Agenda: Planning session for this semester's seminar.
Speaker:
Chris Hall, University of Texas at Austin.
Title: Families of Galois extensions with big groups.
Speaker: Nigel Boston, USC.
Title: Random Galois Groups.
Speaker:
Mihran Papikian, Stanford University.
Title: Drinfeld modular varieties and the Weil-Deligne bound.
Speaker: Matt Boylan, USC.
Title: Half-integral weight modular form coefficients and
special values of L-functions.
Speaker: Michael Filaseta, USC.
Title: Applications of Pade approximants to number theory, Part I.
Speaker: Michael Filaseta, USC.
Title: Applications of Pade approximants to number theory, Part II.
Speaker:
Matt Darnall, University of
Wisconsin at Madison.
Title: Bounds on the Star Discrepancy in the Unit Square.
Speaker:
Carrie Finch, Columbia College/Washington and Lee University.
Title: Finite groups with exactly n elements of order n.
Speaker: Konstantin Oskolkov, USC.
Title: Quantum mechanics of Gauss' sums.
Speaker: Nigel Boston, USC.
Title: On the equation x^4+m*x^2*y^2+y^4=z^2.
Speaker: Matt Boylan, USC.
Title: Non-vanishing of weakly holomorphic modular form
coefficients modulo l and applications.
Speaker: Ognian Trifonov, USC.
Title: On covering systems of the integers.
Speaker: Alice
Silverberg, University of California at Irvine.
Title: Diffie-Hellman and beyond.
Department of Mathematics main page
Abstract: Given a finite group G and a field K, the inverse Galois
problem asks whether there is a Galois extension of K with group G. When K is
the field of rational numbers Q, the answer is not known for general G. We
will discuss one approach to the problem when G is a (general) symplectic
group via the arithmetic of ell-torsion of an abelian variety. We will show
how to construct geometric families of varieties where most of the members
give rise to a family of Galois extensions, one for every prime ell >> 0,
whose groups are all as big as possible.
Abstract: In a recent Inventiones paper, Dunfield and Thurston
considered the distribution of the finite quotients of a random
g-generator g-relator group versus those of fundamental groups of
3-manifolds. There is an analogous question for certain Galois groups, which I
will address.
Abstract: We show that appropriate quotients of Drinfeld modular
varieties and the modular varieties of D-elliptic sheaves have many rational
points over certain finite fields compared to their Betti numbers. This is a
generalization to higher dimensions of some well-known results for modular
curves.
Abstract: We survey the connection between half-integral weight
modular form coefficients and central critical values of modular L-functions
(taking care to define what these objects are). In particular, we highlight
the important work of Shimura, Waldspurger, and Kohnen and Zagier. We will
also discuss applications to some of the speaker's work (joint with Scott
Ahlgren, appearing in the American Journal of Mathematics).
Palmetto Number
Theory Series (PANTS) II,
Clemson University,
Clemson, South Carolina.
Abstract: Pade approximants arise in, for example, the study of
integral equations, scattering theory, the diffusion equation, and the
inversion of Laplace transforms, but in this talk we won't talk about that.
After beginning with the speaker's own introduction to the subject as a
graduate student, who stubbornly worked out a related linear algebra homework
assignment by throwing it into the realm of a combinatorial lattice path
question, we will give a two part excursion into ways that Pade
approximations have been used in a variety of number theoretic contexts.
These applications include results in irrationality measures, Galois groups
associated with polynomials over the rationals, solutions to Diophantine
equations, Waring's problem, prime divisors of binomial coefficients, the
abc-conjecture, k-free numbers in short intervals, and k-free values of
polynomials and of binary forms.
Abstract:
Abstract: (t,m,s) nets are sets of points in the unit cube that
are "well distributed" in a precise sense, i.e they achieve the best known
star discrepancy. In this talk, I will show that the discrepancy at any
point of a random (0,m,2) net in base 2 behaves no worse than a simple
symmetric random walk with O(m) steps.
Abstract: We discuss finite groups that contain exactly n elements
of order n (for a particular fixed n). While this is a group theoretic
question, we use number theory to provide the solution.
Abstract: The basic relation of quauntum mechanics - Schroedinger
equation of a free particle - in the case of periodic initial data generates
a full set of identities for the complete and incomplete Gauss' sums. It will
be demonstrated in the talk, how one can derive the Genocci - Schaar type
identities for the complete Gauss sums, and how one can estimate the
incomplete sums using a functional equation, of the Jacobi's elliptic
theta-function type, for the periodized Green's function of the
Schroedinger equation.
Abstract: In this talk, we discuss a paper of Bremner and Jones on
an old problem considered by Euler and how it relates to the conjecture of
Birch and Swinnerton-Dyer.
Abstract: Let f(z) be a half-integral weight modular form with
integer coefficients a(n) whose poles (if it has any) are supported at the
cusps. Fix a prime l. In this talk, we estimate the number of a(n)'s not
divisible by l and give applications to the study of the ordinary partition
function, p(n), and other functions of arithmetic interest whose generating
functions are of this type.
(Joint work with Scott Ahlgren, Univ. of Illinois).
Abstract: We will survey some old and recent results, discuss open
problems, and prove a simple new result on coverings of the integers.
Abstract: Number Theory is the backbone of public key cryptography.
We will discuss Diffie-Hellman key agreement and some improvements on it,
and show how number theory and algebraic geometry can be used to give new
cryptosystems and a deeper understanding of old ones.
Southeast Regional Meeting on
Numbers (SERMON),
Wake Forest University,
Winston - Salem, North Carolina.