Previous Special Year Seminar

Given a triple (X,π,s) consisting of a homogeneous space X=G/P, a dominant weight π giving a projective embedding of X, and a reduced expression s for the minimal coset representative of w_0 in the parabolic quotient W/W_P, we construct a polytope...

We will discuss combinatorial and algebraic aspects of regular Hessenberg varieties, a large class of subvarieties of the flag variety G/B. For the special case of Peterson varieties, we show their equivariant structure constants are non-negative...

A real plane algebraic curve C is called expressive if its defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of C. We give a necessary and sufficient criterion for expressivity (subject...

We present a combinatorial approach to the Deligne-Mumford-Knudsen compactification of the moduli space of n distinct points on the projective line $P^1$. The idea is to choose a totally symmetric embedding of the orbits of generic points into a...

We show that various classical theorems of linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a general result that involves a tiling of a closed oriented surface by...

Abstract: In this talk I will construct a “quasisymmetric flag variety”, a subvariety of the complete type A  flag variety built by adapting the BGG geometric construction of divided differences to the newly introduced “quasisymmetric divided...

Postnikov's divided symmetrization, introduced in the context of volume polynomials of permutahedra, possesses a host of remarkable ``positivity'' properties. These turn out to be best understood using a family of operators we call quasisymmetric...

The ring of symmetric functions has a linear basis of Schur functions $s_{\lambda}$ indexed by partitions $\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq 0 )$. Littlewood-Richardson coefficients $c^{\nu}_{\lambda, \mu}$ are the structure...

A remarkable result of Brändén and Huh tells us that volume polynomials of projective varieties are Lorentzian polynomials. The dual notion of covolume polynomials was introduced by Aluffi by considering the cohomology classes of subvarieties of a...

K-theory arose in the 1950s from Grothendieck’s formulation of the Riemann-Roch theorem – that is, from attempts to calculate spaces of sections of vector bundles on a variety X via intersection theory on X.  An equivariant version was introduced...