Abstract
Abstract: Categories of perverse sheaves on a possibly singular space X were introduced in a seminal work of Beilinson, Bernstein and Deligne. They are abelian subcategories arising as the hearts of t-structures on the constructible derived category, where the choice of t-structure is determined by a perversity function. The algebraic properties of these categories depend on the topology and geometry of X; for example, under suitable finiteness assumptions, they are equivalent to categories of modules over finite-dimensional algebras.
In this talk, I will focus on the case of complex projective space equipped with its standard affine stratification. The corresponding category of perverse sheaves with respect to the middle perversity enjoys several remarkable algebraic properties. I will explain how these properties can be exploited to describe the stability space of the constructible derived category of P^n. The results presented are based on previous and ongoing joint work with Jon Woolf.
Abstract: Enumerative geometry, as formulated in Gromov--Witten theory, encodes curve-counting information on smooth projective varieties. Such data can be organized in different ways, giving rise to rich geometric structures and invariants, including quantum cohomology and quantum spectra. In the work G. Cotti, ``Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers,'' IMRN, 2022, an unexpected connection was observed between the quantum cohomology of Grassmannians and the distribution of prime numbers. In this talk, I will present recent progress extending this perspective to the enumerative geometry of more general partial flag varieties, highlighting how the relation with prime numbers persists in a broader setting.
Abstract: A century-old conjecture of Hopf predicts the sign of the Euler characteristic of every closed, even-dimensional Riemannian manifold with nonpositive sectional curvature. In the special case of nonpositively curved piecewise Euclidean cubical manifolds, this conjecture leads to a purely combinatorial problem. Charney and Davis showed that this special case is governed by an invariant of flag simplicial spheres, and proposed the Charney--Davis conjecture: every flag simplicial sphere of odd dimension should have nonnegative Charney--Davis invariant, with the appropriate sign. This remains one of the central open problems in geometric and topological combinatorics.
Further strengthenings, formulated in terms of so-called gamma-vectors, were proposed by Gal and by Nevo--Petersen. Somewhat surprisingly, by moving from spheres to matroids, one can formulate analogues of the conjectures of Charney--Davis, Gal, and Nevo--Petersen. In this setting, the role of flag simplicial spheres is played by matroids equipped with flag building sets, in the sense of De Concini and Procesi. Relying on the theory of building sets, as well as on recent breakthroughs in matroid theory due to June Huh and collaborators, we prove matroidal analogues of the conjectures of Charney--Davis and Gal.
Joint work with Basile Coron and Shiyue Li.
Abstract: Kähler geometry lies in the intersection of complex geometry, Riemannian geometry and symplectic geometry. From the complex geometric point of view it is therefore natural to consider special non-Kähler Hermitian metrics on complex manifolds. Among them, pluriclosed metrics deserve particular attention. These metrics always exist on compact complex surfaces but the situation in higher dimension is very different. We will discuss several properties concerning these metrics also in relation with the Bismut connection having Kähler-like curvature. This is joint work with G. Barbaro and F. Pediconi.
Abstract: A brick in an abelian category is simply an object with an endomorphism ring which is a skew-field. In the representation theory of finite dimensional algebras, bricks in the module category play an important role in studying both torsion classes and thick (also known as wide) subcategories. In this talk we will overview some recent results concerning bricks in representation theory and we will discuss an approach to understanding their role in the more abstract setting of noetherian abelian categories. This talk is partly based on ongoing joint work with Jan Stovicek.