilization of the trace formula for unitary groups includes the construction of the endoscopic transfer from the group U(a)xU(b) to U(a+b). When b = 1, the theta correspondence provides an alternative and more intuitive construction of this transfer. I review the known properties of this correspondence as well as the similar correspondence between two unitary groups of the same size, which is closely related to central values of L-functions.
14 April 2008
21 April 2008	Atsushi Ichino, IAS Princeton	Trilinear forms and the central values of triple product L-functions	We give an explicit formula for certain global trilinear forms which appear in Jacquet's conjecture in terms of local trilinear forms and the central values of triple product L-functions.
28 April	Daniel Erman, UC Berkeley	8 points in A⁴	In A⁴, there exist nonreduced 0-dimensional schemes which cannot be deformed into a smooth scheme. The minimal degree of such a nonsmoothable 0-scheme is 8. I will explore why this happens for 8 points in A⁴, and I will explain how to determine when a given 0-scheme of degree 8 can be deformed into a smooth scheme. A key tool in this construction is the GL₄ action on the Hilbert scheme of 8 points in A⁴.
28 April	Bianca Viray, UC Berkeley/UMD	Existence of rational points on smooth projective varieties	Note: special time 3pm, special Room 1308 (ANT seminar double feature!!) Let K be a number field. We prove some results of Poonen, namely: 1) If there exists an algorithm for determining whether a smooth projective geometrically integral variety has a K-point, then there exists an algorithm for determining whether an arbitrary K-variety has a K-point. 2) Given a projective K-variety X and an open set U, there exists an algorithm to construct a smooth projective variety Y and a map f:Y->X such that f(Y(K))=U(K). We also give an explicit example of (Y, f) for a specific (X,U).
5 May	No seminar (Thesis defences)
6 May	Gerard Freixas, Universite de Paris, Paris-Sud	Arithmetic Riemann-Roch and Hilbert-Samuel type formulae for pointed stable curves	Note: special day (Tuesday) The aim of the talk is to formulate arithmetic Riemann-Roch and Hilbert-Samuel type formulae, and explain some consequences. We will focus on special values of Ruelle and Selberg zeta functions for modular curves and (maybe) on volumes of spaces of integral cusp forms for the Petersson metric.
12 May	Jason Starr (SUNY, Stony Brook)	Rational simple connectedness and Serre's "Conjecture II"	Rational simple connectedness is an algebro-geometric analogue of simple connectedness replacing continuous maps from the unit interval with morphisms from the projective line. Using this notion, de Jong, He and I proved the "split case" of Serre's "Conjecture II" for function fields of surfaces over an algebraically closed field: every principal bundle over a surface for a semisimple, simply connected algebraic group has a rational section. In particular, the split case implies the E8 case. Combined work of Merkurjev-Suslin, Ojanguren-Parimala, Colliot-Thélène - Gille - Parimala and Gille reduced the general case of Serre's Conjecture II for function fields to the E8 case. Thus Serre's Conjecture II is now settled for function fields of surfaces over an algebraically closed field. I will explain rational simple connectedness and how it applies to Serre's Conjecture II.






(Old) Fall 2007
3 Sept	Labor Day	No seminar
10 Sept	Niranjan Ramachandran, UMCP	Conjectures on algebraic cycles
17 Sept	Anders Buch, Rutgers University	Quiver coefficients of Dynkin type
24 Sept		Field Committee Meeting Discussion of courses for next year
1 Oct	Niranjan Ramachandran, UMCP	Arithmetic and non-commutative Geometry
8 Oct	Kartik Prasanna, UMCP	Algebraic cycles and exotic Heegner points I
15 Oct
22 Oct	Philip Boalch (IAS/ENS Paris)	Simple examples of moduli spaces of irregular connections Geometry/Topology seminar.
26 Oct (Friday)	Charles Weibel (Rutgers)	The Norm Residue is an isomorphism! (more specialized talk at 2:00, followed by a more general colloquium at 3:00)
29 Oct		No seminar
5 Nov		No seminar
12 Nov	Izzet Coskun, UIC	Positive rules for multiplying Schubert cycles in partial flag varieties
19 Nov	Jinhyun Park, Purdue University	Additive higher Chow groups in Euclidean scissors congruence, cyclic homology, etc.	Abstract: S. Bloch's higher Chow groups provide the motivic cohomology groups for smooth varieties. In this talk, a relatively new approach called the additive higher Chow groups will be explained with several concrete motivational goals in mind. In particular, I will explain some partial results and conjectures on how they can be related to, for instance, 1) motivic cohomology over the dual numbers, 2) Euclidean scissors congruence groups, 3) cyclic homology (additive K-theory), to name a few.
26 Nov		No seminar
3 Dec	Prakash Belkale, University of North Carolina	Generalizations of the Horn and saturation conjectures	I will first review the classical picture for GL(n), where objects from geometry (intersection theory of Grassmannians) and representation theory (invariant theory and sums of Hermitian matrices) are related in many ways. I will talk about some of these generalizations.
10 Dec	Kartik Prasanna	Periods, congruences and special values of L-functions I	In the first talk, I will give an introduction to congruences of modular forms (for the group GL(2,Q)) and their relation to adjoint L-values and Petersson inner products. In the second talk, I will discuss some results on ratios of Petersson inner products of modular forms related by the Jacquet-Langlands correspondence, the interpretation of the primes dividing such ratios as certain specific congruence primes and their relation to certain Rankin-Selberg L-values. Finally, I will propose a conjecture on Petersson inner products in the case of modular forms over totally real fields that makes precise (up to $p$-units) an algebraicity conjecture of Shimura.
	JS??

Organizers: Niranjan Ramachandran, Larry Washington (Math Dept of UMCP)

Last year's seminar schedule

Abstracts

Anders Buch Title: Quiver coefficients of Dynkin type
Abstract: The quiver coefficients for equioriented quivers of type A describe
the Grothendieck classes of orbit closures in the affine space of
quiver representations. These coefficients are now well understood;
they have signs that alternate with degree, are described by nice
combinatorial formulas, they generalize the monomial coefficients of
Schubert and Grothendieck polynomials, and are themselves special
cases of the K-theoretic Schubert structure constants on flag
varieties. The quiver coefficients can also be interpreted as a
formula for very general degeneracy loci. In my talk, I will define
quiver coefficients for arbitrary quivers without oriented loops, and
discuss to what extent these coefficients satisfy positivity
properties. I will also give a formula for quiver coefficients of
quivers of Dynkin type (in K-theory, only for orbit closures with
rational singularities). This formula shows that quiver coefficients
of type A3 have alternating signs.

Kartik Prasanna (several talks this year!)

Title: Periods, congruences and special values of L-functions I,II
In the first talk, I will give an introduction to congruences of
modular forms (for the group GL(2,Q)) and their relation to adjoint
L-values and Petersson inner products. In the second talk, I will
discuss some results on ratios of Petersson inner products of modular
forms related by the Jacquet-Langlands correspondence, the
interpretation of the primes dividing such ratios as certain specific
congruence primes and their relation to certain Rankin-Selberg
L-values. Finally, I will propose a conjecture on Petersson inner
products in the case of modular forms over totally real fields that
makes precise (up to $p$-units) an algebraicity conjecture of
Shimura.

Title: Algebraic cycles and exotic Heegner points I, ...

This series of talks will be an introduction to work in progress,
joint with Massimo Bertolini and Henri Darmon.

Let A be an elliptic curve over Q and K an imaginary quadratic field.
If K satisfies a certain condition with respect to A, called the
Heegner hypothesis, a famous article of Gross and Zagier (Inv. Math.
1986) constructs points on A rational over (abelian extensions of) K
and further relates the nontriviality of such points in the
Mordell-Weil group (tensor Q) to the central derivative of certain
Rankin-Selberg L-functions. The Heegner hypothesis is however not
satisfied by the pair (A,K) when A is a curve with complex
multiplication by (the same) K, and consequently no construction of
Heegner points was known before in this case.

We propose a construction of complex points in this latter setting
using the Abel-Jacobi map on higher dimensional cycles and conjecture
that these points (which we call exotic Heegner points) are algebraic
and even rational over suitable abelian extensions of K. In the first
talk, I will present some relevant background material and describe
this construction. In the later talks, I will outline a p-adic analog
of this construction, for which we can prove (in a rather convoluted
manner) that the points constructed are indeed rational. Finally, I
will explain how our rationality conjecture is implied by the Tate
conjecture for certain varieties. In some cases, we can check the
Tate conjecture explicitly, thus giving a direct ("motivic") proof of
rationality in both the complex and p-adic settings.

Izzet Coskun Title: Positive rules for multiplying Schubert cycles in partial flag varieties
Abstract: The program of giving positive, geometric formulae for multiplying Schubert cycles in homogeneous varieties was initiated in the nineteenth century by classical geometers like Schubert and Pieri. Pieri's rule for multiplying special Schubert cycles with arbitrary Schubert cycles in the Grassmannian is one of the fundamental results of the theory. In this talk we will generalize Pieri's rule to arbitrary Schubert cycles in arbitrary partial flag varieties. The rule has many applications to questions of reality, saturation results in partial flag varieties and quantum cohomology of Grassmannians and flag varieties.

Directions to Campus

By car from I-95

From I-95, take 495 East (directions to Route 1= Baltimore Avenue) and head south on Route 1 towards College Park. Stay on the right lane. Soon after Berwyn Road, there will be a new right turn lane which brings you to the entrance of campus. An immediate right and a STOP sign or two brings you to a paid parking lot.

The mathematics department is straight from the entrance, on the right, just before the big "M" circle. It is a building with a fountain and a bus stop in front and (mysteriously) it is marked "Glenn Martin Institute of Technology".

By train

New Carrolton Station is the closest station. A taxi ride or a short bus ride (F-6) will get you to campus. From Washington, DC Union Station, take the DC Metro Red line (in the direction of Glenmont) to Fort Totten and change to the Green line (direction of Greenbelt) and get off at the College Park Metro Station. Take a (free!) University Shuttle bus and get off at the first stop after entering the campus. The building behind the bus stop is the mathematics department.

For more detailed directions as well as a campus map, click here and scroll down.