Titles and Abstracts
Brent Gorbutt (George Mason University)
Title: Positivity and combinatorics of Peterson Schubert calculus
Abstract: TBA
Hiraku Abe (Okayama University of Science)
Title: Geometry of Peterson Schubert calculus and its computations
Abstract: Harada-Tymoczko constructed an additive basis of the cohomology ring of the Peterson variety in type A, and they gave a Monk-type formula for the multiplication rule. Recently, Goldin-Gorbutt established a positive formula for all structure constants.
In this talk, I will explain a geometric background of these calculus. I will also explain a combinatorial game which effectively computes the structure constants. This is joint work with Hideya Kuwata, Tatsuya Horiguchi, and Haozhi Zeng.
Vasu Tewari (University of Hawaii at Manoa)
Title: Remixed Eulerian numbers: probability, combinatorics, and some combinatorial algebraic geometry.
Abstract: I will discuss a polynomial analogue of Postnikov's mixed Eulerian numbers from multiple perspectives, relating it to work of Klyachko in an algebraic context, Diaconis-Fulton in the probabilistic context, Harada-Horiguchi-Masuda-Park in a geometric context, among others. Time permitting, I will also introduce an even more general (multivariate) analogue of these numbers.
Rahul Singh (ICERM, Brown University)
Title: Positivity in Peterson Schubert Calculus
Abstract: The Peterson variety is a subvariety of the flag manifold equipped with one-dimensional torus action. We present dual bases for the homology and cohomology of the Peterson variety, and prove positivity and stability properties of these bases. We also develop a type-independent proof of the Giambelli formula for the Peterson variety, and use this formula to compute the intersection multiplicity of a Peterson variety with an opposite Schubert variety corresponding to a Coxeter word. Finally, we develop an equivariant Chevalley formula for the cap product of a divisor class with a fundamental class, and a dual Monk rule, for the Peterson variety. Based on joint works (https://arxiv.org/abs/2106.10372 and https://arxiv.org/abs/2111.15663) with Mihalcea and Goldin.
Jongbaek Song (KIAS)
Title: Regular Hessenberg varieties and toric varieties.
Abstract: A Hessenberg variety is a subvariety of the flag variety ($G/B$) determined by two parameters: one is an element of the Lie algebra of $G$ and the other is a $B$-submodule containing the Lie algebra of $B$, known as a Hessenberg space. In this talk, we focus on elements in the regular locus of the Lie algebra and the Hessenberg space determined by negative simple roots. Then, we aim to figure out cohomological relationship of these Hessenberg varieties with a certain class of toric varieties having orbifold singularities. The main result raises an interesting topic concerning toric varieties with symmetries by reflections. This is a joint work with M. Masuda, T. Horiguchi and J. Shareshian.