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

Graphons are limit objects that are associated with convergent sequences of graphs. Problems from extremal combinatorics and theoretical computer science led to a study of graphons determined by finitely many subgraph densities, which are referred to as finitely forcible graphons. In 2011, Lovasz and Szegedy asked several questions about the complexity of the topological space of so-called typical vertices of a finitely forcible graphon can be. In particular, they conjectured that the space is always compact. We disprove the conjecture by constructing a finitely forcible graphon such that the associated space of typical vertices is not compact. In fact, our construction actually provides an example of a finitely forcible graphon with the space which is even not locally compact. This is joint work with Roman Glebov and Dan Kral.

If A is a set of n integers such that the sumset A+A = {a+b : a,b in A} has size 2n-1, then it turns out to be relatively easy to prove that A is an arithmetic progression {c, c+d, c+2d, c+3d, ..., c+(n-1)d}. But what if you only know something a bit weaker, say |A+A| < 10 n, say? Well, then there is a famous theorem due to G. Freiman that says that A is a "dense subset of a generalized arithmetic progression" (whatever that is -- you'll find out). Recently, this subject has been revolutionized by some remarkable results due to Tom Sanders. In this talk I will discuss what these are.

There are many instances in Coding Theory when codewords must be restored from partial information, like defected data (error correcting codes), or some superposition of the strings.These lead to superimposed codes, close relatives of group testing problems.There are lots of versions and related problems, likeSidon sets, sum-free sets, union-free families, locally thin families, cover-free codes and families, etc.We discuss two cases {\it cancellative} and {\it union-free} codes.A family of sets $\mathcal F$ (and the corresponding code of0-1 vectors) is called {\bf union-free} if $A\cup B = C\cup D$ and $A,B,C,D\in \mathcal F$ imply $\{ A,B\}=\{ C, D \}$.$\mathcal F$ is called $t$-{\bf cancellative}if for all distict $t+2$ members $A_1, \dots, A_t$ and $B,C\in \mathcal F$ $$A_1\cup\dots \cup A_t\cup B \neq A_1\cup \dots A_t \cup C. $$Let $c_t(n)$ be the size of the largest $t$-cancellative code on $n$elements. We significantly improve the previous upper bounds of Alon, Monti, K\"orner and Sinaimeri, and introduce a method to deal with such problems, namely to investigate first the constant weight case (i.e., uniform hypergraphs).