Georgia Institute of Technology College of Sciences

We consider the strong parity edge coloring problem, which aims to color the edges of a graph G so that in each open walk on G, some color appears an odd number of times.  We show that this problem is equivalent to the problem of embedding a graph in a vector space over F2 so that the number of difference vectors attained at the edges is minimized. Using this equivalence, we achieve the following:

1. We characterize graphs on n vertices that can be embedded with ceil(log_2 n) difference vectors, answering a question of Bunde, Milans, West, and Wu.

2. We show that the number of colors needed for a strong parity edge coloring of K_{s,t} is given by the Hopf-Stiefel function, confirming a conjecture of Bunde, Milans, West, and Wu.

3. We find an asymptotically optimal embedding for the power of a path.

This talk is based on joint work with Sergey Norin and Doug West.

A codimension-1 submanifold embedded in a symplectic manifold is called “contact type” if it satisfies a certain convexity condition with respect to the symplectic structure. Given a symplectic manifold X it is natural to ask which manifolds Y can arise as contact type hypersurfaces. We consider this question in dimension 4, which appears much more constrained than higher dimensions; in particular we review evidence that no homology 3-sphere can arise as a contact type hypersurface in R^4 except the 3-sphere. We exhibit an obstruction for a contact 3-manifold to embed in certain closed symplectic 4-manifolds as the boundary of a Liouville domain---a slightly stronger condition than contact type---and explore consequences for the symplectic topology of small rational surfaces and potential applications to smooth 4-dimensional topology.

A Springer fiber is the set of complete flags in Cn which are fixed by a given nilpotent matrix. It is a fundamental object of study in geometric representation theory and algebraic combinatorics. The irreducible components of a Springer fiber are indexed by combinatorial objects called standard Young tableaux. It is an open problem to describe geometric properties of these components (such as their singular loci and cohomology classes) in terms of the combinatorics of tableaux. We initiate a new approach to this problem by characterizing which irreducible components are equal to Richardson varieties, which are comparatively much better understood. Another motivation comes from Lusztig's recent study of the cell decomposition of the totally nonnegative part of a Springer fiber into totally positive Richardson cells. This is joint work in progress with Martha Precup.

Given an $r$-uniform hypergraph $H$, the random Turán number $\mathrm{ex}(G^r_{n,p},H)$ is the maximum number of edges in an $H$-free subgraph of $G^r_{n,p}$, where $G^r_{n,p}$ is the Erdős-Rényi random hypergraph with parameter $p$. In the case when $H$ is not r-partite, the problem has been essentially solved independently by Conlon-Gowers and Schacht. In the case when $H$ is $r$-partite, the degenerate case, only some sporadic results are known.

The Sidorenko conjecture is a notorious problem in extremal combinatorics. It is known that its hypergraph analog is not true. Recently, Conlon, Lee, and Sidorenko discovered a relation between the Sidorenko conjecture and the Turán problem. 

 In this talk, we introduce some recent results on the degenerate random Turan problem and its relation to the hypergraph analog of the Sidorenko conjecture.