Previous Seminars

Abstract:

Abstract: Demazure modules are submodules of irreducible modules of Lie algebras generated by extremal weight spaces under the action of a Borel subalgebra. For the general linear group, their characters form a basis of polynomials that containing Schur polynomials, the characters of irreducible modules. The celebrated Littlewood-Richardson rule gives a beautiful combinational model for tensoring irreducible modules or, equivalently, multiplying Schur polynomials. The analogous problem for Demazure modules is to give an excellent filtration, but this exists only in certain cases. One sees this computationally since the product of Demazure characters cannot, in general, be written as a nonnegative sum of Demazure characters. In this talk, I’ll present a nonnegative Littlewood-Richardson rule expanding the product of a Demazure character and Schur polynomial into Schubert characters, certain polynomials indexed by lower order ideals in Bruhat order. This combinatorial statement suggests that the tensor product of Demazure modules admits a Schubert filtration, a still open conjecture of Polo from 1989.

No familiarity with Demazure modules or characters is assumed for this talk.

Abstract: Interesting moduli spaces don’t have many integral points. More precisely, if X is a variety over a number field, admitting a variation of Hodge structure whose associate period map is injective, then the number of S-integral points on X of height at most H grows more slowly than H^ε, for any positive ε. This is a sort of weak generalization of the Shafarevich conjecture; it is a consequence of a point-counting theorem of Broberg, and the largeness of the fundamental group of X. Joint with Ellenberg and Venkatesh (arxiv:2109.01043).

[Special time and day]

Abstract: After giving a historical overview of classical and topological Hochschild homology, I will introduce a motivic variant valued in the stable A¹-homotopy category. Over the field of complex numbers, computations reveal that motivic Hochschild homology of F_p exhibits a rich pattern of “τ-torsion” elements. I will describe these computations and their significance, concluding with a no-go theorem regarding three potential motivic analogues of classical theorems. The talk will be accessible to those without prior experience in A¹-homotopy theory. This is joint work with Bjorn Dundas, Mike Hill, and Paul Arne Ostvaer.

Abstract: Thomason proved that Bott-inverted (or, better, KU-local) algebraic K-theory satisfies étale descent. This was decisive in clarifying the Quillen-Lichtenbaum conjectures in algebraic K-theory. It turns out that KU-local Grothendieck-Witt theory is not an etale sheaf in general but instead satisfies Scheiderer’s b-descent. I will explain this result and its connections with Real algebraic geometry. This is joint work with Nardin, Shah and Yang.

Abstract: The classical coinvariant algebra of a reflection group is a quotient of a polynomial ring by the ideal generated by non-trivial invariants. Coinvariant algebras have their roots in Hilbert’s Basis Theorem and are well-understood from a variety of perspectives using tools such as Schubert calculus, symmetric function theory, and monomial bases. A recent conjecture of Zabrocki introduced differential or “super” coinvariant algebras by adding an additional set of *anti*-commuting variables, or equivalently by considering coinvariant differential forms. We will describe some of the rich combinatorial and algebraic structure of these differential coinvariant algebras, focusing on the Type A case. Topics include their bi-graded Hilbert series, their alternants, and the homology of certain generalized exterior derivative complexes. Joint work with Nolan Wallach.

Abstract: A couple of years ago, S. Gukov and C. Manolescu conjectured that the Melvin-Morton-Rozansky expansion of the colored Jones polynomials can be resummed into a two-variable series F_K(x,q), as part of a bigger program to construct a 3d TQFT \hat{Z} that had been predicted from physics.
In this talk, I will explain how to prove their conjecture for a big class of links by “inverting” a state sum.

Abstract: In this talk, I will outline my work on complex rank 3 topological vector bundles on CP^5. I will describe a classification of such bundles using twisted, topological modular form-valued invariants. This builds a parallel with Atiyah and Rees’ classification of rank 2 topological vector bundles on CP^3, using real k-theory. I will also discuss methods for producing topological vector bundles of rank 3 on CP^5. As time allows, I will outline future directions, including: possible connections between higher real K-theories and vector bundles on projective spaces, and algebraic generalizations of my current work.