Rutgers

Recently, it has been more clearly understood that the N=4 superconformal index leads to the microstate counting of the BPS black holes in AdS_5 X S^5. The leading N^2 behavior of the free energy was shown in various ways to match that of the known BPS black hole in the gravity side, but this correspondence has not been realized at the level of the saddle point analysis of the full matrix model for the N=4 index. Here, I will try to clarify how such saddles corresponding to the BPS black holes arise as 'areal' distributions. The talk is based on https://arxiv.org/abs/2111.10720 with Sunjin Choi, Seok Kim, and Eunwoo Lee; https://arxiv.org/abs/2103.01401 with Sunjin Choi and Seok Kim.

The classical Turán problem asks: given a graph $H$, how many edges can an $4n$-vertex graph have while containing no isomorphic copy of $H$? By viewing $(k+1)$-uniform hypergraphs as $k$-dimensional simplicial complexes, we can ask a topological version (first posed by Nati Linial): given a $k$-dimensional simplicial complex $S$, how many facets can an $n$-vertex $k$-dimensional simplicial complex have while containing no homeomorphic copy of $S$? Until recently, little was known for $k > 2$. In this talk, we give an answer for general $k$, by way of dependent random choice and the combinatorial notion of a trace-bounded hypergraph. Joint work with Jason Long and Bhargav Narayanan.

I'll discuss our recent proof of a conjecture of Talagrand, a fractional version of the "expectation-threshold" conjecture of Kahn and Kalai. As a consequence of this result, we resolve various (heretofore) difficult problems in probabilistic combinatorics and statistical physics.

The second talk will present conjectural motivic generalizations

of ADHM sheaf invariants as well as their wallcrossing formulas.

It will be shown that these conjectures yield recursive formulas

for Poincare and Hodge polynomials of moduli spaces of Hitchin

pairs. It will be checked in many concrete examples that this recursion relation is in agreement with previous results of Hitchin, Gothen, Hausel and Rodriguez-Villegas.

The first talk will present a construction of equivariant

virtual counting invariants for certain quiver sheaves on a curve, called ADHM sheaves. It will be shown that these invariants are related to the stable pair theory of Pandharipande and Thomas in a specific stability chamber. Wallcrossing formulas will be derived using the theory of generalized Donaldson-Thomas invariants of Joyce and Song.

Abstract: In this talk, I will give an overview of the new developments in the AdS_4/CFT_3 correspondence. I will present in detail an N=6 Chern-Simons-matter theory with gauge group U(N) x U(N) that is dual to N M2 branes in the orbifold C^4/Z_k. This theory can be derived from a construction involving D3 branes intersecting (p,q) fivebranes. I will also discuss various quantum mechanical aspects of this theory, including an enhancement of its supersymmetry algebra at Chern-Simons levels 1 and 2, and some novel phenomenon that arise in the U(N) x U(M) theory dual to configurations with N-M fractional branes. A generalization to N=3 CSM theories dual to AdS_4 x M_7, where M_7 is a 3-Sasakian 7-manifold, will be explained. The seminar will be based primarily on Aharony, Bergman, DJ, Maldacena; Aharony, Bergman, DJ; DJ, Tomasiello.