Weekly Bulletin

Isoparametric submanifolds and foliations, in particular in round spheres, have been a historically fundamental concept in submanifold geometry that linked geometry, topology, and algebra. In this talk, I will survey these concepts underlying the rich history and applications, and will describe new results in joint work with A. Lytchak and M. Krannich. In the new work, we analyze the space of all isoparametric submanifolds on compact manifolds with uniformly bounded diameter, volume, and sectional curvature, and show that up to foliated diffeomorphism there are only finitely many such foliations. The talk will be tailored for topologically-minded people. Or at least, the tailor will do his best.

An important question with many practical applications in enumerative geometry is to understand how enumerative invariants attached to moduli spaces of sheaves/complexes/quiver representations change when we vary the stability condition. Wall-crossing for motivic invariants has been a central tool in Donaldson-Thomas theory of CY 3-folds. In a non-CY setting, more recently, Joyce developed a wall-crossing framework for cohomological invariants and Liu adapted his ideas to K-theoretic invariants. In this talk, I will explain a new approach to such wall-crossing formulas and an alternative way to define enumerative invariants in the presence of strictly semistable objects. This new framework extends the range of wall-crossing formulas we can prove, allowing us to deal for example with objects in the derived category of sheaves. This is joint work with Ivan Karpov.

The Fourier transform has played a crucial role in the cohomological study of Jacobians over the decades. Recently, in efforts to understand the P=W conjecture for the Hitchin system, a theory of Fourier transform was developed for (possibly singular) compactified Jacobians, where the perverse filtration comes into play naturally. The purpose of these lectures is to explain some applications of this circle of ideas in the study of cohomology of the more classical compactified Jacobians associated with stable curves. The (in)dependence of the cohomology ring on the degree and the stability condition will be discussed. Based on joint work in progress with Younghan Bae and Davesh Maulik.

The Fourier transform has played a crucial role in the cohomological study of Jacobians over the decades. Recently, in efforts to understand the P=W conjecture for the Hitchin system, a theory of Fourier transform was developed for (possibly singular) compactified Jacobians, where the perverse filtration comes into play naturally. The purpose of these lectures is to explain some applications of this circle of ideas in the study of cohomology of the more classical compactified Jacobians associated with stable curves. The (in)dependence of the cohomology ring on the degree and the stability condition will be discussed. Based on joint work in progress with Younghan Bae and Davesh Maulik.