Upcoming Events and
Speakers:
SERMON
2011 in Savannah, GA at
Armstrong
Atlantic
State
Universty
April 16-17,
2011
Wednesday, April 20.
Speaker: Andrew Vincent, University of South Carolina
Title: Applications of Extremal Graph Theory to Number Theory:
A Theorem of Rainer Dietmann, Christian Elsholtz, Katalin Gyarmati, and
Miklos Simonovits
Abstract: This is an expository talk about a theorem
concerning coprime Diophantine powersets. A set A of positive integers
is called a coprime Diophantine powerset if the shifted product ab + 1
of two different elements a and b of A is always a pure power, and the
occurring pure powers are all coprime. The statement of the theorem is
of particular interest to the analytic number theorist that is also
interested in issues dealing with transcendence theory. The proof, on
the other hand, depends entirely on some standard results in
combinatorics, namely on Ramsey theory. The way in which this theory is
utilized is delightful, in the humble but accurate opinion of the
speaker.
Previous Seminar
Talks:
Wednesday, April 6.
Speaker: Matt Boylan, University of South Carolina
Title: Congruences for p(n).
Abstract: Last week's talk was on congruences between values
of the ordinary partition function, p(n). This week, we will
survey some results from the last 10 years on congruences of type p(An
+ B) = 0 mod M.
Wednesday, March 30.
Speaker: John Webb, University of South Carolina
Title: Applications of modular
forms to the study of integer
partitions.
Abstract: This talk will survey a collection of the speaker's
recent
results on congruence properties of partition functions.
Particular attention will be placed on the methods coming from the
theory of modular forms.
Wednesday, March 2.
Speaker: Jeremy Rouse, Wake Forest University
Title: On the sums of
two cubes.
Abstract: In this talk, we will classify solutions to f(x,y)3
+ g(x,y)3 = x3 + y3,
where
f and g are homogeneous rational
functions. The roots of this question appear in work of Viete in 1591,
and the solution is a clever application of the theory of elliptic
curves. Fans of the ring Z[e(2πi/3)] will enjoy this talk.
Wednesday, February 23.
Speaker: Michael Filaseta, University of South Carolina
Title: Four Seemingly Unrelated
Problems, Part II.
Abstract: We quickly review the four different problems
discussed last week and proceed
to explain how these seemingly unrelated problems are connected to each
other. As noted in
last week's abstract, one of the problems is: Is it possible to
find an irrational number q such
that the infinite geometric sequence 1, q, q2,
...
has
4
terms
in
arithmetic progression?
Last week, we indicated that the answer is, "No." Our talk will
include a fairly detailed
description of the proof of this result.
Wednesday, February 16.
Speaker: Michael Filaseta, University of South Carolina
Title: Four Seemingly Unrelated
Problems.
Abstract: After putting some finishing touches on some
questions asked
of the speaker last week, we turn to the main (not completely
unrelated) topic
of this seminar. We begin this part by discussing the history
associated with
four different problems that are number theoretic or combinatorial in
nature.
Two of these problems remain open and the other two have known
solutions.
Our goal this week and next is to explain how these seemingly unrelated
problems are connected to each other. To disclose a little more
information,
we mention one of the problems with a known solution here. Is it
possible to
find an irrational number q such that the infinite geometric sequence
1, q, q2, ...
has 4 terms in arithmetic progression?
Wednesday, February 9.
Speaker: Michael Filaseta, University of South Carolina
Title: A polynomial conjecture
of Turan rerevisited again
Abstract: This is a survey talk on a conjecture of Turan which
asserts that
there is a constant C such that every polynomial f(x) in Z[x] is within
C of
being an irreducible polynomial (meaning that there is a w(x) in Z[x]
with
the sum of the absolute value of its coefficients at most C and with
f(x) + w(x) irreducible over the rationals). We will give a
history of results
on this still open conjecture, ending with a proof of a result
established
by Mossinghoff and the speaker just last month. Although some of
you
will have heard related talks by Mossinghoff and have been in a course
that
the speaker gave last year that included a discussion of this topic,
note that the
speaker will put a slightly different perspective on the history of
this
problem and introduce some new ideas for possible future
investigations.
Wednesday, February 2.
Speaker: Matthew Boylan, University of South Carolina
Title: An overview of p-adic
modular forms.
Abstract: We survey Serre's classical theory of p-adic modular
forms.
Some of these ideas play an important role in the speaker's recent
joint work with J. Webb on the l-adic fractal behavior of p(n), the
ordinary partition function.
Wednesday,
January 26.
Speaker: Frank Thorne, Stanford University.
Title: The Davenport-Heilbronn
Theorem.
Abstract: We will consider the problem of counting number
fields when ordered by
discriminant, with particular attention to the cubic case. An
asymptotic
formula for the number of cubic fields was proved in a 1971 paper of
Davenport and Heilbronn, and we will discuss this proof in some detail.
In
particular, we will explain how Davenport and Heilbronn reduced this to
a
problem of counting lattice points, and we will explain how the same
techniques may be used to address related problems as well.
We will conclude by presenting some numerical data, obtained with
algorithms which were developed much later than 1971, and investigating
the match of the theorems to the actual data.
Friday, January 28
Speaker: Frank Thorne, Stanford University
and USC.
Title: More on the
Davenport-Heilbronn Theorem.
Abstract: Continued from above.
Wednesday, January 19.
Speaker:
John Webb,
University
of South Carolina
Title: Regarding the l-adic fractal behavior of partition
numbers
Abstract: Folsom, Kent, and Ono have recently proved that for
any prime l and any positive integer m, values of the partition
function display a certain periodicity modulo
lm.
I
will
describe
their
result,
give
a
proof
for
an
improved "zoom rate",
and discuss implications of their work. I will also present
calculations which illustrate this periodicity and suggest that this
improved "zoom rate" is sharp. This is joint work with Matt
Boylan.
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