Speaker: Nigel Boston, USC.
Title: An overview of the proof of Fermat's Last Theorem.
Speaker:
Gretchen Matthews, Clemson University.
Title: Multipoint codes as subcodes and implications for decoding.
Speaker:
Sebastian Pauli, University of North Carolina at Greensboro.
Title: Construction of Class Fields of Local Fields.
Speaker:
Ken Ono , University of Wisconsin at Madison.
Title: Modular Forms, Infinite Products, and Singular Moduli.
Department colloquium,
LC 412, 3:30 pm.
Speaker: Ken Ono, University of Wisconsin at Madison.
Title: Freeman Dyson's "Challenge for the Future":
The mock theta functions.
"The mock theta-functions give us tantalizing hints of a grand synthesis
still to be discovered. Somehow it should be possible to build them into a
coherent group-theoretical structure, analogous to the structure of modular
forms which Hecke built around the old theta-functions of Jacobi. This
remains a challenge for the future."
-Freeman Dyson, 1987
Here we announce a solution to Dyson's "challenge for the future" by
providing the "coherent group-theoretical structure" that Dyson desired.
In joint work with Kathrin Bringmann, we show that Ramanujan's mock theta
functions are special examples of exotic Maass forms. We use these results to
prove theorems about Dyson's partition ranks, and to solve the 1966
Andrews-Dragonette Conjecture, whose history dates to Ramanujan's last letter
to Hardy. If time permits, I will describe maps which Bringmann and I define
which map classical cusp forms to Maass forms. These maps induce a canonical
isomorphism between such spaces of forms. This picture is the general
framework desired by Dyson.
Speaker: Josh Cooper, USC.
Title: Collinear triple hypergraphs and the finite plane Kakeya
Problem.
Speaker: Nigel Boston, USC.
Title: Arboreal Galois representations.
Speaker: Carrie Finch, USC.
Title: Sequences of reducible 0,1-polynomials with exponents in
arithmetic progression.
Speaker: Patrick Corn,
University of Georgia.
Title: The Brauer-Manin obstruction on surfaces.
Speaker: Nigel Boston, USC.
Title: Class numbers of p-groups and an application of elliptic
curves.
Speaker:
Eric Mortenson, Penn State University.
Title: On the broken 1-diamond partition.
Speaker: Matt Boylan, USC.
Title: The partition function modulo small primes.
Speaker: Michael Filaseta, USC.
Title: Irreducibility and gcd algorithms for sparse polynomials.
Speaker:
Title:
Department of Mathematics main page
Abstract: Algebraic geometry codes are error-correcting codes
defined using points on a curve over a finite field. Typically, one point
is used, and the code is called a one-point code. However, using more points
may result in a more efficient code with better error-correcting capabilities.
Even so, these multipoint codes are thought to be less amenable to practical
use. In particular, many of the algorithms developed for decoding one-point
codes do not readily extend to the multipoint case. In this talk, we view
multipoint codes as subcodes of one-point codes and discuss
implications for decoding.
Abstract: Let K be a p-adic field. We give an explicit
characterization of the abelian extensions of K of degree p by relating the
coefficients of the generating polynomials of extensions L/K of degree p to
generators of the norm group N_{L/K}(L^*). This is applied in the
construction of class fields of degree p^m.
Abstract: Modular forms play many roles in mathematics. In number
theory, modular forms often arise as generating functions for interesting
quantities such as representation numbers of integers by quadratic forms,
partition functions, values of L-functions, and also degrees of characters
of sporadic simple groups like the Monster. In his 1994 ICM lecture,
Borcherds found a striking new phenomenon. He proved that certain modular
forms of half-integral weight serve as generating functions for the infinite
product exponents of other modular forms, thereby greatly generalizing some
of the prettiest q-series dating back to works of Euler and Jacobi on
classical theta functions. His work pertained to an exceptionally rich
family of modular forms, those with a `Heegner divisor'. Zagier later found
a beautiful number theoretic explanation of the Borcherds phenomenon, one
involving singular moduli, complex multiplication, and elliptic curves. In
this lecture, we provide a general framework which includes Zagier's
reformulation of Borcherds' theory as a special case. We show that all of
these results follow from beautiful properties of a delightfully rich
sequence of modular forms, the weak Maass-Poincare series of half-integral
weight.
Abstract:
In his last letter to Hardy, Ramanujan defined 17 peculiar functions
which are now referred to as his mock theta functions. Although these
mysterious functions have been investigated by many mathematicians over the
years, many of their most basic properties remained unknown. This inspired
Freeman Dyson to proclaim:
Abstract: We show that the problem of counting collinear points in a
permutation (previously considered by the author and J. Solymosi) and
the well-known finite plane Kakeya problem are intimately connected.
Via counting arguments and by studying the hypergraph of collinear
triples we show a new lower bound (5q/14 + O(1)) for the number of
collinear triples of a permutation of GF(q) and a new lower bound (q(q +
1)/2 + 5q/14 + O(1)) on the size of the smallest Besicovitch set in
GF(q)^2. Several intriguing questions about the structure of the
collinear triple hypergraph are presented.
Abstract: We describe various results on Galois groups of iterates
of a given quadratic polynomial. Joint work with Rafe Jones (U. of Wisconsin).
Abstract:
Fix positive integers n and d and consider the sequence of reducible
polynomials 1+x^n+x^{n+d}, 1+x^n+x^{x+d}+x^{n+2d},
1+x^n+x^{x+d}+x^{x+2d}+x^{n+3d}, etc. In this talk, we will discuss the
length of this sequence of reducible polynomials based on the particular
choices for n and d.
Abstract: Let X be a variety over a number field k. One might hope
that X has a k-rational point whenever X has a k_v-rational point for
every completion k_v of k, but unfortunately this statement--the so-called
"Hasse principle"--is false in general. In 1971, Manin observed that many,
if not all, of the existing counterexamples to the Hasse principle could
be explained in terms of an obstruction to rational points coming from the
Brauer group of X. In this talk, we describe this obstruction and how it
can be computed on various classes of surfaces, with special attention to
Del Pezzo surfaces of degree 2.
Abstract: Let k(G) be the number of conjugacy classes of G and D(m) =
#{k(G) : #G = m}. In joint work with Marty Isaacs I show that D(p^9) is an
unbounded function of the prime p by reducing it to a question on elliptic
curves.
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 were
able to produce interesting arithmetic theorems and conjectures for the
related partition functions. Here we investigate the broken 1-diamond
partition and discuss a statistic and its role in congruence properties.
Abstract: Let p(n) denote the ordinary partition function, the
function which counts the number of ways to write the positive integer n as a
sum of positive integers. Arithmetic properties of p(n) have been studied
since the work of Ramanujan, who famously proved, for example, that
p(5n+4) is divisible by 5 for all n. In more recent work, Ahlgren and Ono
proved that p(n) satisfies similar linear congruences modulo every positive
integer M coprime to 6. By contrast, it is widely believed that
p(n) does not satisfy any linear congruence modulo 2 or 3, though less is
known. In this talk, we use modular forms and their connections with Galois
representations to prove that, for all positive integers s, there are
infinitely many n in every residue class r mod 3^s with p(n) not divisible
by 3. We also prove similar results on the indivisibility of p(n) by the
primes 2, 5, 7, and 11.
Abstract: Let f(x) be a polynomial with integer coefficients. Let
n be the degree of f, let r be its number of terms and let H be its
height (the largest absolute value of a coefficient). A sparse polynomial is
one in which r is small compared to n (a more precise definition will not
be necessary). Our main objective is to describe algorithms that run fast
with r and H being fixed. Known algorithms, such as the L^3 algorithm for factoring and the
Euclidean algorithm for gcd's, run in time that are polynomial in n. We
describe algorithms that run in time that are polynomial in log n. The
results are from joint work with Andrzej Schinzel.
Abstract: