**Miranda Cheng**, University of Amsterdam

*Optimal Jacobi Forms and Mock Theta Functions*

**John Duncan**, Case Western Reserve University

*Umbral Moonshine Modules*

Umbral moonshine attaches mock modular forms and meromorphic Jacobi forms to automorphisms of the Niemeier lattices. It is now known that this association can be recovered from specific, graded modules for the Niemeier lattice automorphism groups. We will describe recent progress in a program to realize these modules explicitly.

**Matthias Gaberdiel,** ETH Zurich

*Higher Spins & Strings*

**Terry Gannon**, University of Alberta

*Thoughts stolen from the enemy*

Subfactors and VOAs should both describe CFT, but what is relatively easy in one formulation can be very difficult in the other. In my talk I'll describe lessons the VOA world can learn from the subfactor one.

**Sarah Harrison,** Harvard University

*Umbral Moonshine and K3 Surfaces*

Recently, 23 cases of umbral moonshine, relating mock modular forms and finite groups, have been discovered in the context of the 23 even unimodular Niemeier lattices. One of the 23 cases in fact coincides with the so-called Mathieu moonshine, discovered in the context of K3 non-linear sigma models. Here we establish a uniform relation between all 23 cases of umbral moonshine and K3 sigma models, and thereby take a first step in placing umbral moonshine into a geometric and physical context. This is achieved by relating the ADE root systems of the Niemeier lattices to the ADE du Val singularities that a K3 surface can develop, and the configuration of smooth rational curves in their resolutions. A geometric interpretation of our results is given in terms of the marking of K3 surfaces by Niemeier lattices.

**Jeffrey Harvey**, University of Chicago

*Traces of Singular Moduli and Moonshine for the Thompson group*

We observe a relationship between the representation theory of the Thompson sporadic group and a weakly holomorphic modular form of weight one-half that appears in Zagier's work on traces of singular moduli and Borcherds products. We conjecture the existence of an infinite dimensional graded module for the Thompson group and use the observed relationship to propose a McKay-Thompson series for each conjugacy class of the Thompson group and then construct weakly holomorphic weight one-half forms at higher level that coincide with the proposed McKay-Thompson series. We also observe a discriminant property in this conjectured moonshine for the Thompson group that is closely related to the discriminant property conjectured to exist in Umbral Moonshine.

**Shamit Kachru**, Stanford University

*Moonshine at c=12*

**Christoph Keller**, Rutgers University

*Modular invariance and holographic CFTs*

Modular invariance plays an important role in AdS3/CFT2 holography. I discuss the structure of non-holomorphic CFT partition functions, namely in what sense the light spectrum determines the heavy spectrum and how to construct example partition functions using Poincare series. This yields necessary conditions on the spectrum of holographic CFTs. Finally I will discuss permutation orbifolds as examples of such theories.

**Heeyeon Kim**, Perimeter Institute

*Quiver Quantum Mechanics and Wall-Crossing *

I will talk about computation of the Witten index of 1d N=4 gauged linear sigma model which describes wall-crossing of BPS states in 4d N=2 theories. In the phase where the gauge group is broken to a finite group, the index is expressed as the JK-residue integral. Using this result, I am going to examine large-rank behaviour of the Kronecker quivers which describes the most simplest wall-crossing phenomena. I will also talk about how the refined Witten indices of quivers are preserved under the mutation process.

**Ching Hung Lam**, Institute of Mathematics, Academia Sinica

*On holomorphic vertex operator algebras of central charge 24*

I will talk about the recent progress on the classification of (strongly regular) holomorphic vertex operator algebras of central charge 24. In particular, I will discuss a construction of certain holomorphic vertex operator algebras of central charge 24 using orbifold construction associated to inner automorphisms. This talk is based on a joint work with Hiroki Shimakura.

**Geoffrey Mason**, University of California, Santa Cruz

*Symplectic automorphisms of some hyperk\"ahler manifolds*

**Greg Moore**, Rutgers University

*Measuring the Elliptic Genus*

This talk is based on the recent paper co-authored with N. Benjamin, M. Cheng, S. Kachru, and N. Paquette.

**Sameer Murthy**, King's College London

*ADE Little string theories, Mock modular forms, and Umbral moonshine*

**Hirosi Ooguri**, California Institute of Technology

*Analytic Bootstrap Bounds*

**Daniel Persson**, Chalmers University of Technology

*U-duality, exotic instantons and automorphic forms on Kac-Moody groups*

**Anne Taormina**, Durham University

*Signatures of Mathieu Moonshine in Z_2-orbifolds of Conformal Field Theories*

**Roberto Volpato**, SLAC & Stanford University

*Fricke S-duality in CHL models*

We consider dual pairs of four dimensional heterotic/type IIA CHL models with 16 space-time supersymmetries. We provide strong evidence for the existence of an S-duality acting on the heterotic axion-dilaton by a Fricke involution S --> -1/NS, where N is the order of the orbifold symmetry. While most models are self-dual, in some cases S-duality relates the CHL model to a compactification of type IIA on an orbifold of T^6. We provide a simple criterion to determine whether a model is self-dual or not. Finally, we argue that in self-dual CHL models the lattices of electric and magnetic charges must be N-modular and verify this prediction.

**Katrin Wendland**, University of Freiburg

*How does extended supersymmetry affect the elliptic genus?*

The elliptic genus of K3 and its decomposition into characters of the N=4 superconformal algebra of associated conformal field theories can be viewed as the outset of Mathieu Moonshine. Thus, extended supersymmetry induces additional properties of the elliptic genus, which so far lack a satisfactory geometric interpretation. We investigate the implications of this decomposition on geometric structures that underlie the elliptic genus.