Geometry, Algebra, and Physics Seminar

Severi varieties are spaces which parametrize plane curves of fixed degree and geometric genus (with at worst nodal singularities), and the definition can be easily extended to projective surfaces other than the plane. In this talk, I will discuss the problem of proving that Severi varieties are irreducible (or of determining their irreducible components, since irreducibility fails for some surfaces). Although irreducibility is known to hold for Severi varieties of \( \mathbb{P}^2 \) and a few other rational surfaces, the problem remains open in most cases.

Partially based on a joint work with Rostislav Devyatov and Alexander Merkurjev, the sharp upper bounds on indexes of twisted flag varieties under the spin groups \( \mathrm{Spin}(n) \) are being established. In equivalent terms, for every integer m of the interval \( [1,n/2] \), we are looking for the sharp upper bound on the greatest common divisor of degrees of the finite base field extensions over which the Witt index of a given n-dimensional quadratic form of trivial discriminant and Clifford invariant becomes at least \( m \).

Modular tensor categories, or more generally nondegenerately braided fusion categories, are objects vital to the modern study of conformal field theory and topological quantum computation. Examples arise as representation categories corresponding to certain vertex operator algebras and quantum groups. A key characteristic of these categories is nondegeneracy, which amounts to the invertibility of a matrix associated to the braiding. But much like the invertibility of matrices in general, nondegeneracy is a nontrivial assumption; there are copious amounts of degenerately braided fusion categories such as the representation categories of finite groups. In this talk I'll discuss the very open problem of embedding degenerately braided fusion categories into nondegenerate ones. The discussion will be elementary and example-driven, and have the feeling of covering space theory in algebraic topology.

Given a reductive group \( G \) acting on a quasi-projective variety \( X \) , it is well known that the derived category of a GIT quotient \( X//G \) is equivalent to a window subcategory of the derived category of the quotient stack \( [X/G] \) . The idea of mirror symmetry suggests an equivalence between the derived category of a variety and the appropriate version of the Fukaya category of its mirror, and raises the question whether, or under what circumstances, there is a geometric construction of a window on the mirror side. I will go over some progress answering this question in the affirmative for certain toric varieties, where the construction is essentially a topological deformation. If time permits, I will introduce other variants of the construction, eyeing on the remaining cases and applications in algebraic geometry.

In 1976, Victor Guillemin published a paper discussing geometric scattering theory, in which he related the Lax-Phillips Scattering matrices (associated to a noncompact hyperbolic surface with cusps) and the sojourn times associated to a set of geodesics which run to infinity in either direction. Later, the work of Guillemin was generalized to locally symmetric spaces by Lizhen Ji and Maciej Zworski. In the case of a \( \mathbb Q \) -rank one locally symmetric space \( \Gamma \backslash X \), they constructed a class of scattering geodesics which move to infinity in both directions and are distance minimizing near both infinities. An associated sojourn time was defined for such a scattering geodesic, which is the time it spends in a fixed compact region. One of their main results was that the frequencies of oscillation coming from the singularities of the Fourier transforms of scattering matrices on \( \Gamma \backslash X \) occur at sojourn times of scattering geodesics on the locally symmetric space. In this talk I will review the work of Guillemin, Ji and Zworski as well as discuss the work from my doctoral dissertation on analogous results for higher rank locally symmetric spaces. In particular, I will describe higher dimensional analogues of scattering geodesics called \( \textbf{Scattering Flat} \) and study these flats in the case of the locally symmetric space given by the quotient \( SL(3,{\mathbb Z}) \backslash SL(3,{\mathbb R}) / SO(3) \). A parametrization space is discussed for such scattering flats as well as an associated vector valued parameter (bearing similarities to sojourn times) called \textbf{sojourn vector} and these are related to the frequency of oscillations of the associated scattering matrices coming from the minimal parabolic subgroups of \( \text{SL}(3,R) \) . The key technique is the factorization of higher rank scattering matrices.

I will discuss some natural geometric constructions in the Fano variety of lines of a smooth complex cubic fourfold \( X \) . The 2-dimensional loci S of lines which have too many tangent 2-planes, and V those which have a triply tangent 2-plane play a special role in the geometry of \( X \) and \( F(X) \). I will discuss these surfaces and their invariants and use this analysis to give a geometric argument of the fact that the number of nodal rational curves of primitive class in \( F(X) \) of a very general cubic fourfold \( X \) is \( 3780 \) .

In this talk I will give first a short introduction and overview of cohomological invariants, focusing on finite groups. After that I will explain the computation of cohomological invariants of degree two of finite groups.

In 1987, Deligne obtained an isometry for families of compact Riemann surfaces and holomorphic vector bundles, which links an object arising from spectral geometry, the determinant line bundle, to intersection bundles. However, an important requirement for this to work is to only consider smooth metrics, which is restrictive for situations linked to number theory.

One such setting is that of modular curves with holomorphic flat unitary vector bundles coming from representations of Fuchsian groups. In that case, the natural metrics are not smooth. The work of Freixas i Montplet and von Pippich has shown that it was possible to overcome the singularities for the trivial line bundle, and prove a Deligne-Riemann-Roch isometry.

In this talk, we will see how to extend this result to a more general class of flat unitary vector bundles over modular curves. Interesting applications in number theory have to do with the first non-zero derivative at 1 of the Selberg zeta function, and the Weil-Petersson norm of weight 1 modular forms with Nebentypus.