Algebra and Number Theory Seminar

Fall 2006

Thursdays, 11:15-12:05, 106 McAllister

September 14

Dan Goldston (San Jose State University)


Primes in Tuples


This talk will describe the ideas behind the recent work of Goldston-Pintz-Yildirim on gaps between primes and primes in tuples. Originally the method was a generalization of earlier work on primes using the circle method and moment methods, but eventually it moved closer to ideas related to the Selberg sieve.

September 21

Nayandeep Deka Baruah (University of Illinois at Champaign-Urbana)


Partition Identities Arising from Ramanujan's Modular Equations and Theta Function Identities


Recently, Bruce C. Berndt and I have shown that certain modular equations and theta function identities of Ramanujan imply elegant partition identities. In this talk, I will present some of our identities.

September 28
Eric Mortenson (PSU)

On The Broken 1-Diamond Partition


Recently, Andrews and Paule initiated the study of broken k-diamond partitions. Their study of the respective generating functions led to an infinite family of modular forms, about which they are able to produce interesting arithmetic theorems and conjectures for the related parition functions. Here we investigate the broken $1$-diamond parition and discuss a statistic and its role in congruence properties.

October 5

Ae Ja Yee (PSU)


The Lecture Hall Theorem and an l-generalization of Euler's theorem


Lecture hall partitions are partitions whose parts satisfy a certain ratio condition. Their enumeration by Bousquet-Melou and Eriksson gives a finite version of an theorem of Euler on strict partitions. In this talk we will discuss the lecture hall theorem and a generalization of Euler's theorem. This is joint work with Carla Savage.

October 12

Peter Schneider (Muenster)


The $p$-adic Satake isomorphism and crystalline Galois representations


Reporting on joint work with J. Teitelbaum and with C. Breuil I will introduce certain Banach algebra completions of the Hecke algebra of a maximal compact subgroup in p-adic GL_n and will compute them explicitly as algebras of p-adic analytic functions. Then I will discuss the emerging picture of a correspondence between crystalline p-adic Galois representations and characters of these p-adic Banach algebras. This is a first glimpse at a conjectural p-adic local Langlands correspondence.

October 19

Dale Brownawell (PSU)


Independence in Positive Characteristic

October 26

Matt Papanikolas (Texas A & M)


Difference Galois groups over function fields and periods of Drinfeld modules


In this talk we will present recent results on algebraic independence over function fields. By introducing a Tannakian formalism for Drinfeld modules and relating it to the Galois theory of certain Frobenius difference equations, we determine the transcendence degrees of fields generated by periods of Drinfeld modules and more generally Anderson t-modules. More precisely, we show that the transcendence degree of the period matrix of a Drinfeld module is equal to the dimension of its Galois group. We will discuss applications of this result to Carlitz logarithms, zeta values, and periods of t-motives.

November 2

Harold Diamond (University of Illinois at Champaign-Urbana)


Generalized Euler Constants and a Question about Monotonicity


We define a family {g_r} of generalized Euler constants and show that g_r tends to exp(-g) as r tends to infinity. (Here gis Euler's constant.) This sequence appears to converge monotonically; we investigate whether it really does.

November 9

Hamza Yesilyurt (University of Florida)


Ramanujan's forty identities for the Rogers-Ramanujan functions


In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. Most of the elementary proofs given for these identities are based on Schroter- type theta function identities in particular, the identities of L. J. Rogers. We give a generalization of some extensions of Rogers's identity due to D. Bressoud and also formulas of H. Schroter. Applications to modular equations, partition identities, Ramanujan's identities for the Rogers-Ramanujan functions as well as new identities for these functions are given.

November 16

Robert Vaughan (PSU)


Bombieri's theorem on primes in arithmetic progressions


Bombieri's theorem plays a crucial rôle in the recent work of Goldston, Pintz and Yildirim on small gaps between primes. In this talk an overview will be given of Bombieri's theorem and some possible generalizations.

November 23


November 30

George Andrews (PSU)


Partitions, Durfee Symbols and the Atkin-Garvan Moments of Ranks


In 1944, Freeman Dyson defined the rank of a partition with the object of providing a combinatorial interpretation of the Ramanujan conguences for the partition function, p(n). Dyson's discoveries and conjectures have led to an extensive field of research with exciting discoveries by Atkin, Garvan, Swinnerton-Dyer, Ono, Bringmann and Mahlburg and others. In this talk, we consider moments that Atkin and Garvan associated with the ranks defined by Dyson. We shall reveal the objects they enumerate and shall discuss implications and possibilities.

December 7

Paul Baum (PSU)


Geometric Structure in the Representation Theory of Reductive P-adic Groups


Let G be a reductive p-adic group. Irr(G) denotes the set of equivalence classes of smooth irreducible representations of G. A conjecture of A.-M. Aubert, P.Baum, and R.Plymen asserts that Irr(G) is a countable disjoint union of complex affine varieties and states what these varieties are. This talk reviews the conjecture and then gives a plausibility argument for its validity. The argument
is based on an equivalence relation between algebras which is less restrictive than Morita equivalence. The new equivalence relation permits the pulling apart of strata in the primitive ideal space in a way which is not allowed by Morita equivalence.

December 14

Stephen Zemyan (PSU)


Prime-Based Entire Functions, the Moments of their Zeroes, and Prime Gaps


The purpose of this paper is to investigate the properties of prime numbers by studying the zeroes of a related family of polynomials.  In particular, we consider the positive and negative power moments of their zeroes, as well as the relationships between the zeroes, and prime gaps.