Mathematics Colloquium Winter 2009

January 13, 2009

Dispersive Properties of Surface Water Waves

RECRUITMENT CANDIDATE: Vera M. Hur
MIT, Cambridge, MA

I will speak on the dispersive character of waves on the interface between vacuum and water under the influence of gravity and surface tension. I will begin by giving a precise account of the formulation of the surface water-wave problem and discussion of its distinct features. They include the dispersion relation, its severe nonlinearity, traveling waves and the Hamiltonian structure. I will describe the recent work of Hans Christianson, Gigliola Staffilani and myself on the local smoothing effect of 1/4 derivative for the fully nonlinear problem under surface tension with some detail of the proof. If time permits, I will explore some open questions regarding long-time behavior and stability.

January 15, 2009

Strichartz Estimates for a Wealth of Schrödinger Operators

RECRUITMENT CANDIDATE: Michael J. Goldberg
Johns Hopkins University, Baltimore, MD

The Strichartz estimates are a fundamental family of L^p inequalities governing solutions to the free Schrödinger equation. They describe the dispersive nature of the evolution -- local concentrations of mass can only exist for a specified finite period of time. The associated function spaces can be used to assess well-posedness and scattering properties of perturbed or nonlinear Schrödinger equations. I will review the efforts to identify other Schrödinger operators onRⁿ that satisfy the same range of Strichartz estimates as the Laplacian. On the technical side, recent progress is driven by improvements in the underlying harmonic and functional analysis. On the motivational side, specific applications often give rise to operators that fall outside (or at the edge) of our current understanding. I will present the orbital stability of solutions in NLS as one such application.

January 20, 2009

Hirzebruch Invariants of Complex Algebraic Varieties

RECRUITMENT CANDIDATE: Laurentiu Maxim
CUNY Lehman College, White Plains, NY

An old theorem of Chern, Hirzebruch and Serre asserts that the signature of closed oriented manifolds is multiplicative in fiber bundles with trivial monodromy action (i.e., bundles for which the fundamental group of the base acts trivially on the cohomology of the fiber). The contribution of monodromy to the signature of a fiber bundle was later described by Atiyah and Meyer. In this talk I will survey various extensions of these results to the singular setting, and discuss parametrized versions of them in the complex algebraic context. The talk will be suitable for a general audience.

January 22, 2009

On Knot Homology Theories

RECRUITMENT CANDIDATE: Elisinda Grigsby
Columbia University, New York, NY

Understanding knots (smoothly-imbedded circles in 3-manifolds, considered up to isotopy) is essential for understanding 3- and 4-dimensional manifolds. I will discuss two recently-developed tools for studying knots, both inspired by ideas in physics: Khovanov homology and Heegaard Floer homology. In the less than ten years since their introduction, they have generated a flurry of activity and a stunning array of applications. There are also intriguing connections between the two theories that have yet to be fully understood.

January 27, 2009

Rigidity of the solutions for focusing mass-critical nonlinear Schrodinger equation

RECRUITMENT CANDIDATE: Xiaoyi Zhang
Institute for Advanced Study, Princeton, NJ

In recent years, there has been lot of work addressing the global wellposedness, the scattering and blowup theory for mass critical nonlinear Schr\"odinger equation $iu_t+\Delta u=\pm |u|^{\frac 4d}u$. In this talk, I will give a review on these work, then I will introduce in more details our recent work about the rigidity of the solutions for the focusing problem when the solution has ground state mass. Here the ground state $Q$ refers to the unique positive radial solution of elliptic equation $\Delta Q-Q+|Q|^{\frac 4d}Q=0$. More precisely, our results show that the only global nonscattering solution is the solitary wave $e^{it}Q$ up to symmetries of the equation when the solution has exactly the ground state mass.

January 29, 2009

Transverse knots via braids

RECRUITMENT CANDIDATE: Keiko Kawamuro
Institute for Advanced Study, Princeton, NJ

In this talk, I will discuss several topics related to transverse knots. I will introduce a conjecture on the maximal self-linking number of a topological knot in the standard contact 3-sphere. I will show how to apply braid theory, the HOMFLY polynomial, and the Khovanov-Rozansky homology in order to address the conjecture. I will further discuss the computation of the self-linking number using open-book decomposition and its application to contact geometry.

February 17, 2009

Breaking Vegas: Beat the Wheel

Debra Lewis,
UCSC Professor, Mathematics

As was memorably demonstrated in Mark Twain's short story "Science vs. Luck", many games of chance are actually games of skill. Even the outcome of a spin of the roulette wheel isn't really random. Join us next Tuesday for a showing of the History Channel documentary "Breaking Vegas: Beat the Wheel" describing the largely successful efforts of a team of UCSC physics graduate students to win at roulette by means of careful observation, dynamical systems theory, and a computer worthy of Maxwell Smart.

February 24, 2009

Projectivizing Taylor's Expansion

Richard Montgomery,
UCSC Professor, Mathematics

In beginning calculus we often start explaining the derivative of a function in terms of tangent lines to its graph. From there we move on to the derivative as a slope, and then as a linear operator. What if we stay true to the original picture, and insist the derivative be a line? Then the first derivative of a plane curve will lie in the projective line. Where do its higher derivatives live? What about the 'derivatives' of a surface in space, or of a function from Euclidean k-space to N-space? One natural compactification of the space of linear operators (k by N matrices) is the Grassmannian of k-planes in (k+N)-space. In Nash blow-up (also known as Cartan prolongation) the pointwise derivative is an element of a certain Grassmannian. In this talk we will report on our ongoing work in this area with visiting distinguished professor Michail Zhitomirskii. The work connects to the contact geometry and to the problem of resolving singularities. Open questions abound, questions regarding what 'derivatives' are possible, what derivatives can be realized, and the nature of the space which 'receives' higher derivatives.

March 3, 2009

On the behavior at s=1 of zeta functions of algebraic number fields

Samit Dasgupta, UCSC Associate Professor
Mathematics & Sloan Research Fellow

The Riemann zeta function has a simple pole at s = 1 with residue 1. Dirichlet generalized this fact by calculating the residues at s = 1 of zeta functions associated to a quadratic field. Dedekind then generalized this formula to the case of a general number field. In the 1970s, Stark stated a series of conjectures generalizing Dedekind's formula. These conjectures outlined a G-equivariant version of the formula, where G is the Galois group of an extension of algebraic number fields. The most explicit of these conjectures, known as the "rank one abelian Stark conjecture", applies when the Galois group in question is abelian. Most of this talk will be geared towards motivating and stating these various formulae. In 1982, Gross stated certain p-adic analogues of Stark's conjectures, including an analogue of the rank one abelian conjecture. We'll conclude the talk by describing a proof of Gross's p-adic rank one abelian conjecture under certain technical hypotheses. This is joint work with Henri Darmon and Rob Pollack.