Georgia Institute of Technology College of Sciences

Fifty years ago Erdos asked to determine the minimum number of $k$-cliques in a graph on $n$ vertices with independence number less than $l$ (we will refer this as $(k,l)$-problem). He conjectured that this minimum is achieved by the disjoint union of $l-1$ complete graphs of size $\frac{n}{l-1}$. This conjecture was disproved by Nikiforov who showed that Erdos' conjecture can be true only for finite many pairs of $(k,l)$. For $(4,3)$-problem, Nikiforov further conjectured that the balanced blow-up of a $5$-cycle achieves the minimum number of $4$-cliques. We first sharpen Nikiforov's result and show that Erdos' conjecture is false whenever $k\ge 4$ or $k=3, l\ge 2074$. After introducing tools (including Flag Algebra) used in our proofs, we state our main theorems, which characterize the precise structure of extremal examples for $(3,4)$-problem and $(4,3)$-problem, confirming Erdos' conjecture for $(k,l)=(3,4)$ and Nikiforov's conjecture for $(k,l)=(4,3)$. We then focus on $(4,3)$-problem and sketch the proof how we use stability arguments to get the extremal graphs, the balanced blow-ups of $5$-cycle. Joint work with Shagnik Das, Hao Huang, Humberto Naves and Benny Sudakov.

A discussion of the papers "Getting started in probabilistic graphical models" by Airoldi (2007) and "Inferring cellular networks using probabilistic graphical models" by Friedman (2004).

We discuss two concepts of low-dimensional topology in higher dimensions: near-symplectic manifolds and overtwisted contact structures. We present a generalization of near-symplectic 4-manifolds to dimension 6. By near-symplectic, we understand a closed 2-form that is symplectic outside a small submanifold where it degenerates. This approach uses some singular mappings called generalized broken Lefschetz fibrations. An application of this setting appears in contact topology. We find that a contact 5-manifold, which appears naturally in this context, is PS-overtwisted. This property can be detected in a rather simple way.

The talk is devoted to the classical problem of estimating the Van der Waerden number W(n,k). The famous Van der Waerden theorem states that, for any integers n\ge 3, k\ge 2, there exists the smallest integer W(n,k) such that in any k-coloring of the set {1,2,...,W(n,k)}, there exists a monochromatic arithmetic progression of length n. Our talk is focused on the lower bounds for the van der Waerden number. We shall show that estimating W(n,k) from below is closely connected with extremal problems concerning colorings of uniform hypergraphs with large girth. We present a new lower bound for W(n,k), whose proof is based on a continuous-time random recoloring process.