## The UC Berkeley Combinatorics SeminarFall 2024 - Wednesdays 11:10am - 12:00pm, Evans 891Introductory pre-talk for graduate students (open to all) 10:40am - 11:05am, Evans 891 Zoom Meeting ID: 953 1397 4237, the password is the name of our favorite combinatorial sequence Organizers: Christian Gaetz, Nicolle González, Mitsuki Hanada, and John Lentfer If you would like to be added to the seminar mailing list, contact Nicolle González. If you would like to view recordings of some of the talks (in a Google drive), contact Mitsuki Hanada for access. |
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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 amplituhedronThe 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 polynomialsKazhdan-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 fibersWe 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 VarietiesGalashin 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 |