The Applied Algebra Seminar

A
Monday afternoon research seminar

The seminar is currently organized by Laura Colmenarejo and Nantel Bergeron.

During 2016-17, the seminar takes place from 15:00-16:00 in Ross Building room N638. If you come by bus, the route 196A, 196B takes you to campus from Downsview subway station. If you come by car, you can find the available parking lots here.

The seminar has been running since 1997. The topics of talks have typically been any mixture of algebra with any other field: combinatorics, geometry, topology, physics, etc. Further down this page you will find links to the seminar webpages for previous years. The audience usually consists of 6–12 people, including several graduate students and post-docs. For this reason, speakers are encouraged to devote a portion of their talk to the suggestion of open problems and the directions for research in their area. If you are interested in speaking at the seminar, contact Laura Colmenarejo or Nantel Bergeron.You may also be interested in the Algebraic Combinatorics Seminar at the Fields Institute.

Dates are listed in reverse-chronological order. Unless otherwise indicated, all talks will take place on Monday from 15:00-16:00 in N638 Ross Building (York University).

Date Speaker Title (click titles for abstract) 5 Dec. 2016 Alex Fink

(Queen Mary U. of London)28 Nov. 2016 Li Ying

(Texas A&W U.)21 Nov. 2016 Farid Aliniaeifard

(York U.)14 Nov. 2016 Alex Woo

(U. Of Idaho)07 Nov. 2016 Zach Teitler

(Boise State U.)Waring ranks of homogeneous forms The Waring rank of a homogeneous form is the number of terms needed to write it as a sum of powers of linear forms. It is related to secant varieties, provides a measure of the complexity of polynomials, and has applications in statistics, sciences, and engineering. I will discuss three topics related to Waring rank. (1) Waring ranks of general forms have been known for some time, but it is also of interest to determine Waring rank of particular forms such as the generic determinant and permanent. I will describe some recent results obtained via algebraic and geometric lower bounds for Waring rank; this is joint work with Jaroslaw Buczynski and with Harm Derksen. (2) A variation of a conjecture of Strassen asserts that the Waring rank of the sum of two forms in independent variables is the sum of the ranks of the summands. I will describe an elementary sufficient condition for a strong version of Strassen's conjecture. (3) It is an open question to determine the maximum Waring rank occurring among forms of a given degree, in a given number of variables. I will describe an upper bound; this is joint work with Gregoriy Blekherman.31 Oct. 2016 Laura Colmenarejo

(York U.)Stability in the combinatorics of symmetric functions In this talk, I will introduce the h-plethysm coefficients, related to the usual plethysm coefficients. I will present a combinatorial description of them. This description will be the tool to give a combinatorial proof of some well-known stability properties of the plethysm coefficients. I will also introduce some families of reduced kronecker coefficients. For these families, I will give their generating function as well as combinatorial descriptions in terms of plane partitions and quasipolynomials. These results are included in my thesis.24 Oct. 2016 Dave Anderson

(Ohio State U.)Diagrams and essential sets for signed permutations The essential set of a permutation, defined via its Rothe diagram by Fulton in 1992, gives a minimal list of rank conditions cutting out the corresponding Schubert variety in the flag manifold. I will describe an analogous notion for signed permutations, giving minimal conditions for Schubert varieties in flag varieties for other classical groups. This is related to a poset-theoretic construction of Reiner, Woo, and Yong, and thus gives a diagrammatic method for computing the latter.17 Oct. 2016 Adeyemo Praise

(Fields Institute)The Three Presentations of T- Equivariant K- Theory of G/B Let G be a simple, simply connected, complex Lie group with a Borel subgroup B. G/B is the flag manifold associated to G. It is acted upon by maximal torus T contained in B by multiplication. We study the T- equivariant K-theory of G/B, the Grothendieck group of coherent sheaves on G/B. In this talk, I will give the three presentations of T-equivariant K-theory of this homogeneous space- Schubert presentation, Borel presentation and GKM ring. I will describe partially how to convert an element in one presentation to another. It represents the ongoing work with Shizuo Kaji.3 Oct. 2016 Georg Merz

(Math. Institut)Newton Okounkov bodies - A bridge between algebraic and convex geometry. Inspired by work of A. Okounkov, Kaveh-Khovanskii and Lazarsfeld-Mustata independently introduced the notion of a Newton-Okounkov body which associates a convex body to a line bundle on a projective variety. This construction is a generalization of the toric case, and it allows to transport algebra geometric questions into such of convex geometry, and vice versa. In this talk I will give an outline of the construction, present some known results and give some generalizations of these in the setting of graded linear series.26 Sep. 2016 John Maxwell Campbell

(York U.)The evaluation of immaculate functions in terms of the ribbon basis There is an elegant combinatorial formula for evaluating elements of the ribbon basis of NSym in terms of immaculate functions, but there is no known cancellation-free formula for expanding elements of the immaculate basis in terms of ribbon functions in NSym. However, recent research suggests that many different classes of immaculate functions may be evaluated in terms of ribbon functions using cancellation-free Jacobi-Trudi-like formulas. In this talk, we offer a sketch of a proof of a surprising Jacobi-Trudi-like formula for expanding immaculate-rectangles as linear combinations of ribbon functions.12 Sep. 2016 Jake Levinson

(U. of Mich.)(Real) Schubert Calculus from Marked Points on P^1 I will describe a family of Schubert problems on the Grassmannian, defined using tangent flags to points of P^1 (or more generally, a stable curve) in its Veronese embedding. For Schubert problems having a finite set of solutions, the Shapiro-Shapiro Conjecture (later proven by Mukhin-Tarasov-Varchenko) proposed that, when the marked points are all real, the solutions would be as real as possible. More recently, Speyer gave a remarkable description of the real topology of the family in terms of Young tableaux and Schützenberger's jeu de taquin. I will go through this story, then give analogous results on the topology of one-dimensional Schubert problems (where the family consists of curves). In this case the combinatorics involves orbits of so-called "evacuation-shuffling", an algorithm related to tableau promotion and evacuation.

Below you will find links to the seminar webpages for previous years.