Talk

We introduce a new operation, double point surgery, on a configuration of surfaces in a 4-manifold, and use it to construct configurations that are smoothly knotted, without changing the topological type or the smooth embedding type of the individual components of the configuration. Taking branched covers, we produce smoothly exotic actions of $Z_m \oplus Z_n$ on simply connected 4-manifolds with complicated fixed-point sets.

Abstract: (joint with Arkadiy Skopenkov). Let $N$ be a closed connected smooth n-manifold and let $E^m(N)$ be the set of isotopy classes of embeddings of $N$ into Euclidean m-space. The set $E^{n+2}(S^n)$ of isotopy classes of codimension-2 embeddings of the n-sphere has been intensively studied. In the 60s and 70s a great deal was also learnt about embeddings of closed manifolds in codimension-3 and higher: key names are Haefliger and Wall amongst others. The results of the 60s and 70s were more complete in the piecewise linear (PL) category and tended to be general results as opposed to detailed analyses of specific examples. Recently there has been important new progress based on a modified surgery algorithm for classifying embeddings discovered by M. Kreck. In this talk I take up the problem of determining $E^7(N)$ for a closed smooth connected orientable 4-manifold N. In the case that the integral homology of groups of N are torsion free I will present a complete description of $E^7(N)$ via readily computable invariants. I will also present a complete description of the set of the PL embeddings of N up to PL isotopy. The case $N = S^1 \times S^3$ is of particular interest: it gives a counter example to the ``informal $\alpha$-invariant hypothesis" as well as exhibiting subtle inertial phenomena.

The motivation of constructing the acid zeta function is to study the distribution of the Riemann zeta zeros. In this lecture, I will present theory of the acid zeta function and the adjoint acid zeta function, particularly, as one of the applications, we have some important reasons to doubt the truth of the Riemann Hypothesis.

About 10 years ago the methods developed by A.Wiles, R. Taylor, and their collaborators led to the proof of the modularity of the elliptic curves defined over the field of rational numbers. In a recent work Dielefait, Gueberooff, and Pacetti developed a new method, allowing to compare two 2-dimensional l-adic Galois representations, and applied their method to prove modularity of three elliptic curves defined over an imaginary quadratic field. To illustrate that method, I shall outline a proof of modularity of another elliptic curve over an imaginary quadratic field, making use of the recent calculations of M.Mink.

We give linear equations that characterize the Fourier Jacobi expansions of paramodular forms from among all convergent series of Jacobi forms. We suspect that these linear equations in fact characterize the Fourier-Jacobi expansions of paramodular forms from among all formal series of Jacobi forms. We use these linear equations to compute small eigenvalues of possible weight two paramodular cusp forms up to level 1000. We compare this data with the Paramodular Conjecture for modularity in genus two using the work on rational abelian surfaces of A. Brumer and K. Kramer.

We describe an abelian group structure on the set of classes of modular categories modulo some equivalence relation. The resulting Witt group of modular categories resembles (and contains) the Witt group of finite abelian groups with quadratic forms. The conjecture of Moore and Seiberg, that all chiral rational conformal field theories come from reductive groups via WZW, coset and orbifold constructions, can be interpreted as a statement about generators of this Witt group.

This is a joint work with Rina Anno (U Chicago). We show how to construct a network of functors which correspond to `generalized braid diagrams', between derived categories of coherent sheaves on cotangent bundles of full and partial flag varieties. For a subclass of these diagrams (which includes all the ordinary braids) we prove that isotopic diagrams correspond to isomorphic functors. We then outline our strategy for proving the general case.

We discuss two new classes of strongly reflective modular forms with respect to orthogonal groups $O(2,n)$. The first class contains 36 functions including the Borcherds form $\Phi_{12}$. The second one has at least 12 functions (this work is in progress) including the Igusa modular form of weight 5. We give some applications of these remarkable modular forms to the algebraic geometry of modular varieties and to the theory of Kac-Moody Lie algebras.