Georgia Institute of Technology College of Sciences

Judge John Dever gives the mathematically ubiquitous concept of adjunction a categorical definition. With the hom functor acting as an "inner product", categorical adjoints may be seen as the analogy of adjoint linear operators of a Hilbert space.

We consider optimal alignments of random sequences of length n which are i.i.d. For such alignments we count which letters get aligned with which letters how often. This gives as for every opitmal alignment the frequency of the aligned letter pairs. These frequencies expressed as relative frequencies and put in vector form are called the "empirical distribution of letter pairs along an optimal alignment". It was previously established that if the scoring function is chosen at random, then the empirical distribution of letter pairs along an opitmal alignment converges. We show an upper bound for the rate of convergence which is larger thatn the rate of the alignement score. the rate of the alignemnt score can be obtained directly by Azuma-Hoeffding, but not so for the empirical distribution of the aligned letter pairs seen along an opitmal alignment: which changing on letter in one of the sequences, the optimal alginemnt score changes by at most a fixed quantity, but the empirical distribution of the aligned letter pairs potentially could change entirely.

It is well known that solutions of compressible Euler equations in general form discontinuities (shock waves) in finite time even when the initial data is $C^\infty$ smooth. The lack of regularity makes the system hard to resolve. When the initial data have large amplitude, the well-posedness of the full Euler equations is still wide open even in one space dimenssion. In this talk, we discuss some recent progress on large data solutions for the compressible Euler equations in one space dimension. The talk includes joint works with Alberto Bressan, Helge Kristian Jenssen, Robin Young and Qingtian Zhang.

The $4$-genus of a knot is an important measure of complexity, related tothe unknotting number. A fundamental result used to study the $4$-genusand related invariants of homology classes is the Thom conjecture,proved by Kronheimer-Mrowka, and its symplectic extension due toOzsvath-Szabo, which say that closed symplectic surfacesminimize genus.Suppose (X, \omega) is a symplectic 4-manifold with contact type bounday and Sigma is a symplectic surface in X such that its boundary is a transverse knot in the boundary of X. In this talk we show that there is a closed symplectic 4-manifold Y with a closed symplectic submanifold S such that the pair (X, \Sigma) embeds symplectically into (Y, S). This gives a proof of the relative version of Symplectic Thom Conjecture. We use this to study 4-genus of fibered knots in the 3-sphere.We will also discuss a relative version of Giroux's criterion of Stein fillability. This is joint work with Siddhartha Gadgil