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Dan Goldston (San Jose State University) |
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Title: |
Primes in Tuples |
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Abstract: |
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. |
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Abstract: |
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) |
| Title |
On The Broken 1-Diamond Partition |
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Abstract: |
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. |
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Title |
The Lecture Hall Theorem and an l-generalization
of Euler's theorem |
Abstract |
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. |
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The $p$-adic Satake isomorphism and crystalline Galois representations |
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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. |
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Independence in Positive Characteristic |
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Difference Galois groups over function
fields and periods of Drinfeld modules |
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Abstract: |
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. |
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Generalized
Euler Constants and a Question about Monotonicity |
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Abstract: |
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. |
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Hamza Yesilyurt (University of Florida) |
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Ramanujan's forty identities for
the Rogers-Ramanujan functions |
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Abstract: |
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. |
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Robert Vaughan (PSU) |
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Bombieri's
theorem on primes in arithmetic progressions |
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Abstract: |
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Title: |
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Abstract: |
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. |
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Paul Baum (PSU) |
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Geometric Structure in the Representation Theory of Reductive P-adic Groups |
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Abstract: |
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) |
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Prime-Based Entire Functions, the
Moments of their Zeroes, and Prime Gaps |
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Abstract: |
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. |