Georgia Institute of Technology College of Sciences

It is well known that the cohomology of the moduli space A_g of g-dimensional principally polarized abelian varieties stabilizes when the degree is smaller than g. This is a classical result of Borel on the stable cohomology of the symplectic group. By work of Charney and Lee, also the stable cohomology of the minimal compactification of A_g, the Satake compactification, is explicitly known.In this talk, we consider the stable cohomology of toroidal compactifications of A_g, concentrating on the perfect cone compactification and the matroidal partial compactification. We prove stability results for these compactifications and show that all stable cohomology is algebraic. This is joint work with S. Grushevsky and K. Hulek.

Tree codes are the basic underlying combinatorial object in the interactive coding theorem, much as block error-correcting codes are the underlying object in one-way communication. However, even after two decades, effective (poly-time) constructions of tree codes are not known. In this work we propose a new conjecture on some exponential sums. These particular sums have not apparently previously been considered in the analytic number theory literature. Subject to the conjecture we obtain the first effective construction of asymptotically good tree codes. The available numerical evidence is consistent with the conjecture and is sufficient to certify codes for significant-length communications. (Joint work with Cris Moore.)

We consider the semi-linear elliptic PDE driven by the fractional Laplacian: \begin{equation*}\left\{%\begin{array}{ll} (-\Delta)^s u=f(x,u) & \hbox{in $\Omega$,} \\ u=0 & \hbox{in $\mathbb{R}^n\backslash\Omega$.} \\\end{array}% \right.\end{equation*}An $L^{\infty}$ regularity result is given, using De Giorgi-Stampacchia iteration method.By the Mountain Pass Theorem and some other nonlinear analysis methods, the existence and multiplicity of non-trivial solutions for the above equation are established. The validity of the Palais-Smale condition without Ambrosetti-Rabinowitz condition for non-local elliptic equations is proved. Two non-trivial solutions are given under some weak hypotheses. Non-local elliptic equations with concave-convex nonlinearities are also studied, and existence of at least six solutions are obtained. Moreover, a global result of Ambrosetti-Brezis-Cerami type is given, which shows that the effect of the parameter $\lambda$ in the nonlinear term changes considerably the nonexistence, existence and multiplicity of solutions.