Mathematics Department - Algebra Seminar - Spring 2017

Algebra Seminar - Spring 2017


Charles Weibel



Past Talks

Wednesday, March 22nd

Ilya Kapovich, Rutgers University

"Dynamics and polynomial invariants for free-by-cyclic groups"

Time: 2:00 PM
Location: Hill 705
Abstract: We develop a counterpart of the Thurston-Fried-McMullen "fibered face" theory in the setting of free-by-cyclic groups, that is, mapping tori groups of automorphisms of finite rank free groups. We obtain information about the BNS invariant of such groups, and construct a version of McMullen's "Teichmuller polynomial" in the free-by-cyclic context.

The talk is based on joint work with Chris Leininger and Spencer Dowdall.

Wednesday, March 8th

Oliver Pechenik, Rutgers University

"Decompositions of Grothendieck polynomials"

Time: 2:00 PM
Location: Hill 705
Abstract: Finding a combinatorial rule for the Schubert structure constants in the K-theory of flag varieties is a long-standing problem. The Grothendieck polynomials of Lascoux and Schützenberger (1982) serve as polynomial representatives for K-theoretic Schubert classes, but no positive rule for their multiplication is known outside of the Grassmannian case. We contribute a new basis for polynomials, give a positive combinatorial formula for the expansion of Grothendieck polynomials in these "glide polynomials", and provide a positive combinatorial Littlewood-Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. A specialization of the glide basis recovers the fundamental slide polynomials of Assaf and Searles (2016), which play an analogous role with respect to the Chow ring of flag varieties. Additionally, the stable limits of another specialization of glide polynomials are Lam and Pylyavskyy's (2007) basis of multi-fundamental quasisymmetric functions, K-theoretic analogues of I. Gessel's (1984) fundamental quasisymmetric functions. Those glide polynomials that are themselves quasisymmetric are truncations of multi-fundamental quasisymmetric functions and form a basis of quasisymmetric polynomials.

(Joint work with D. Searles).

Wednesday, March 1st

Charles Weibel, Rutgers University

"The Witt group of surfaces and 3-folds"

Time: 2:00 PM
Location: Hill 705
Abstract: If V is an algebraic variety, the Witt group is formed from vector bundles equipped with a nondegenerate symmetric bilinear form. When it has dimension $<4$, it embeds into the more classical Witt group of the function field (Witt 1934). When V is defined over the reals, versions of the discriminant and Hasse invariant enable us to determine W(V).

Wednesday, February 22nd

Ryan Shifler, Virginia Tech

"Equivariant Quantum Cohomology of the Odd Symplectic Grassmannian "

Time: 2:00 PM
Location: Hill 705
Abstract: The odd symplectic Grassmannian IG:=IG(k, 2n+1) parametrizes k dimensional subspaces of C2n+1 which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on IG with two orbits, and IG is itself a smooth Schubert variety in the submaximal isotropic Grassmannian IG(k, 2n+2). We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of IG, i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case k=2, and it gives an algorithm to calculate any quantum multiplication in the equivariant quantum cohomology ring.

The current work is joint with L. Mihalcea.

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