Speaker: Matt Boylan, USC.
Title: Integer weight mock theta analogues.
Speaker: Dan Baczkowski, USC.
Title: On rational numbers associated with arithmetic functions
evaluated at factorials.
Speaker: Matt Boylan, USC.
Title: Differential operators on harmonic weak Maass forms.
Speaker: Vladimir Temlyakov, USC.
Title: Some problems where approximation theory meets
number theory.
Speaker:
Frank Thorne,
University of Wisconsin at Madison.
Title: Extensions of results on the distribution of primes.
Speaker: Michael Filaseta, USC.
Title: Diophantine Equations Arising from the Study of
Galois Groups I.
Speaker: Michael Filaseta, USC.
Title: Diophantine Equations Arising from the Study of
Galois Groups II.
Speaker: Vsevelod Lev,
Georgia Tech and University of Haifa.
Title: Additive bases in abelian groups.
NO SEMINAR.
Speaker: Karl Mahlburg, MIT.
Title: Asymptotics for partitions without sequences
Speaker: Hui Xue, Clemson University.
Title: Central derivatives of certain L-series.
NO SEMINAR.
Speaker: Jennifer Larson, USC.
Title: The partition function and Dyson's rank conjecture.
Department of Mathematics main page
Abstract: Recent work of Bringmann and Ono explains how the
"mock theta" functions of Ramanujan fit in the broader theory of automorphic
forms. While a mock theta function F(q) is not a modular form, by adding
to it the period integral of a suitable theta series (a modular form of
half-integral weight 1/2 or 3/2), we obtain a harmonic weak Maass form.
Since period integrals of modular forms are non-holomorphic, we say that
the mock theta function is the "holomorphic part" of the resulting Maass form.
In this talk, we discuss harmonic weak Maass forms whose non-holomorphic
parts are given by period integrals of modular forms of integer weight
like the classical Delta function. We will study arithmetic properties of
the holomorphic parts, which play the role of "mock theta"-type
functions in this setting.
Abstract: Florian Luca established that for a fixed rational
number r, there are a finite number of positive integers n and m for
which f(n!) = r * m!$ where f is one of the following arithmetic
functions: the number of divisors function, Euler's phi-function, or the sum
of the divisors function. We establish a generalization of these results.
In particular, a consequence of our work is the following:
Let k be a fixed positive integer. Then there are finitely many
positive integers n, m, a, and b such that
b * f(n!) = a * m!, where gcd(a,b) = 1 and the number of
distinct prime divisors of a*b <= k. (Joint work with Michael Filaseta,
Florian Luca, and Ognian Trifonov.)
Abstract: In this talk, I will give an overview of differential
operators on Maass forms. In particular, I will discuss recent work of
Bruinier, Ono, and Rhoades characterizing the images of some of these
operators, and I will discuss applications of their results.
Palmetto Number Theory
Series (PANTS) V,
Furman University,
Greenville, South Carolina.
Joint with the Compressed Sensing seminar.
Note that the date is different than usual, but the time and location are
the same.
Note that the room and time are different than
usual.
Abstract: The distribution of the prime numbers is a very classical
subject in number theory, and (arguably) one that is still poorly understood.
After giving a general introduction to the subject, we will discuss recent
work of Goldston-Graham-Pintz-Yildirim, Maier, Shiu, and
Granville-Soundararajan concerning the distribution of primes in short
intervals. We will then present several adaptations and generalizations
of their work. In doing so, we will obtain distribution theorems for
class numbers, ranks of elliptic curves, irreducible polynomials over
finite fields, and prime "bubbles" in imaginary quadratic fields.
Abstract: The first part of this talk will focus on the general
topic of studying Galois groups, in particular of Laguerre polynomials,
that led to some Diophantine investigations by the speaker and several
others. Some background will be given, including introductory
discussions on Galois groups and the use of Newton polygons.
One main goal of this part is to lead up to two Diophantine problems,
in particular, one that has been the focus of joint research with
M. Bennett and O. Trifonov and the other involving more
recent investigations with S. Laishram and N. Saradha.
The second part of the talk will be a close analysis of the
more recent work with Laishram and Saradha, eplaining in some
detail how the general Diophantine problem is resolved.
Abstract: The first part of this talk will focus on the general
topic of studying Galois groups, in particular of Laguerre polynomials,
that led to some Diophantine investigations by the speaker and several
others. Some background will be given, including introductory
discussions on Galois groups and the use of Newton polygons.
One main goal of this part is to lead up to two Diophantine problems,
in particular, one that has been the focus of joint research with
M. Bennett and O. Trifonov and the other involving more
recent investigations with S. Laishram and N. Saradha.
The second part of the talk will be a close analysis of the
more recent work with Laishram and Saradha, eplaining in some
detail how the general Diophantine problem is resolved.
Abstract: Consider two eight-element sets in R^3, each of which
is the set of vertices of a parallelepiped. What is the smallest possible
size of the Minkowski sum of such two sets? How the answer changes if we
replace R^3 with R^n or consider a vector space over a field of finite
characteristic?
These problems arise naturally in connection with the Additive Basis
Conjecture. Extending the main known result towards the proof of this
conjecture, a theorem of Alon, Linial, and Meshulam, we show that if
B_1,...,B_k are generating subsets of a finite abelian group G,
with k sufficiently large, then the multiset union of the B_i's
is an additive basis of G.
Joint work with Rom Pinchasi
Abstract: Partitions without sequences (i.e., containing no
adjacent parts) were recently studied in connection to a wide variety of
applications, including probability distributions and cellular automata. The
main result of this talk is an asymptotic series expansion for the number of
such partitions of size n. As shown by Andrews, the generating series for
these partitions is the product of a theta function and one of Ramanujan's
mock theta functions, and thus does not have the type of simple modular
transformation properties that are typically used to make such estimates. In
particular, any modular transformation introduces a non-holomorphic error
integral, which are in fact part of the main term in the asymptotic expansion.
Abstract: The famous Gross-Zagier formula reveals the miraculous
relation between the central derivatives of L-functions and heights of
certain CM points. In this talk I will give an introduction to the
Gross-Zagier formula and its various generalizations.
Southeast Regional Meeting on Numbers (SERMON),
Clemson University,
Clemson, South Carolina.
Abstract: Let p(n) be the ordinary partition function; i.e., p(n)
counts the number of ways to write the positive integer n as a sum of
smaller positive integers. Ramanujan proved, for all n, that
p(5n+4) = 0 mod 5, p(7n+5) = 0 mod 7, and p(11n+6) = 0 mod 11. All known
proofs of these congruences require facts from classical analysis.
Nevertheless, in 1944, Dyson (while an undergraduate at Cambridge) conjectured
a combinatorial explanation for these congruences. To each partition of
n, he associated a statistic called a "rank". He noticed that the partitions
of 5n+4 fall into 5 equinumerous sets according to their rank values modulo 5.
This was proved in 1954 by Atkin and Swinnerton-Dyer. In this talk, following
recent works of Zwegers, Bringmann and Ono, and Ahlgren and Treneer,
we discuss how Dyson's rank conjecture fits into the modern framework of
the theory of harmonic weak Maass forms. In particular, we outline how one
would prove the rank conjecture by using this theory to reduce it to a
certain finite computation. We will then discuss issues affecting the size
of this computation.