The seminars are held on Wednesdays from 10:10 to 11:00 in room LeConte 312
If you would like to speak, please contact Samuel Gross at ssgross@math.sc.edu. Open dates are highlighted in the schedule below.
Upcoming:
Speaker: Jon Hanke, University of Georgia.
Title: The 290-Theorem and Representing Numbers by Quadratic Forms.
Abstract: This talk will describe several finiteness theorems for quadratic forms, and progress on the question: "Which positive definite integer-valued quadratic forms represent all positive integers?". The answer to this question depends on settling the related question "Which integers are represented by a given quadratic form?" for finitely many forms. The answer to this question can involve both arithmetic and analytic techniques, though only recently has the analytic approach become practical.
We will describe the theory of quadratic forms as it relates to answering these questions, its connections with the theory of modular forms, and give an idea of how one can obtain explicit bounds to describe which numbers are represented by a given quadratic form.
The Following Dates are Still Open:
Previous Seminar Talks:
Speaker: Matthew Boylan, University of South Carolina.
Title: An overview of modular forms in the study of partitions.
Abstract: Classical modular forms play important roles in many different areas of mathematics. These include, for example, arithmetic geometry (elliptic curves, L-functions, and connections to Fermat's Last Theorem and the Birch and Swinnerton-Dyer Conjecture), algebraic number theory (class numbers, singular moduli), Lie theory, representation theory, geometry, and mathematical physics.
In this talk, I will discuss the role that modular forms play in recent work on the ordinary partition function, p(n). The function p(n) counts the number of ways to write the integer n as a sum of integers <= n. As such, it is a fundamental object in additive and combinatorial number theory. Moreover, it serves as a "testing ground" for developing and exploring techniques in the theory of automorphic forms.
Speaker: John Webb , University of South Carolina..
Title: Partition Values and Central Critical Values of Certain Modular L-Functions.
Abstract: I will present a new result which relates the central critical values of certain modular L-functions to values of the partition function. Part of the proof required a substantial calculation which will be described.
Speaker:Michael Mossinghoff, University of South Carolina.
Title: The distance to an irreducible polynomial.
Abstract: Given an integer polynomial f(x), how many coefficients do you need to adjust before you are assured of finding an irreducible polynomial of the same degree? More precisely, does there exist an absolute constant C so that for every f(x) in Z[x] there exists an irreducible g(x) in Z[x] with deg(f) = deg(g) and L(f-g) <= C, where L(h(x)) denotes the sum of the absolute values of the coefficients of h(x)? This question was first posed by Turan in 1962, and it remains unsolved. We discuss some algorithms designed to investigate this question, and report on the results of some recent computations.
Speaker: Dan Yasaki , University of North Carolina at Greensboro..
Title: Exploring modularity over a CM quartic field
Abstract: I will report on an on-going project investigating a Taniyama-Shimura type correspondence between elliptic curves and cuspforms over CM quartic fields. The Hecke data for the cusp forms is computed using a technique which generalizes the modular symbol algorithm. I will give the data collected so far, as well as an overview of the reduction algorithm.
Speaker: Greg Dresden , University of Georgia.
Title: Resultants of Cyclotomic Polynomials.
Abstract: There are several ways to find the resultant of cyclotomic polynomials. We present a new method that relies only on the combinatorical properties of the cyclotomics. We also provide a constructive formula for the gcd of two cyclotomics and show how this problem is equivalent to the problem on resultants. Similar work has appeared in recent articles by M. Mossinghoff and M. Filaseta (among others).
Speaker: Michael Mossinghoff, University of South Carolina..
Title: Wieferich Madness
Abstract: A Barker sequence is a finite sequence of integers, each 1 or -1, whose off-peak aperiodic autocorrelations are all 0, 1, or -1. No Barker sequences with length n > 13 are known, and it is widely conjectured that no long Barker sequences exist. We describe how an extensive search for Wieferich prime pairs (q,p), which are defined by the property that q^(p-1) = 1 mod p^2, allows one to show that if a Barker sequence of length n > 13 exists, then n must be quite large.
Speaker:Andrew Sills, Georgia Southern University..
Title: A generalization of the Euler–Glaisher partition bijection.
Abstract: In 1748, Euler published his Introductio in Analysin Infinitorum. Chapter 16 of this work is the first systematic study of integer partitions in the mathematical literature. In it, he introduces infinite product generating functions and uses them to derive what is now known as Euler’s partition identity, an English translation of which reads as follows:
“The number of different ways a given number can be expressed as the sum of different whole numbers is the same as the number of ways in which the same number can be expressed as the sum of odd numbers, whether the same of different.” In modern terminology, the preceding is rephrased as “the number of partitions of n into distinct parts equals the number of partitions of n into odd parts.”
In 1883, J.W.L. Glaisher published the first bijective proof of Euler’s partition identity, along with a natural generalization: “the number of partitions of n where no part appears more than m − 1 times equals the number of partitions of n where no part is divisible by m.” By combining a construction of P.A. MacMahon called “partitions of infinity” and knowledge of G.E. Andrews’ “partition ideals of order 1” with Glaisher’s bijective proof of Euler’s identity, we are led to discover a large class of partition identities with straightforward bijective proofs.
This is joint work with James Sellers and Gary Mullen of Penn State. All terms will be defined and illustrated with concrete examples, so the required mathematical background will be minimal, and the talk should be accessible to all graduate students.
Speaker: Michael Filaseta, University of South Carolina.
Title: Open problems on covering systems.
Abstract: On the same general subject as last week but with a different speaker, this talk will focus more on open problems while at the same time still surveying various classical results and a variety of recent progress on the subject. Particular emphasis will be given to the contributions of Erdős and his inquisitive spirit.
A copy (pdf) of Michael's slides from this talk are available here.
Speaker: Ognian Trifonov, University of South Carolina.
Title: Coverings of the integers.
Abstract: We will prove some basic properties of the coverings and discuss some open problems and recent progress on coverings.
Speaker: Daniel Baczkowski, University of South Carolina.
Title:A Smorgasbord of Problems from Number Theory
Abstract: This talk will be a review of unsolved and solved problems in number theory pertaining to the speaker's research. The level will be for the novice as the talk's intended purpose is to satisfy a general audience; that is, if you have never had a number theory course, then you are encouraged and welcome to come and see what is happening. The work to be discussed has rudimentary roots in numerous other subjects including Approximation Theory, Combinatorics, and Geometry. Mostly a summary of results will be stated and one or two short, mellifluous proofs will be given to satisfy the appetite of the most die hard mathematicians.
Speaker: John Webb, University of South Carolina.
Title: Heegner Systems and the Birch, Swinnerton-Dyer Conjecture.
Abstract: We'll go over the proof of the known case of the Birch, Swinnerton-Dyer Conjecture, focusing on work by Gross, Zagier, and Kolyvagin.
Speaker: Matt Boylan, University of South Carolina.
Title: Infinite products and CM traces of modular forms.
Abstract: One of the goals of this talk is to discuss an important theorem of Borcherds (Invent., 1995). In the case of interest, the theorem says that the exponents in the naive infinite product expansion of a classical integer weight modular form of a certain type (having a "Heegner divisor") arise as the Fourier coefficients of a modular form of weight 1/2. Definitions and examples will be given.
In reproving a special case of Borcherds result, Zagier initiated a study of traces (over Galois conjugates of CM points) of values of modular functions, and he observed a duality between canonical bases of modular forms of weights 1/2 and 3/2. A second goal of this talk is to state some of Zagier's results. The work of Zagier has inspired a tremendous amount of activity (of more recent vintage) that continues to the present and includes work of the presenter.