The UC Berkeley Combinatorics Seminar

DATE	SPEAKER	TITLE (click to show abstract)
August 28th (No pre-talk)	Lauren Williams, Harvard	Cyclic partial orders, Parke-Taylor identities, and the magic number conjecture for the m=2 amplituhedron The magic number conjecture says that the cardinality of a tiling of the amplituhedron An,k,m is the number of plane partitions which fit inside a k by (n-k-m) by m/2 box. (This is a generalization of the fact that triangulations of even-dimensional cyclic polytopes have the same size.) I'll explain how we prove the magic number conjecture for the m=2 amplituhedron; we also show that all positroid tilings of the hypersimplex have the same cardinality. Along the way, we give volume formulas for Parke-Taylor polytopes in terms of circular extensions of cyclic partial orders, and we prove new variants of the classical Parke-Taylor identities. This is joint work with Matteo Parisi, Melissa Sherman-Bennett, and Ran Tessler.
September 4th	Christian Gaetz, UC Berkeley	Hypercube decompositions and combinatorial invariance for Kazhdan-Lusztig polynomials Kazhdan-Lusztig polynomials are of foundational importance in geometric representation theory. Yet the Combinatorial Invariance Conjecture, due to Lusztig and to Dyer, suggests that they only depend on the combinatorics of Bruhat order. I'll describe joint work with Grant Barkley in which we adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veličković-Williamson to prove this conjecture for Kazhdan-Lusztig R-polynomials in the case of elementary intervals in the symmetric group. This significantly generalizes the main previously known case of the conjecture, that of lower intervals.
September 11th	Joshua Turner, UC Davis	Haiman ideals, link homology, and affine Springer fibers We will discuss a class of ideals in a polynomial ring studied by Mark Haiman in his work on the Hilbert scheme of points, and ask some purely algebraic questions about them. It turns out that these questions are very closely tied to homology of affine Springer fibers, Khovanov-Rozansky homology of links, and to the ORS conjecture. We will discuss which cases are known and unknown, and compute some simple examples.
September 18th	Tonie Scroggin, UC Davis	Splicing Positroid Varieties Galashin and Lam established the connection between the torus equivariant cohomology on $\Pi_{k,n}^\circ$ and Khovanov-Rozansky homology HHH of the torus link $T(k,n-k)$. Additionally, the multiplication of braids $T(k,s)\cdot T(k,t)\to T(k,s+t)$ suggests that there should be a map of open positroid varieties. In this talk, I will describe an explicit isomorphism which decomposes an open subset in the open positroid variety $\Pi_{k,n}^{\circ}$ in the Grassmannian $\mathrm{Gr}(k,n)$ into the product of two open positroid varieties $\Pi_{k,n-a+1}^{\circ}\times \Pi_{k,a+k-1}^{\circ}$. The isomorphism is given by freezing a certain subset of cluster variables in $\Pi_{k,n}^\circ$ and constructing a cluster quasi-homomorphism. This is joint work with Eugene Gorsky.
September 25th
October 2nd	Matthias Beck, SFSU
October 9th
October 16th	John Lentfer, UC Berkeley
October 23rd
October 30th	Michelle Wachs, University of Miami
November 6th	Rebecca Whitman, UC Berkeley
November 13th	Monica Vazirani , UC Davis
November 20th	Sarah Brauner, Brown
November 27th	No seminar - Thanksgiving
December 4th	Mary Claire Simone, UC Davis
December 11th

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