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We propose central charges for 3d branes in 3d mirror symmetry and formulate a mirror symmetry statement for them in terms of Fourier-type transforms. We show that the central charges of 2d branes can be recovered from those of 3d branes through their Hom spaces, which leads to the mirror symmetric Gamma conjecture in 2d mirror symmetry. Several explicit examples will be discussed. This is based on ongoing joint work with Conan Leung.

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Let X be a compact complex manifold in the Fujiki class (i.e., bimeromorphic to a compact Kähler manifold) with torsion canonical bundle, and let E be a holomorphic vector bundle over X. Assume that H^2(X, End^0 E) = 0, where End^0 E denotes the trace-free subbundle of End(E). We prove that the Kuranishi space of joint deformations of the pair (X, E) is smooth. In particular, when E is of rank 1, the pair (X, L) admits unobstructed deformations. Our result recovers and refines the known results of K. Chan--Y.-H. Suen (2016), S. Li--X. Pan (2019) and of D. Iacono--M. Manetti (2021). The approach we use is analytic and follows the Tian--Todorov method more faithfully, which is completely different from theirs.

On the other hand, we construct examples showing that when the assumption H^2(End^0 E) = 0 is dropped, Kur(X, E) may be singular, thereby answering a question raised by Simon Felten.

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The Atiyah-Floer conjecture relates gauge theory (Instanton Floer homology of 3-manifolds) and symplectic geometry (Lagrangian Floer homology of moduli spaces of flat connections). There are several versions of this conjecture, some of which have been established when some moduli spaces do not contain reducible connections. The cases when these exist are substantially harder: defining the symplectic side still remains out of reach.

Following work of Manolescu-Woodward and Jeffrey, one can instead consider an equivariant version of the conjecture, for which it is possible to define the symplectic case, even in the above hard cases.

I will present a model for equivariant Morse homology (arXiv:2404.17393) that I believe is applicable to both side of the conjecture, and some results within Morse theory (arXiv:2505.01362) that should permit to relate both sides.

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This is an expository talk about several recent works on the Fukaya category of Coulomb branches, which is a topic related to 2D mirror symmetry, 3D mirror symmetry, symplectic Khovanov homology, and the higher representation theory of quantum groups. The main part of the talk is based on arXiv:2406.04258 by Mina Aganagic, Ivan Danilenko, Yixuan Li, Vivek Shende, and Peng Zhou, as well as ongoing work by Mina Aganagic, Vivek Shende, Peng Zhou, and Elise LePage.

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In this talk, we define a trigonometric DAHA action on the equivariant quantum cohomology of T*G/P via shift operators. As applications, we give a new short proof of the Peterson's isomorphism, as well as a conjecture of Braverman-Finkelberg-Nakajima regarding the spherical subalgebra of trigonometric DAHA.

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In joint work with Matessi-Ruddat-Zharkov, we prove the existence of Lagrangian torus fibrations on Calabi-Yau hypersurfaces in toric Fano manifolds given by a reflexive polytope. The result is motivated by the Strominger-Yau-Zaslow conjecture, which predicts the existence of these fibrations on Calabi-Yau manifolds near large complex structure limits. In this talk, I will outline the main setup of the construction. The idea is to replace the ordinary algebraic equation with a new one involving "ironing coefficients" and a convex (but not strictly convex!) potential which has the effect of breaking the manifold into local models. Over these models, we apply the Liouville flow technique in the style of Evans-Mauri.