The power of symmetries lies at the heart of interaction between modern mathematics and theoretical physics. A classic instance of such an interaction is the parallel development of Quantum Mechanics and that of the theory of finite dimensional representations of complex semi-simple lie algebras in the early 20th century. Natural questions arising from physics served as motivation for the development of the mathematical theory. The initiation of the theory of infinite dimensional representations of non-compact groups was also motivated by the need to understand the Representation Theory of the Poincare Group, the symmetry group of Special Relativity.

Quantum Theory and Representation Theory have since flourished independently but have continued to benefit from a cross fertilization of ideas. In recent years, there has been renewed interest in this interaction centered especially around various Supersymmetric Quantum Field Theories and Geometric aspects of Representation Theory. These topics turn out to also exhibit surprising connections to Conformal Field Theory and the theory of Classical and Quantum Integrable Systems.

This discussion meeting will bring together physicists and mathematicians working on exciting problems at this interface. There will be several mini-courses by leading experts and there will also be a small component involving research talks by some of the participants. The program is aimed at helping young and established researchers learn the foundational aspects of these research fields and also at establishing new and lasting conversations between physics and mathematics communities. We also expect fruitful interactions with the parallel ICTS program on Integrable Systems.

On the mathematical side, some of the topics that will be covered in the meeting are Geometric Langlands, Moduli spaces of Vector/Principal bundles, Higgs bundles, D-modules, Riemann-Hilbert correspondences, Homological Mirror Symmetry, Perverse Sheaves, Non-Abelian Hodge Theory, Character Varieties, Knot invariants, Quiver Representations and Quiver Varieties.

On the physical side, the topics will include the Gauge theory approach to Geometric Langlands, Topological Quantum Field Theories, Supersymmetric field theories in various dimensions, especially Seiberg-Witten solutions to 4d N=2 theories, the Hitchin system, Geometry of Monopole and Instanton moduli spaces, the AGT correspondence, spectrum of BPS states and physics approaches to Knot Invariants.

**Speakers**:

The mini courses will be delivered by

- Sergey Cherkis (University of Arizona)
- Tudor Dimofte (University of California, Davis)
- Jacques Distler (University of Texas, Austin)
- Ron Donagi (University of Pennsylvania)
- Sergei Gukov (Caltech)
- Lotte Hollands (Heriot-Watt University)
- Michiaki Inaba (Kyoto University)
- Takuro Mochizuki (Research Institute for Mathematical Sciences, Kyoto)
- Tony Pantev (University of Pennsylvania)
- Masa-Hiko Saito (Kobe University)
- Piotr Sulkowski (University of Warsaw)

We will also have research talks by,

- Sujay Ashok (IMSc, Chennai)
- Satoshi Nawata (Fudan University, Shanghai)
- Du Pei (CQGMS, Aarhus)
- P. Ramadevi (IIT Bombay, Mumbai)
- Meng-Chwan Tan (NUS, Singapore)
- Lakshya Bhardwaj (Perimeter Institute for Theoretical Physics)

The mini-courses will be on:

**Week 1 (July 16-20)**

- Instantons and Monopoles, Sergey Cherkis
- An Introduction to Class-S and Tinkertoys, Jacques Distler
- The Geometric Langlands conjecture and non-abelian Hodge theory, Ron Donagi
- Introduction into spectral networks, Lotte Hollands
- Lectures on Parabolic bundles, Parabolic Connections and Geometric Langlands, Michiaki Inaba and Masa-Hiko Saito
- Kobayashi-Hitchin correspondence for wild harmonic bundles, Takuro Mochizuki

**Week 2 (July 23-27)**

- Higher algebra in SUSY QFT, Tudor Dimofte
- Geometric Langlands and S-duality in $\mathcal{N}=4$ SYM followed by a lectures on "VOA[M4]" , Sergei Gukov
- Modular spectral covers and Hecke eigensheaves on interesections of quadrics, Tony Pantev
- Topological strings, knots, and quivers, Piotr Sulkowski

Detailed abstracts for the courses and research talks can be found here.

Full schedule is available here.

This program also overlaps with the parallel ICTS program on Integrable systems in Mathematics, Condensed Matter and Statistical Physics and we encourage interactions across the two meetings.