# Algebra and Combinatorics Colloquium

## Spring 2023

Date: | January 27, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Ayah Almousa, University of Minnesota |

Title: | GL-equivariant resolutions over Veronese subrings |

Abstract: | We construct explicit GL-equivariant minimal free resolutions of certain (truncations of) modules of relative invariants over Veronese subrings in arbitrary characteristic. The free modules in the resolution correspond to certain skew Schur modules associated to "ribbon" or "skew-hook" diagrams, and the differentials at each step are surprisingly uniform. We then utilize the uniformity of these resolutions to explicitly compute information about tensor products, Hom, and Tor between these modules and show that they also have rather simple descriptions in terms of ribbon skew-Schur modules. I will emphasize the hidden role of symmetric function theory in detecting the answer to this question and guiding our intuition to build new tools to prove our results in arbitrary characteristic. This is joint work with Mike Perlman, Sasha Pevzner, Vic Reiner, and Keller VandeBogert. |

Date: | February 10, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Laura Matusevich, Texas A&M University |

Title: | Differential operators and algebra retracts |

Abstract: | I will define rings of differential operators, and discuss the few explicitly known examples. Using algebra retracts (subrings that are also quotients) I will show how to compute the rings of differential operators on toric face rings. All the terminology will be introduced in the talk, no background on differential operators or toric face rings is necessary. This is joint work with Berkesch, Chan, Klein, Page and Vassilev. |

Date: | February 17, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Mahrud Sayrafi, University of Minnesota |

Title: | Bounding the Multigraded Regularity of Powers of Ideals |

Abstract: | Building on a result of Swanson, Cutkosky--Herzog--Trung and Kodiyalam described the surprisingly predictable asymptotic behavior of Castelnuovo--Mumford regularity for powers of ideals on a projective space P^n: given an ideal I, there exist integers d and e such that for large enough n the regularity of I^n is exactly dn+e. Through a medley of examples we will see why asking the same question about an ideal I in the total coordinate ring S of a smooth projective toric variety X is interesting. After that I will summarize the ideas and methods we used to bound the region reg(I^n) as a subset of Pic(X) by proving that it contains a translate of reg(S) and is contained in a translate of Nef(X), with each bound translating by a fixed vector as n increases. Along the way will see some surprising behavior for multigraded regularity of modules. This is joint work with Juliette Bruce and Lauren Cranton Heller. |

Date: | March 3, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Chun-Hung Liu, Texas A&M University |

Title: | Proper conflict-free coloring and maximum degree |

Abstract: | A conflict-free coloring of a hypergraph is a coloring on the vertices such that for every hyperedge, some color appears exactly once on the vertices of this edge. This notion is motivated by a frequency assignment problem of cellular networks and is a generalization of a number of variants of coloring notions of graphs. We prove a general upper bound for the number of colors for (proper) conflict-free coloring involving the maximum degree and rank. It provides improvements about linear coloring and star coloring of graphs with bounded maximum degree and addresses a conjecture of Caro, Petrusevski and Skrekovski on proper conflict-free coloring of graphs. They conjectured that for every d \geq 3, every connected graph with maximum degree at most d has a proper conflict-free coloring with d+1 colors. We prove that (1.655083+o(1))d colors suffice. We also prove that the fractional coloring version of this conjecture is asymptotically true. This is joint work with Daniel Cranston. |

Date: | March 24, 2023 |

Time: | 3:00pm |

Location: | BLOC 506A |

Speaker: | Avery St. Dizier , UIUC |

Title: | A Polytopal View of Schubert Polynomials |

Abstract: | Schubert polynomials are a family of multivariable polynomials whose product can be used to solve problems in enumerative geometry. Despite their many known combinatorial formulas, there remain mysteries surrounding these polynomials. I will describe Schubert (and the special case of Schur) polynomials with a focus on polytopes. From this perspective, I will address questions such as vanishing of Schubert coefficients, relative size of coefficients, and interesting properties of their support. Time permitting, I'll talk about my current work on generalizing the Gelfand–Tsetlin polytope, and its connections with representation theory and Bott–Samelson varieties. |

Date: | March 31, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Hongdi Huang, Rice University |

Title: | |

Date: | April 5, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Yuri Bahturin, Memorial University of Newfoundland |

Title: | From groups to algebras and back |

Abstract: | We analyze and extend the classical Malcev correspondence between the divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. The new setting is arbitrary finite-dimensional nilpotent algebras. We prove the implicit function theorem for the polynomial functions on such algebras. This allows us to produce various correspondences between these algebras and (quasi)groups, each built on the same underlying set. Applications are provided for the commensurators of nilpotent groups, filiform Lie algebras and partially ordered algebras. (joint work with Alexander Olshanskii) |

Date: | April 14, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Lauren Snider, Texas A&M University |

Title: | |

Date: | April 21, 2023 |

Time: | 12:40pm |

Location: | BLOC 302 |

Speaker: | Jesus de Loera, University of California, Davis |

Title: | Who discovered Ramsey theory? An algebraic re-examination of Ramsey theory. |

Abstract: | It is indisputable Ramsey numbers are among the most mysterious and fascinating in Combinatorics. My talk focuses on Arithmetic Ramsey numbers and Diophantine problems, I discuss Rado numbers. These numbers are actually older than the usual graph theory version. For a positive integer k and linear equation E the Rado number R_k(E) is the smallest integer number n such that every k-coloring of [n] it contains a monochromatic solution to the equation E. A very famous example are Schur numbers, which are the Rado numbers for the equation E (X+Y=Z). I will discuss computation, bounds, and verification of Rado numbers and the fascinating history of Ramsey theory connected with names like Hilbert, Schur, van der Waerden, appearing along the way. I will not assume you know anything from the audience but I hope I will show, not just history, but also some new algebraic results. Our work combines discrete geometry, logic, algebraic geometry, an combinatorial number theory to investigate the behavior of Rado numbers. First, we computed many new exact values for Rado numbers using SAT solvers. In particular, we give a method for computing infinite families of Rado numbers, solving a few open questions. Regarding complexity and verification: Suppose someone suggests to you the value of R_k(E) . How can you certify that this value is correct and not a lie? We encode the problem as a system of polynomial equations and show that the degrees of Nullstellensatz certificates are actually bounded above by another Ramsey-number arising in a two-player game. The extremal k-colorings are in fact the solutions of this system which says that any proof that the proposed value of R_k(E) is not correct may require a doubly exponential certificate. At the heart is how combinatorial algebraic geometry relates to Ramsey numbers. This is joint work with Jack Wesley. |

Date: | April 28, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Patricia Klein, Texas A&M University |

Title: | |