Georgia Institute of Technology College of Sciences

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

Small group discussions of "Time Management for New Faculty" by Ailamaki & Gehrke lead by Matt Baker, Dan Margalit, Brett Wick, and Haomin Zhou.

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.

This is a will be a panel made of two senior grad students, a post doc and a faculty member. The panelists will answer questions and give advice to younger graduate students on a range of topics including how to be a good citizen of the department and choosing an advisor. The panelists are Dr. Kang, Dr. Kelly Bickel, Albert Bush, and Chris Pryby.

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.

Following Kato, we define the sum, $H=H_0+V$, of two linear operators, $H_0$ and $V$, in a fixed Hilbert space in terms of its resolvent. In an abstract theorem, we present conditions on $V$ that guarantee $\text{dom}(H_0^{1/2})=\text{dom}(H^{1/2})$ (under certain sectorality assumptions on $H_0$ and $H$). Concrete applications to non-self-adjoint Schr\"{o}dinger-type operators--including additive perturbations of uniformly elliptic divergence form partial differential operators by singular complex potentials on domains--where application of the abstract theorem yields $\text{dom}(H^{1/2})=\text{dom}((H^{\ast})^{1/2})$, will be presented. This is based on joint work with Fritz Gesztesy and Steve Hofmann.

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.

Gabai has a nice criteria for recognizing fibered knots in 3-manifolds. This criteria is best described in terms of sutured manifolds and simple sutured hierarchies. We will introduce this terminology and prove Gabai's result. Given time (or in subsequent talks) we might discuss generalizations concerning constructing foliations on knot compliments and 3-manifolds in general. Such results are very useful in understanding the minimal genus representatives of homology classes in the manifold (in particular, the minimal genus of a Seifert surface for a knot).

The Erdos-Szekeres (happy ending) theorem claims that among any N points in general position in the plane there are at least log_4(n) of them that are the vertices of a convex polygon. I will explain generalizations of this result that were discovered in the last 30 years involving pseudoline arrangements and families of convex bodies. After surveying some previous work I will present the following results: 1) We improve the upper bound of the analogue Ramsey function for families of disjoint and noncrossing convex bodies. In fact this follows as a corollary of the equivalence between a conjecture of Goodman and Pollack about psudoline arrangements and a conjecture of Bisztrinsky and Fejes Toth about families of disjoint convex bodies. I will say a few words about how we show this equivalence. 2) We confirm a conjecture of Pach and Toth that generalizes the previous result. More precisely we give suffcient and necesary conditions for the existence of the analogue Ramsey function in the more general case in which each pair of bodies share less than k common tangents (for every fixed k). These results are joint work with Andreas Holmsen and Michael Dobbins.