Mathematics

Department of Mathematics Retiree's Luncheon

Conformal Bootstrap in the Lightcone Limit

Daliang Li (Johns Hopkins Physics)

Abstract: The modern revival of conformal bootstrap has provided a new window into the strongly coupled dynamics of quantum field theories and the dual quantum gravitational theories. In this talk, we will explore universal structures of conformal field theories emerging from the conformal bootstrap in the lightcone limit. We start by reviewing the analysis of a 4-point function of real scalars, showing the existence of large spin, double-twist operators in unitary CFTs in d>2. These operators correspond to two-particle bound states in AdS and their anomalous dimensions correspond to the binding energies between the two particles. I will demonstrate the features of the binding energies arising from gauge and gravitational interactions. Then, using the 4-point function involving conserved currents and the stress energy tensor, I will establish a relation between the energy flux positivity in CFT and the attractiveness of gravity at large distances in AdS. Finally, I will use the bootstrap method to prove the conformal collider bounds for all unitarity CFTs in d>2.

Convexity in Tree Spaces

Bo Lin   [email] (UC Berkeley)

Abstract: We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all ultrametrics. The ${\rm CAT}(0)$-metric of Billera-Holmes-Vogtman arises from the theory of orthant spaces. While its geodesics can be computed by the Owen-Provan algorithm, geodesic triangles are complicated. We show that the dimension of such a triangle can be arbitrarily high. Tropical convexity and the tropical metric behave better. They exhibit properties desirable for geometric statistics, such as geodesics of small depth. This is a joint work with Bernd Sturmfels, Xiaoxian Tang and Ruriko Yoshida.

The Gopakumar-Vafa Invariants

Lutian Zhao (UIUC Math)

Abstract: In 1998, Gopakumar and Vafa argued from M-theory that BPS counts (now known as Gopakumar-Vafa invariants) have the same "generating function" as the Gromov-Witten invariants. In particular, these invariants are integral, and they agree with naive curve counting in many cases. Also, it explains the contribution of multicovering and bubbling phenomena. The basic idea of this counting is to use Lefschetz decomposition on the moduli space of D-Branes to "virtually count" the number of abelian varieties. In this talk, I will discuss why it is a promising counting invariant and give some easy cases of this counting. The serious difficulty of this counting is the definition of moduli of D-Branes, which only have a satisfactory description at g=0. If time permits, I will describe some attempts by Hosono-Saito-Takahashi, Kiem-Li and Maulik-Toda on this theory.

Castelnuovo-Mumford regularity of power products

Winfried Bruns (University of Osnabrück)

Abstract: We show that the Castelnuovo-Mumford regularity and related invariants of products of powers of ideals in a standard graded polynomial ring are affine-linear in the exponents if these are large enough, provided that each ideal is generated by elements of constant degree. A counterexample shows that linearity is false without the condition, but the regularity is always given by the maximum of finitely many affine-linear polynomials. This is joint work with Aldo Conca (Genoa).

The smallest parts function and the third order mock theta function $\omega(q)$

Ae Ja Yee (Pennsylvania State University)

Abstract: The smallest parts function spt$(n)$, counting the total number of appearances of the smallest parts in all partitions of $n$, has received great attention since it was first introduced by George Andrews in 2007. In this talk, I will give an introduction to and brief history of this subject. I will also discuss some recent discoveries in connection to the mock theta function $\omega(q)$. This is joint work with G. E. Andrews, A. Dixit, D. Passary, D. Schultz, J. Sellers and L. Wang.