# Algebra and Combinatorics Seminar

## Fall 2023

Date: | September 15, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Laura Matusevich, TAMU |

Title: | Combinatorics of Gorenstein affine semigroup rings |

Abstract: | Affine semigroup rings are algebras that are generated by finitely many monomials. They are very suitable for combinatorial treatment, so people in commutative algebra like to translate algebraic properties into combinatorial terms (and vice versa) if possible. In this talk, I will describe the combinatorial mechanics of the Gorenstein property for affine semigroup rings. I will give a definition of "Gorenstein" in the talk that is useful for computations, but is a bit technical. As for an intuitive definition, let me just say that assuming your ring is Gorenstein has a habit of making theorems work... This is joint work with Byeongsu Yu. |

Date: | September 29, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Youngho Yoo, TAMU |

Title: | Path odd-covers of graphs |

Abstract: | We study the minimum number of paths needed to express the edge set of a given graph as the symmetric difference of the edge sets of the paths. This problem sits in between Gallai's path decomposition problem and the linear arboricity problem. It is also motivated by the study of the diameter of partition polytopes, and we adapt some techniques therein to prove bounds on the path odd-cover number of graphs. Joint work with Steffen Borgwardt, Calum Buchanan, Eric Culver, Bryce Frederickson, and Puck Rombach. |

Date: | October 13, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Jianping Pan, North Carolina State University |

Title: | Polynomials from Schubert Calculus via Diagrams |

Abstract: | Polynomials are powerful tools in many fields, for example, representation theory, geometry, and topology. Understanding the combinatorics of the polynomials may reveal important information in these fields. This talk will focus on four polynomials from Schubert calculus: Schubert, key, Grothendieck, and Lascoux polynomials. I will discuss diagrams related to these polynomials, including Kohnert diagrams, snow diagrams, and rook diagrams. This is joint work with Tianyi Yu. |

Date: | October 27, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Chun-Hung Liu, TAMU |

Title: | Assouad-Nagata dimension of minor-closed metrics |

Abstract: | Assouad-Nagata dimension addresses both large-scale and small-scale behaviors of metric spaces and is a refinement of Gromov’s asymptotic dimension. A metric space is a minor-closed metric if it is defined by the distance function on the vertices of an edge-weighted graph that satisfies a fixed graph property preserved under vertex-deletion, edge-deletion, and edge-contraction. In this talk, we determine the Assouad-Nagata dimension of every minor-closed metric. It is a common generalization of known results about the asymptotic dimension of H-minor free unweighted graphs, about the Assouad-Nagata dimension of complete Riemannian surfaces with finite Euler genus, and about their corollaries on weak diameter coloring of minor-closed families of graphs and asymptotic dimension of minor-excluded groups. |

Date: | November 17, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Shixuan Zhang, TAMU (ISEN) |

Title: | Certification of sums of squares via low-rank optimization |

Abstract: | To certify a sum of k squares on a real projective variety, one can minimize the distance of the sum of squares of k linear forms from it in the space of quadrics. When k is smaller than the dimension of linear forms, the certification problem can be applied in low-rank semidefinite relaxation of polynomial optimization, similar to the Burer-Monteiro method. We discuss the existence of spurious local minima in this nonconvex certification problems, and show that in some interesting cases, there is no spurious local minima, or any spurious local minimum would lie on the boundary of the sum-of-square cone. These characterizations could potentially lead to efficient algorithms for polynomial and combinatorial optimization. |

Date: | December 1, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Catherine Yan, TAMU |

Title: | Combinatorial Identities for Vacillating Tableaux |

Abstract: | Vacillating tableaux are sequences of integer partitions that satisfy specific conditions. The concept of vacillating tableaux stems from the representation theory of the partition algebra and the combinatorial theory of crossings and nestings of matchings and set partitions. In this talk we discuss the enumeration of vacillating tableaux and present multiple combinatorial identities and integer sequences relating to the number of vacillating tableaux and limiting vacillating tableaux. |