Georgia Institute of Technology College of Sciences

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).

The aim of this talk is to give fairly self contained proof of the following result due to Eliashberg. There is exactly one holomorphically fillable contact structure on $T^3$. If time permits we will try to indicate different notions of fillability of contact manifolds in dimension 3.