Title:  Derangements, discrete Morse theory, and the homology of the boolean complex

Abstract:  The boolean complex is a construction associated to finite simple graphs. We summarize a matching which shows that this complex is homotopy equivalent to a wedge of spheres, and the number of these spheres is related to the boolean number, a graph invariant. To better understand this structure, we use a correspondence between derangements and basis elements and compute the homology of the boolean complex for several specific examples. A basic knowledge of discrete Morse theory may be helpful but is not necessary.

Pizza at 4:00 p.m., talk at 4:15 p.m.

Title:  Bouquet algebra of toric ideals

Abstract: To any integer matrix A one can associate a toric ideal I_A, whose sets of generators are basic objects in discrete linear optimization, statistics, and graph/hypergraph sampling algorithms. The basic algebraic problem is that of implicitization: given the matrix A, find a set of generators with some given property (minimal, Groebner, Graver, etc.). Then there is a related problem of complexity: how complicated can these generators be? In general, it is known that Graver bases are much more complicated than minimal generators. But there are some classical families of toric ideals where these sets actually agree, providing very nice results on complexity and sharp degree bounds.

This talk is about combinatorial signatures of generating sets of I_A. For the special case when A is a 0/1 matrix, bicolored hypergraphs give the answer. It turns out that such hypergraphs give an intuition for constructing basic building blocks for the general case too. Namely, we introduce the bouquet graph and bouquet ideal of the toric ideal I_A, whose structure determines the Graver basis. This, in turn, leads to a complete characterization of toric ideas for which the following sets are equal: the Graver basis, the universal Groebner basis, any reduced Groebner basis and any minimal generating set. This generalizes many of the classical examples.

Title: Self-improvement properties for nonlocal equations

Abstract:  I will present some results related to generalization of Meyers result to nonlocal equation. It happens that any weak solution of a nonlocal equation with data in L2 is automatically better at the integrability AND differentiability scale. This is a completely new phenomenon relying on the nonlocality of the operator. The proof is based on a new stopping time argument and a suitable generalization of Gehring lemma.

In his talk, Sabar will weave the remarkable story of the Kurdish Jews and their dying Aramaic tongue with the moving tale of how a consummate California kid came to write a book about his family's past in Iraqi Kurdistan. The book, "My Father's Paradise: A Son's Search for his Jewish Past in Kurdish Iraq," won the National Book Critics Circle Award for Autobiography, one of the highest honors in American letters.

Sponsored by the Jewish Studies Program

This will count as a Wired Event!