Mathematics

Fourier coefficients of modular forms

Dipendra Prasad   [email] (Tata Institute of Fundamental Research)

Abstract: Beginning with modular forms of integral and half integral weights on the upper half-plane where many properties of Fourier coefficients have been topics of large studies for centuries, I discuss, in this mostly expository lecture, some of these aspects for modular forms (or, automorphic forms) for larger groups, inspired by a famous theorem of Waldspurger on Fourier coefficients of half-integral weight forms.

Stable Commutator Length in Baumslag-Solitar groups

Matt Clay (University of Arkansas)

Abstract: I will talk about computing stable commutator length in Baumslag-Solitar groups and about the spectrum of values it takes. For a certain elements, stable commutator length is computable and takes only rational values. There is a gap in the stable commutator length spectrum: no element of a Baumslag-Solitar group has stable commutator length between 0 and 1/12. Some of these techniques apply more generally to other groups acting on trees. This is joint work with Max Forester and Joel Louwsma. View talk at http://youtu.be/rbDe4Fh0x50

Some geometric mechanisms for Arnold Diffusion

Rafael de la Llave (Georgia Tech, Math)

Abstract: We consider the problem whether small perturbations of integrable mechanical systems can have very large effects. It is known that in many cases, the effects of the perturbations average out, but there are exceptional cases (resonances) where the perturbations do accumulate. It is a complicated problem whether this can keep on happening because once the instability accumulates, the system moves out of resonance. V. Arnold discovered in 1964 some geometric structures that lead to accumulation in carefully constructed examples. We will present some other geometric structures that lead to the same effect in more general systems and that can be verified in concrete systems. In particular, we will present an application to the restricted 3 body problem. We show that, given some conditions, for all sufficiently small (but non-zero) values of the eccentricity, there are orbits near a Lagrange point that gain a fixed amount of energy. These conditions (amount to the non-vanishing of an integral) are verified numerically. Joint work with M. Capinski, M. Gidea, T. M-Seara

Supercongruences and CM modular forms

Detchat Samart (UIUC Math)

Abstract: In this talk, we will present some results and conjectures about supercongruences related to Fourier coefficients of weight 3 modular forms. More precisely, one can use a result due to Chrisholm et al. to deduce some supercongruences between truncated $_3F_2$-hypergeometric series and $p$th-Fourier coefficients of weight 3 CM newforms, which were originally conjectured by Rodriguez-Villegas and Mortenson. These examples exhibit some interesting connections between the newforms and certain families of elliptic curves.

Numerical Algebraic Geometry: Theory and Practice

Andrew Sommese (Notre Dame)

Abstract: The goal of Numerical Algebraic Geometry is to carry out algebraic geometric calculations in characteristic zero using numerical analysis algorithms. This comes down to numerical algorithms to compute and manipulate solution sets of polynomial systems. Numerical Algebraic Geometry is a natural outgrowth of the continuation methods to compute isolated complex solutions of systems of polynomials with complex coefficients.  There are a wide range of applications including solution of chemical systems, kinematics, numerical solution of systems of nonlinear differential equations, and computation of algebraic geometric invariants. Bertini is open-source C software, developed by Bates (Colorado State U.), Hauenstein (Notre Dame), Sommese (Notre Dame), and Wampler (General Motors R. & D.), to carry out Numerical Algebraic Geometry computations. Bertini will be rewritten to make it a better tool for users.  Bertini dates from over a decade ago, and from this experience we have identified several possibilities for significant improvements. One goal is to change some of the data structures and add internal functionality that will give the user the ability to write scripts and interface with other software. In this talk, I will give an overview of Numerical Algebraic Geometry with an especial focus  on applications to the numerical solution of systems of nonlinear differential equations. I will consider the theoretical algorithms underlying the area in the light of the practical issues that arise when implementing the algorithms in the current and the future Bertini.

A refined notion of arithmetically equivalent number fields, and curves with isomorphic Jacobians

Dipendra Prasad (Tata Institute of Fundamental Research)

Abstract: Number fields with the same zeta functions have been of interest for a long time, inspiring similar constructions in a variety of contexts, the most common of which is the construction of Riemannian manifolds with the same spectrum. By a variant of this, we construct examples of number fields which are not isomorphic but for which their idele class groups are isomorphic. We also construct examples of projective algebraic curves which are not isomorphic but whose Jacobians are.