The UC Berkeley combinatorics seminar |
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DATE |
SPEAKER |
TITLE (click to show abstract) |

January 24th | Andrew Gitlin, UC Berkeley |
## Vertex models for LLT polynomials and super ribbon functionsIntegrable vertex models have recently been used to study various families of symmetric and nonsymmetric polynomials. In this talk, we will define Yang-Baxter integrable vertex models, from which we will construct a class of partition functions that equal the LLT polynomials of Lascoux, Leclerc, and Thibon and a class of partition functions that equal the super ribbon functions of Lam. Using the vertex model formalism, we can prove many properties of these polynomials, including (super)symmetry and Cauchy identities. Finally, we will discuss applications of our vertex models to domino tilings of the Aztec diamond. This is based on joint work with Sylvie Corteel, David Keating, and Jeremy Meza. |

January 31st | Christopher Ryba, UC Berkeley |
## Stable Centres of Various Combinatorial AlgebrasThe centres of the group algebras of the symmetric groups S_n turn out to have a rich combinatorial structure. They come with a basis indexed by conjugacy classes (equivalently, partitions of n), and in an appropriate sense, the multiplication rules are "polynomial in n". Framing this stability property in the language of symmetric functions provides us with a powerful tool to both understand and apply this theory. After explaining this classical example, we will report on progress towards replicating these results for other combinatorial families of algebras such as wreath products, Hecke algebras, and finite general linear groups. This is partly joint work with Arun Kannan. |

February 7th | Chris Eur, Harvard | ## Tautological classes of matroidsWe introduce a new way of geometrically studying matroids that unifies, recovers, and extends several recent developments on the interaction between matroid theory and algebraic geometry. A key step that we establish is a formula relating Ehrhart polynomials of generalized permutohedra to their volumes in a way that is different from previous methods. Joint work with Andrew Berget, Hunter Spink, and Dennis Tseng. |

February 14th | Eleonore Bach, FU Berlin | ## A concrete construction of the cographic hyperplane arrangementGeometrically carrying a trove of information about the underlying simple graph, the graphic hyperplane arrangement $H_G$ yields an interesting mathematical object to study a simple graph $G$. For example, one proves that the normal vectors of $H_G$ are linearly independent if and only if they induce forests on $G$ and the regions of $H_G$ are in a one-to-one correspondence to the acyclic orientations of $G$. With the graphic hyperplane arrangement we can associate the graphic matroid whose bases are spanning forests of $G$. What information do we obtain if we apply duality, i.e., if we start with the dual of the graphic matroid, called the cographic matroid whose bases are complements of spanning forests of $G$? In this talk we are going to start answering the above question by constructing the normal vectors of the cographic hyperplane arrangement associated with the cographic matroid for simple, connected and bridgeless graphs. |

February 21st | Academic Holiday - No Seminar | |

February 28th | Galen Dorpalen-Barry, Ruhr-Universität Bochum | ## Cones of Hyperplane Arrangements and the Varchenko-Gel’fand ringIn 1943, J.L. Woodbridge of Philadelphia submitted the following problem to American Mathematical Monthly: “Show that n cuts can divide a cheese into as many as (n+1)(n^2 - n + 6)/6 pieces.” This question, and its solution, are deeply connected to the study of collections of lines in \mathbb{R}^2, planes in \mathbb{R}^3, and more generally hyperplanes in \mathbb{R}^n. We will explore the solution and a more general version: given n hyperplanes in a real, d-dimensional vector space, how can we figure out the number of chambers of an arrangement of hyperplanes, without necessarily being able to see and count them?There are many wonderful solutions to this question. We present one provided by the Varchenko-Gel’fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise addition and multiplication. Varchenko and Gel’fand gave a simple presentation for this ring, which can be computed using simple facts about linear algebra, but which takes on deeper meaning when one knows a bit more about oriented matroids. We use tools from Gröbner basis theory to reframe and generalize their results to cones of hyperplane arrangements, which are intersections of halfpsaces defined by some of the hyperplanes of the arrangement. Time permitting, we will discuss an interesting consequence of this approach regarding Koszulity and supersolvable arrangements. |

March 7th | Sophie Rehberg, FU Berlin |
## Rational Ehrhart TheoryThe Ehrhart quasipolynomial of a rational polytope P encodes fundamental arithmetic data of P, namely, the number of integer lattice points in positive integral dilates of P. Ehrhart quasipolynomials were introduced in the 1960s, satisfy several fundamental structural results and have applications in many areas of mathematics and beyond. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by Linke (2011), Baldoni-Berline-Koeppe-Vergne (2013), and Stapledon (2017). We introduce a generating-function ansatz for rational Ehrhart quasipolynomials, which unifies several known results in classical and rational Ehrhart theory. In particular, we define y-rational Gorenstein polytopes, which extend the classical notion to the rational setting. This is joint work with Matthias Beck and Sophia Elia. |

March 14th | Bennet Goeckner, University of Washington |
## Type cones and products of simplicesA polytope $P$ is the convex hull of finitely many points in Euclidean space. For polytopes $P$ and $Q$, we say that $Q$ is a \emph{Minkowski summand} of $P$ if there exists some polytope $R$ such that $Q+R=P$. The \emph{type cone} of $P$ encodes all of the (weak) Minkowski summands of $P$. In general, combinatorially isomorphic polytopes can have different type cones. We will first consider type cones of polygons, and then show that if $P$ is combinatorially isomorphic to a product of simplices, then the type cone is simplicial. This is joint work with Federico Castillo, Joseph Doolittle, Michael Ross, and Li Ying. No specialized knowledge of polytopes will be assumed. |

March 21st | Sping Break - No Seminar | |

March 28th | Stacey Law, Cambridge |
## On plethysms and Sylow branching coefficientsWe give a recursive formula for plethysm coefficients involved in Foulkes’ Conjecture, a long-standing open problem lying at the intersection of the theory of symmetric functions, the representation theory of symmetric groups and algebraic combinatorics. From this we deduce a stability result and resolve two conjectures of de Boeck concerning plethysms. We also obtain new results on Sylow branching coefficients for symmetric groups for the prime 2, the divisibility properties of which were recently used to characterise whether a Sylow subgroup of a finite group is normal and confirm a prediction of Malle and Navarro. This is joint work with Y. Okitani. |

April 4th | Serkan Hoşten, San Francisco State University |
## Symmetry adapted Gram spectrahedraI will report on the geometric and combinatorial structure of symmetry adapted PSD cones and symmetry adapted Gram spectrahedra of symmetric polynomials. I will focus on symmetry adapted Gram spectrahedra of symmetric binary forms, quadrics, ternary quartics and sextics. In particular, I will present a characterization of extreme points of these spectrahedra for symmetric binary forms that are of rank two. This is complementary to a classical result in the non-symmetric case. I will also report what we know about the facial structure and combinatorics of the same spectrahedra. |

April 11th | Yibo Gao, Massachusetts Institute of Technology |
## Balanced Shifted TableauxWe introduce balanced shifted tableaux, as an analogue of balanced tableaux of Edelman and Greene, from the perspective of root systems of type B and C. We show that they are equinumerous to standard Young tableaux of the corresponding shifted shape by presenting an explicit bijection. This is joint work with J. Gao and S. Gao. |

April 18th | Oleg Karpenkov, University of Liverpool |
## Geometry of continued fractions.In this talk we introduce a geometrical model of continued fractions and discuss its appearance in rather distance research areas:-- values of quadratic forms (Perron Identity for Markov spectrum) -- the 2nd Kepler law on planetary motion -- Global relation on singularities of toric varieties |

April 25th | Julianne Vega, Kennesaw State University |
## Matching complexes of (general) polygonal line tilingsIn this talk, we introduce matching complexes. The talk will include a brief history of what is known about matching complexes, including a discussion of their relation to independence complexes. We will end with results that further our understanding of matching complexes in relation to polygonal line tilings. Joint work with Marge Bayer and Marija Jelić Milutinović. |