► Tamas Hausel 1

Title: Mirror symmetry for Hitchin systems

Abstract: After a short introduction to mirror symmetry phenomena for Hitchin systems, I will explain how to model the Hitchin system on certain very stable upward flows by the spectrum of equivariant cohomology of a Grassmannian. Then we find  its mirror as the spectrum of the Kirillov algebra of a minuscule representation of  the Langlands dual group. 

► Tamas Hausel 2

Title: Big algebras and real forms

Abstract: We generalise the construction of the first talk to non-minuscule representations using a big commutative subalgebra of the Kirillov algebra, ringifying the equivariant intersection cohomology of affine Schubert varieties. We finish discussing the finer stucture of these big algebras and in particular a set of results and conjectures on involutions induced from quasi-split real forms. At the end we mention expectations of principal endoscopy and transfer for big algebras.

► Mirko Mauri

Title: Hodge-to-singular correspondence

Abstract: We show that the cohomology of moduli spaces of Higgs bundles decomposes in elementary summands depending on the topology of the symplectic singularities on a (fixed!) master object and/or the combinatorics of certain posets and lattice polytopes. This is based on a joint work with Luca Migliorini and Roberto Pagaria.

► Michael McBreen

Title: Microlocal Sheaves on Affine Slodowy Slices

Abstract: I will describe certain moduli of wild higgs bundles on the line, and explain why they are affine analogues of Slodowy slices. I will then describe an equivalence between microlocal sheaves on a particular such space and a block of representations of the small quantum group. Joint work with Roman Bezrukavnikov, Pablo Boixeda Alvarez and Zhiwei Yun.

► Yuuji Tanaka 1

Title: On the moduli spaces of semistable Higgs sheaves on projective surfaces

Abstract: The moduli spaces of semistable Higgs bundles on a curve have been attracting lots of researchers in Geometry, Topology, Representation Theory, Number Theory, and Mathematical Physics. They have been indeed one of the wealthiest sources of research such as in the studies of non-abelian Hodge theory, the P=W conjecture, the SYZ style topological Mirror symmetry for the Hitchin fibrations, the Langlands programme and its geometric one, and so on. I'll talk about interesting analogues of these semistable Higgs bundles for the complex projective surface case and their surprising outcome, strongly motivated by the Kapustin-Witten and Vafa-Witten theories on toplogically twisted versions of N=4 super Yang-Mills theory more broadly over smooth four-manifolds. This talk is partly based on joint work with Chih-Chung Liu and Steven Rayan and other joint work with Richard Thomas.

► Yuuji Tanaka 2

Title: On a blowup formula for sheaf-theoretic virtual enumerative invariants on projective surfaces and its applications

Abstract: In the second talk, I'll speak about a blowup formula for sheaf-theoretic virtual enumerative invariants on projective surfaces, which include the Donaldson-Mochizuki invariant (a virtual analogue of the Donaldson invariant), the virtual Euler characteristic or virtual chi_y genus of the moduli space of semistable sheaves on a project surface (they are the instanton parts of the ordinary or K-theoretic Vafa-Witten invariant, respectively), and the virtual Verlinde number and virtual Segre one of the moduli space (the former is a generalisation of the K-theoretic Donaldson invariant and the latter is a generalisation of the Donaldson invariant with fundamental matters).

Our blowup formula has exciting applications for these virtual enumerative invariants. For example, we obtain blowup formulae for the generating series of the virtual Euler characteristics and virtual chi_y-genera of the moduli spaces, in which modular forms appear in the same way as in Vafa-Witten's original paper in '94, as Goettsche and Goettsche-Kool conjectured. These enable one to compute some of universal functions in the generating series of the instanton part of the Vafa-Witten invariants on a projective surface, provided they existed as Goettsche-Kool and Goettsche-Kool-Laarakker conjectured.

This talk is based on joint work with Nikolas Kuhn and another joint work with Nikolas Kuhn and Oliver Leigh.

► Yaoxiong Wen

Title: Mirror symmetries for parabolic Hitchin systems, from classical to global

Abstract: We study the parabolic Hitchin systems under the Langlands dual, focusing on types B and C. We aim to understand the SYZ and topological mirror symmetries for the Langlands dual parabolic Hitchin systems. We find three levels of dualities/symmetries: 

  1. Classical level deals with the parabolic structures, which relate to nilpotent orbits. The duality here is the Springer dual.

  2. Local level plays a crucial role; it serves as a bridge between the classical level and the global level. 
      It deals with affine Spaltenstein fiber; the symmetry here is Lusztig's canonical quotient.

  3. Global level deals with the moduli space of parabolic Higgs bundles. Mirror symmetries here are SYZ and topological mirror symmetries.

This talk is partly based on the joint work with B. Fu and Y. Ruan (arXiv:2207.10533) and the in-progress work with X. Su, B. Wang, and X. Wen.

► Sang-Bum Yoo

Title: HIGGS BUNDLES WITH A FIXED DETERMINANT ON AN IRREDUCIBLE NODAL CURVE

Abstract: We introduce a construction of the moduli space of Higgs bundles with a fixed determinant on an irreducible nodal curve. To construct the moduli space, we modify the construction of the moduli space of vector bundles with a fixed determinant on an irreducible nodal curve given by U. Bhosle. Then we define the Hitchin map on the moduli space and describe their fibers. This work is in progress.