Schedule

All talks will be held in CBEC 130, in the Chemistry building. (The link takes you to room info and a campus map.)

Wednesday 5/9	Thursday 5/10	Friday 5/11	Saturday 5/12
9:20: Yong	9:00: Lam	9:00: Tymoczko	9:00: Mihalcea
11:00: Shimozono	10:30: Pechenik	10:30: Pawlowski	10:30: Buch
2:00: Rimanyi	1:30: Ramadas	1:30: Woo	1:30: Billey
3:30: Fulton	3:00: Chan	3:00: Tarasca	3:00: wrap-up
	4:30: Knutson

Abstracts

Sara Billey: Boolean product polynomials and Schur-positivity

We study a family of symmetric polynomials that we refer to as the Boolean product polynomials. The motivation for studying these polynomials stems from the computation of the characteristic polynomial of the real matroid spanned by the nonzero vectors in Rⁿ all of whose coordinates are either 0 or 1. To this end, one approach is to compute the zeros of the Boolean product polynomials over finite fields.

The zero loci of these polynomials cut out hyperplane arrangements known as resonance arrangements, which show up in the context of double Hurwitz polynomials. By relating the Boolean product polynomials to certain total Chern classes of vector bundles, we establish their Schur-positivity by appealing to a result of Pragacz relying on earlier work on numerical positivity by Fulton-Lazarsfeld. Subsequently, we study a two-alphabet version of these polynomials from the viewpoint of Schur-positivity. As a special case of these polynomials, we recover symmetric functions first studied by Desarmenien and Wachs in the context of descents in derangements.

This is based on joint work with Lou Billera and Vasu Tewari.

Anders Buch: Quantum K-theory

The quantum K-theory ring of a flag variety is a generalization of the quantum cohomology ring that encodes information about the arithmetic genera of Gromov-Witten varieties in its structure constants. I will speak about recent results about this ring, based on joint papers with Chaput, Chung, Li, Mihalcea, and Perrin.

Melody Chan: Brill-Noether varieties and set-valued tableaux

Joint work with Nathan Pflueger. A set-valued tableau (due to Buch) is like a usual tableau, but boxes are filled with finite sets instead of single elements. They arose in a recent Euler characteristic formula for Brill-Noether varieties. I will explain how, and I will talk about an RSK-type formula we derive to relate counting set-valued semistandard tableaux to counting skew tableaux of related shapes.

William Fulton: Schubert Calculus with a Twist

In order for Schubert polynomials in types B, C, and D to provide formulas for classical loci where symmetric and skew-symmetric matrices drop rank, one needs to allow symplectic and quadratic forms on vector bundles with values in a line bundle. Carrying this out, without ignoring torsion or dividing by 2's, requires making some new geometric constructions, which lead to some new algebra with interesting combinatorial properties. Untwisted, the new polynomials recover the single Schubert polynomials of Billey-Haiman and the double Schubert polynomials of Ikeda-Mihalcea-Naruse. This talk will describe some of this story, which is joint work with Dave Anderson.

Allen Knutson: Quiver varieties, the Yang-Baxter equation, and Schubert calculus

In 2003 Tao and I gave a formula for equivariant Schubert calculus on Grassmannians, in terms of "puzzles". In 2006 Zinn-Justin (with whom this work is joint) observed that a certain 9x9 matrix constructed from puzzles satisfies the "Yang-Baxter equation". Solutions to this equation were constructed representation-theoretically by Drinfel'd and Jimbo, and in 2012 given a geometric interpretation by Maulik and Okounkov using quiver varieties.

Maulik-Okounkov classes on cotangent bundles of d-step flag manifolds (a subclass of quiver varieties) extend Schubert classes. I'll define these, and explain how puzzles compute the products of (not quite) these classes, for d up to 4, in particular discovering and proving rules for K_T(2-step) and K(3-step). Unfortunately the 4-step rule is not positive.

Thomas Lam: Back-stable Schubert calculus and bumpless pipedreams

This is a continuation of Mark Shimozono's talk. I will discuss a class of "bumpless" pipedreams and use them to give formulae for double Schubert polynomials, Stanley-Edelman-Greene-coefficients, double Schur functions, and back stable double Schubert polynomials.

The combinatorics of bumpless pipedreams is derived from Knutson's interval positroid pipedreams. We will explain some of the geometry (affine and infinite flag varieties and Grassmannians, positroid varieties and graph Schubert varieties) behind this connection.

(Joint with Seung-Jin Lee and Mark Shimozono.)

Leonardo Mihalcea: Motivic Chern classes, Hecke algebras, and stable envelopes

The Chern-Schwartz-MacPherson (CSM) class of a compact (complex) variety X is a homology class which provides an analogue of the total Chern class of the tangent bundle of X, for X singular. Its K-theoretic version, the motivic Chern class, was defined by Brasselet, Schurmann and Yokura. It is a class with good functorial properties, and for smooth X it normalizes to the Hirzebruch's lambda-y class of the cotangent bundle of X. In the talk I will introduce these classes, and I will explain how one can use Bott-Samelson resolutions and a Verdier-Riemann Roch theorem to calculate motivic Chern classes of Schubert cells. It turns out that these classes can be recursively obtained using the Demazure-Lusztig operators in the Hecke algebra. I will also discuss relations to K theoretic envelopes of Maulik and Okounkov, as studied in the recent work of Su, Zhao and Zhong, and a positivity conjecture for the expansion of a motivic Chern class into appropriate K-theoretic Schubert classes. This recovers and extends beyond Lie type A recent results obtained by Fejer, Rimanyi and Weber, using localization techniques. The talk is based on ongoing joint work with Paolo Aluffi, Changjian Su, and Jorg Schurmann.

Brendan Pawlowski: Generalized coinvariant algebras as cohomology rings

Haglund, Rhoades, and Shimozono introduced a generalization of the coinvariant algebra of S_n whose dimension is the number of ordered set partitions of n into k blocks for a fixed k ≤ n, and whose graded Frobenius characteristic is a symmetric function arising in the Delta conjecture of Haglund-Remmel-Wilson, a strengthening of the (former) shuffle conjecture. We show that this generalized coinvariant algebra (with its S_n-module structure) is the ordinary cohomology ring of the space of n-tuples of lines spanning C^k, and construct a Schubert-like affine paving of this space, giving Schubert bases for the generalized coinvariant algebras. This is joint work with Brendon Rhoades.

Oliver Pechenik: Unique rectification in d-complete posets: towards the K-theory of Kac-Moody flag varieties

I'll talk about an REU from last summer with Rahul Ilango and Michael Zlatin. The jeu-de-taquin-based Littlewood-Richardson rule of Thomas and Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, Buch and Samuel (2016) developed a combinatorial theory of "unique rectification targets" in minuscule posets to extend the rule to K-theoretic Schubert structure constants for minuscule varieties. Separately, Chaput and Perrin (2012) used the combinatorics of Proctor's "d-complete posets" to extend the rule to a broader class of cohomological structure constants for Kac-Moody flag varieties. We begin to address the unification of these theories. Our main result is the identification of unique rectification targets in a large class of d-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain K-theoretic Schubert structure constants in the Kac-Moody setting.

Rohini Ramadas: Dynamics on the moduli space M_0,n

The moduli space M_0,n parametrizes configurations of n points on the Riemann sphere P¹, up to change of coordinates on P¹. M_0,n is non-compact but has many compactifications with interesting combinatorial structure.

Hurwitz correspondences are certain natural (multi-valued) self-maps of M_0,n.

I will introduce M_0,n, its compactifications, and Hurwitz correspondences. I will describe how the dynamics of Hurwitz correspondences can be studied by studying induced linear actions on homology groups of compactifications of M_0,n.

Richard Rimanyi: Generating functions of degeneracy loci formulas

Degeneracy loci formulas express equivariant characteristic classes of invariant subvarieties of representations. They occur in different cohomology theories (H, K, Ell) and flavors (e.g. fundamental class, Segre-Schwartz-MacPherson class, motivic Chern class, and more). Even within the same theory and flavor one often faces natural infinite sequences of classes---e.g. such examples occur for quivers, matroid representation spaces, and singularities. We will review a non-obvious way of encoding these infinite sequences of polynomials by (residue) generating functions.

Mark Shimozono: Back-stable Schubert calculus and homology of the infinite Grassmannian

We will discuss Knutson's back-stable limit of double Schubert polynomials, double Stanley symmetric functions and positivity of equivariant Stanley-Edelman-Greene coefficients. These coefficients, which satisfy a form of Graham positivity, appear in Peterson's j-basis, a form of the Schubert basis of equivariant homology of the infinite Grassmannian. Under the isomorphism with double symmetric functions the j-basis maps to Molev's dual Schur functions. We introduce a new divided difference operator for the zero-th reflection and use it to create the dual Schurs. This construction also works for affine root systems, creating the double k-Schur functions, which are series whose lowest degree term in nonequivariant variables are the k-Schur functions. (Joint with Thomas Lam and Seung-Jin Lee.)

Nicola Tarasca: K-classes of Brill-Noether varieties and a determinantal formula

Brill-Noether varieties for pointed curves parametrize linear series on curves with prescribed vanishing at marked points. I will present a formula for the Euler characteristic of the structure sheaf of Brill-Noether varieties for curves with at most two marked points. The formula recovers the classical Castelnuovo number in the zero-dimensional case, and previous work of Eisenbud-Harris, Pirola, Chan-López-Pflueger-Teixidor in the one-dimensional case. The result follows from a new determinantal formula for the K-theory class of certain degeneracy loci of maps of flag bundles. This is joint work with Dave Anderson and Linda Chen.

Julianna Tymoczko: Geometry of Springer fibers

The Springer fiber of a linear operator X consists of the flags fixed by X, in the sense that the image of each nested subspace of the flag under the linear operator is contained in itself. Springer fibers were constructed as one of the classic examples of geometric representation theory: their cohomology carries an action of the symmetric group, and varying X over nilpotent conjugacy classes recovers all irreducible representations of the symmetric group. Like Schubert varieties, important aspects of their geometry are encoded combinatorially; however, much less is known about Springer fibers than Schubert varieties. We give some new results classifying singular and smooth components for certain Springer fibers (joint with Joan Kim, Haley Hoech, and Nicole Magill), and time permitting, describe how certain intersections of Schubert cells and Springer fibers are isomorphic to affine Schubert cells (joint with Linda Chen).

Alexander Woo: Nash blowups of cominuscule Schubert varieties and combinatorial Peterson translates

The Nash blowup is a natural construction of a partial resolution of singularities for any variety. We identify the Nash blowup of a Schubert variety in G/P for any cominuscule P as a Schubert variety in G/Q for some larger Q. One motivation for studying the Nash blowup comes from the Peterson translate, a tool described by Carrell and Kuttler for identifying the singular locus of an arbitrary Schubert variety, and we show that in this case the possible Peterson translates correspond to the T-fixed points of the Nash blowup. For Grassmannians, the Nash blowup is the fibered product of the usual and dual Kempf-Laksov resolutions, a characterization that we conjecture extends to the covexillary case.

This is joint work with Ed Richmond (Oklahoma State) and William Slofstra (Waterloo).

Alexander Yong: Tableaux, polynomials and Schubert calculus

I will recall the role Young tableaux and Schur polynomials play in textbook Schubert calculus. By analogy, I will then describe a model for equivariant Schubert calculus of Grassmannians in terms of edge-labelled Young tableaux and factorial Schur polynomials, developed in collaboration with Hugh Thomas (UQAM).

One extension of these ideas, with Oliver Pechenik (Michigan), gave the first combinatorial rule for equivariant K-theory of Grassmannians. Another used edge-labelled tableaux to help extend the results of Alexander Klyachko and of Allen Knutson-Terence Tao concerning the Horn conjecture on eigenvalues of Hermitian matrices; this was joint work with David Anderson (Ohio State U.) and Ed Richmond (Oklahoma State U). Building on this, we recently presented a polynomial time algorithm to decide the vanishing of the structure constants (with Anshul Adve (UCLA) and Colleen Robichaux (UIUC)). Finally, I will conclude by reporting on some related conjectures that appear in the new doctoral thesis of Cara Monical (UIUC).