Geometry, Combinatorics, and Integrable SystemsAutumn 2019Time: Thursdays 3-4pmLocation: MA 317 |
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| August 29
Thurs, 3pm | Sjuvon Chung
(OSU) |
A formula for quantum K-theory of projective space |
| September 26
Thurs, 3pm | Mohamed Omar
(Harvey Mudd) |
Convex Intersection via Algebra |
| October 3
Thurs, 3pm | Leonardo Mihalcea
(Virginia Tech) |
Cotangent Schubert Calculus |
| November 14
Thurs, 3pm | Federico Castillo
(Kansas) |
Todd Class of Permutohedral Variety |
| November 21
Thurs, 3pm | Vasu Tewari
(Penn) |
Divided symmetrization, quasisymmetric functions and the Peterson variety |
(Chung): The quantum K-ring of the flag variety G/P has become better understood in recent years. When G/P is complex projective space, a considerable amount can be said about this ring. In particular, there is a recursive formula which describes the ring structure relative to a Schubert basis. We’ll take a look at this formula and explore its consequences.
(Omar): What intersection patterns can arise from an arrangement of convex sets? This question has received considerable attention as of late, prompted by recent discoveries in the sciences. In this talk, we show how algebra can help us in answering this question, or at the very least unearth obstructions.
(Mihalcea): A natural question with roots in representation theory and microlocal analysis is to find good analogues of Schubert classes in the intersection rings of the cotangent bundle of a flag manifold. One answer is given in terms of the characteristic classes of singular subvarieties in the flag manifold, such as the Chern-Schwartz-MacPherson classes (in cohomology) or motivic Chern classes (in K theory). I will give the definition of these classes, and I will discuss some (known or conjectural) properties of the transition matrix to the ordinary Schubert basis. Joint work with P. Aluffi, J. Schurmann, C. Su.
(Castillo): Berline and Vergne described a precise relation between the number of integer points of a polytope and the volumes of its faces. This relation can be seen as a higher dimensional analogue of Pick's theorem. We study the specific case of the permutohedron via the connection with toric varieties. This is joint work with Fu Liu.
(Tewari): The procedure of divided symmetrization was introduced by A. Postnikov in the context of computing volume polynomials of various classes of permutahedra. This procedure takes a multivariate polynomial as input and outputs a scalar, which in many cases is a combinatorially interesting quantity. In this talk, I will describe how performing divided symmetrization is equivalent to reducing multivariate polynomials modulo the ideal generated by the homogeneous quasi-symmetric polynomials of positive degree in a fixed number of variables. I will subsequently discuss how divided symmetrization can be used to understand the Schubert expansion of the Anderson-Tymoczko class of the Peterson variety. Along the way, we will encounter familiar combinatorial objects such as flagged tableaux, reduced pipe dreams, P-partitions and various Catalan objects. This is joint work with Philippe Nadeau at Institut Camille Jordan.