We explore possible stable properties of the sequence of
zeta functions associated to a geometric Z_p-tower of curves over
a finite field of characteristic p, in the spirit of Iwasawa theory.
Several fundamental questions and conjectures will be discussed,
and some supporting examples will be given. This introductory talk
is accessible to graduate students in number theory and arithmetic
geometry.
I will discuss various geometric gluing constructions. First I will discuss constructions for Constant Mean Curvature hypersurfaces in Euclidean spaces including my earlier work for two-surfaces in three-space which settled the Hopf conjecture for surfaces of genus two and higher, and recent generalizations in collaboration with Christine Breiner in all dimensions. I will then briefly mention gluing constructions in collaboration with Mark Haskins for special Lagrangian cones in Cn. A large part of my talk will concentrate on doubling and desingularization constructions for minimal surfaces and on applications on closed minimal surfaces in the round spheres, free boundary minimal surfaces in the unit ball, and self-shrinkers for the Mean Curvature flow. Finally I will discuss my collaboration with Simon Brendle on constructions for Einstein metrics on four-manifolds and related geometric objects.
Based on D. Catlin's work, Property $(P_q)$ of the boundary implies the compactness of the $\bar{\partial}$-Neumann operator $N_q$ on smooth pseudoconvex domains. We discuss a variant of Property $(P_q)$ of the boundary of a smooth pseudoconvex domain for certain levels of $L^2$-integrable forms. This variant of Property $(P_q)$ on the one side, implies the compactness of $N_q$ on the associated domain, on the other side, is different from the classical Property $(P_q)$ of D. Catlin and Property $(\widetilde{P_q})$ of J. McNeal.
Free boundary minimal surfaces in the ball are proper branched minimal
immersions of a surface into the ball that meet the boundary of the ball
orthogonally. Such surfaces have been extensively studied, and they arise as
extremals of the area functional for relative cycles in the ball. They also
arise as extremals of an eigenvalue problem on surfaces with boundary. In
this talk I will describe uniqueness (joint work with R. Schoen) and
compactness (joint work with M. Li) theorems for such surfaces.
We consider holomorphic Poisson structures as a special kind of
generalized geometry in the sense of Hitchin and Gaultieri.
A consideration on local deformation leads us to compute their associated
Lie algebroid cohomology spaces. As this cohomology is represented by the
limit of a bi-complex, we consider various situations early degeneration of
the associated spectral sequence of the bi-complex occurs. Cases for
discussion include Kahlerian manifolds and nilmanifolds with abelian complex
structures.
I will discuss a heuristic that predicts that the ranks of all but finitely many elliptic curves defined over Q are bounded above by 21. This is joint work with Bjorn Poonen, John Voight, and Melanie Matchett Wood.
Two permutations are conjugate if and only if they have the same cycle structure, and two complex unitary matrices are conjugate if and only if they have the same set of eigenvalues. Motivated by the large literature on cycles of random permutations and eigenvalues of random unitary matrices, we study conjugacy classes of random elements of finite classical groups. For the case of GL(n,q), this amounts to studying rational canonical forms. This leads naturally to a probability measure on the set of all partitions of all natural numbers. We connect this measure to symmetric function theory, and give algorithms for generating partitions distributed according to this measure. We describe analogous results for the other finite classical groups (unitary, symplectic, orthogonal). We were excited to learn that (at least for GL(n,q)), exactly the same random partitions arise in the “Cohen-Lenstra heuristics” of number theory.
In this talk we will discuss various recent claims of algorithms which solve certain instances of the elliptic curve discrete logarithm problem (ECDLP) over finite fields in sub-exponential time. In particular, we will discuss approaches which use Groebner basis algorithms to solve systems coming from summation polynomials. The complexity of these approaches relies on the so-called first fall degree assumption. We will raise doubt to this first fall degree assumption and hence to the claimed complexity.
A classical result of H.J Brascamp and E.H. Lieb says that the ground state eigenfunction for the Laplacian in convex regions (and of Schr ̈odinger operators with convex potentials on Rn) is log-concave. A proof can be given (interpreted) in terms of the finite dimensional distributions of Brownian motion. Some years ago the speaker raised similar questions (and made some con- jectures) when the Brownian motion is replaced by other stochastic processes and in particular those with transition probabilities given by the heat kernel of the fractional Laplacian–the rota- tionally symmetric stable processes. These problems (for the most part) remain open even for the unit interval in one dimension. In this talk we elaborate on this topic and outline a proof of a result of M. Kaßmann and L. Silvestre concerning superharmonicity of eigenfunctions for certain fractional powers of the Laplacian. Our proof is joint work with D. DeBlassie.