Explore advanced mathematical physics through this comprehensive workshop focusing on recent developments in gauge theories and their applications to differential invariants. Delve into the active interplay between theoretical physics and mathematics, examining supersymmetric quantum field theories, superstrings on curved backgrounds, and explicit calculations of equivariant Donaldson invariants in higher rank scenarios. Study the mathematical foundations of higher rank Donaldson invariants, Vafa-Witten invariants, and Donaldson-Thomas invariants while investigating BPS spectra of non-Lagrangian N=2 gauge theories and their connections to new four-dimensional invariants. Master key technical tools including equivariant localization for extracting exact results, and explore connections to matrix models, topological strings, and quantum Hitchin systems. Examine gauge theories on complex toric surfaces, their instanton moduli spaces, surface operators in gauge theory, and moduli spaces of principal bundles with parabolic structure. Investigate quantum cohomology of instanton moduli spaces, K-theory of quiver varieties, quantum integrability, and quantum Yang-Baxter equations. Discover relationships to knot theory, Khovanov homology, higher dimensional supersymmetric gauge theories, and M-theory through lectures covering Coulomb branches of 3d N=4 gauge theories, virtual topological invariants, geometric R-matrices for Hilbert schemes, u-plane integrals, character varieties, chiral algebras from three dimensions, and invariants from non-gauge theories.

Syllabus

Coulomb branches of 3d N=4 gauge theories 1
Invariants of manifolds from 6 dimensions 1
Virtual topological invariants of moduli spaces of sheaves on surfaces 1
On the log-local principle for (refined) Gromov--Witten theory
A geometric R-matrix for the Hilbert scheme of points on a general surface
Coulomb branches of 3d N=4 gauge theories 2
Virtual topological invariants of moduli spaces of sheaves on surfaces 2
Mocking the u-plane integral and Donaldson invariants 1
Coulomb branches of 3d N=4 gauge theories 3
Mocking the u-plane integral and Donaldson invariants 2
Virtual topological invariants of moduli spaces of sheaves on surfaces 3
Invariants of manifolds from 6 dimensions 2
The curious hard Lefschetz property for character varieties
Invariants of manifolds from 6 dimensions 3
Virtual topological invariants of moduli spaces of sheaves on surfaces 3
Invariants of Manifolds from Non-Gauge Theories 1
Invariants of Manifolds from Non-Gauge Theories 1
Chiral Algebras from Three Dimensions
MTC[M3] and Twisted Hilbert Spaces
Invariants of manifolds from 6 dimensions 4