Abstract: Very stable and wobbly bundles have been known for 4 decades,  proving meaningful in many different contexts.  These notions were recently generalised by Hausel and Hitchin to Higgs bundles fixed under a natural C* action.  Depending on the existence of very stable points, irreducible components of the fixed-point locus can be classified into very stable and wobbly, the former being better behaved than the latter. For example, Hausel—Hitchin's methods allow to compute the multiplicities of very stable components. Given the role of the nilpotent cone, to which fixed points belong, it is important to determine the existence or not of very stable components. Hausel and Hitchin proved that all components with a generically regular nilpotent Higgs field are very stable. In this talk, I will show that components with generic nilpotent order two are wobbly, with very few low rank exceptions. Time permitting, I will explain ongoing work on other nilpotent orbits.  

Abstract: In the context of algebraic geometry, decomposition and inertia groups are special subgroups of the Cremona group which preserve a certain subvariety of $\mathbb{P}^n$ as a set and pointwise, respectively. These groups were and still are classic objects of study in the area, with explicit descriptions in several instances. In the particular case where this fixed subvariety is a hypersurface of degree $n+1$, we have the notion of Calabi-Yau pair which allows us to use new tools to deal with the study of these groups and one of them is the so-called volume preserving Sarkisov Program. Using this approach we prove that an appropriate algorithm of the Sarkisov Program in dimension 2 applied to an element of the decomposition group of a nonsingular plane cubic is automatically volume preserving. From this, we deduce some properties of the (volume preserving) Sarkisov factorization of its elements. Regarding now a 3-dimensional context, we give a description of which toric weighted blowups of a point are volume preserving and among them, which ones will initiate a volume preserving Sarkisov link from a Calabi-Yau pair $(\mathbb{P}^3,D)$ of coregularity 2. In this case, $D$ is necessarily an irreducible normal quartic surface having canonical singularities. This last result enhances and extends the recent works of Guerreiro and Araujo, Corti and Massarenti in a log Calabi-Yau geometrical perspective, and it is a possible starting point to study the decomposition group of such quartics. 

Abstract: I will explain how to construct kernels for differential equations using Master formula. Then I will introduce generalized Heun's equations following Boalch-Katz-Simpson, and show how two equations in BKS correspondence share the same kernel.

This is joint work in progress with Vasily Golyshev

Abstract: We discuss several recent results of ongoing work (in collaboration with I. Gajur and D. Van Straten and with V. Buchstaber and I. Gajur) on interesting properties of multiplicative generalised Bessel kernels, which include the well-known Clausen and Sonin-Gegenbauer formulas, examples of Kontsevich discriminant locus polynomials, given as addition laws for special two-valued formal groups (Buchshtaber-Novikov-Veselov), and the connection with "period functions", solving some Picard-Fuchs-type equations for the Calabi-Yau cases and related to analogues of Landau-Ginzburg superpotentials.

Abstract: Kontsevich's deformation quantization formula associates to any Poisson manifold an algebra of "quantum observables", defined as a noncommutative deformation of the product of functions. The formula is Feynman-style series expansion, whose coefficients are multiple zeta values, making it intractable for direct calculation.  Following a suggestion of Kontsevich, I will explain how K-theory and mixed Hodge structures can be used to construct natural "period coordinates" on the moduli space of smooth Poisson varieties, in which the quantization can often be computed simply, explicitly and nonperturbatively as the exponential map for a complex torus. This gives a conceptual explanation for the appearance of various classical transcendental functions in the relations defining well-known noncommutative algebras. This talk is based on forthcoming joint work with A. Lindberg. 

Abstract: I will outline a connection between the geometry of the space of Bridgeland stability conditions of a triangulated category and its semiorthogonal decompositions, explaining the main concepts using the example of coherent sheaves on the projective line. Time permitting, I will discuss forthcoming work on a modular partial compactification of the space of stability conditions. This talk is based on joint work with Daniel Halpern-Leistner. 

Abstract: I will introduce a program aiming at constructing lifts of differential equations in a systematic way.

The talk is based on joint work with Ilia Gaiur.

Abstract: Hypergeometric motives offer a tool to construct mirrors to certain Fano varieties out of the known constructions. Among these Fanos is the main family of anticanonical del Pezzo weighted hypersurfaces. In this talk, I will present a mirror construction for these surfaces and describe the derived category of the associated stacks. This last result is part of an ongoing project to study Homological Mirror Symmetry for the surfaces. The talk is based on joint works with A. Corti and F. Rota.