Georgia Institute of Technology College of Sciences

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.

By the 4-color theorem, every planar graph on n vertices has an independent set of size at least n/4. Finding a simple proof of this fact is a long-standing open problem. Furthermore, no polynomial-time algorithm to decide whether a planar graph has an independent set of size at least (n+1)/4 is known. We study the analogous problem for triangle-free planar graphs. By Grotzsch' theorem, each such graph on n vertices has an independent set of size at least n/3, and this can be easily improved to a tight bound of (n+1)/3. We show that for every k, a triangle-free planar graph of sufficiently large tree-width has an independent set of size at least (n+k)/3, thus giving a polynomial-time algorithm to decide the existence of such a set. Furthermore, we show that there exists a constant c < 3 such that every planar graph of girth at least five has an independent set of size at least n/c.Joint work with Matthias Mnich.

Allowing formal desuspensions of maps and objects takes the category of topological spaces to the category of spectra, where cohomology is naturally represented. The EHP spectral sequence encodes how far one can desuspend maps between spheres. It's among the most useful tools for computing homotopy groups of spheres. RP^infty has a cell structure with a cell in each dimension and with attaching maps of degrees ...020202... Note that this sequence is periodic. In fact, it is more than the degrees of these maps which are periodic and a map of Snaith relates this periodicity to the EHP sequence.We will develop the EHP sequence, James periodicity and the relationship between the two.