Abstracts

(Corey): Let Gr₀(d,n) denote the open subvariety of the Grassmannian Gr(d,n) consisting of d-1 dimensional subspaces of P^n-1 meeting the toric boundary transversely. We prove that Gr₀(d,n) is schoen in the sense that all of its initial degenerations are smooth. The main technique we will use is to express the initial degenerations of Gr₀(3,7) as a inverse limit of thin Schubert cells. We use this to show that the Chow quotient of Gr(d,n) by the maximal torus H in GL(n) is the log canonical compactification of the moduli space of 7 lines in P² in linear general position.

(Sagnier): We are used to seeing integers with the usual structure of an ordered ring. A.Connes and C.Consani proposed in 2014 to look at them with another structure which is an idempotent semiring with an action of the positive integers by multiplication. With the eyes of algebraic geometry, it is a semiringed topos whose points are linked with Riemann zeta function. The hope in the long term is that this new framework coming from algebraic geometry could help translating ideas of the demonstration of the analog of the Riemann hypothesis for zeta functions associated to smooth projective curves over a finite field to the actual Riemann zeta function. I will explain A.Connes' and C.Consani's point of view in the first part of my talk. However, this point of view heavily relies on the natural order on R compatible with addition and multiplication so one may wonder if, for Gaussian integers, where nothing of this sort exists, one can adapt the ideas and the methods of A.Connes and C.Consani. This is what I have done in my Ph.D. thesis and what I will explain in the second part of my talk.

(Weigandt): A prism tableau is an overlay of semistandard tableaux. In joint work with A. Yong, prism tableaux were used to provide a formula for Schubert polynomials. This expression directly generalizes the tableau rule for Schur polynomials. We will discuss fillings of more general prism shapes. The resulting polynomials are multiplicity free sums of Schubert polynomials. The terms in this sum are determined by an associated alternating sign matrix. We use this connection to give a prism formula for the multidegrees of alternating sign matrix varieties.

(Fraser): I will formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. This map sends a network to a linear combination of SLr tensor invariants known as webs, and is defined using the r-fold dimer model on the network. When r equals 1, our map is Postnikov's boundary measurement used to coordinatize positroid strata. The main result is that the higher rank map factors through Postnikov's map. As a corollary, we deduce a complete set of diagrammatic relations between SLr webs, reproving a result of Cautis-Kamnitzer-Morrison in the classical (q=1) setting. We establish compatibility between our map and restriction to positroid strata. This is joint with Thomas Lam and Ian Le.

(Precup):

(Kubjas):

Past Seminars

Autumn 2017

Spring 2017