Algebra and Combinatorics Seminar
Fall 2018
Date: | August 31, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Catherine, Sarah, Yue |
Title: | Organizing meeting |
Date: | September 7, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Sarah Witherspoon, Texas A&M University |
Title: | Lie algebra structure on derivations and Hochschild cohomology |
Abstract: | Derivations are linear operators on rings that obey the Leibniz rule, for example, differentiation on rings of functions. The space of all derivations on a ring is a Lie algebra. Generalizing from linear to multilinear operators naturally leads to the subject of Hochschild cohomology, a source of important algebraic invariants for rings. It arises in many settings, for example, in representation theory, in noncommutative geometry, and in algebraic deformation theory. In this talk, we will give a brief introduction to Hochschild cohomology and mention some of its applications and some recent work on its structure as a graded Lie algebra. |
Date: | September 14, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Jurij Volcic, Texas A&M University |
Title: | Multipartite rational functions: the universal skew field of fractions of a tensor product of free algebras |
Abstract: | A commutative ring embeds into a field if and only if it has no zero divisors; moreover, in this case it admits a unique field of fractions. On the other hand, the problem of noncommutative localization and embeddings into skew fields (that is, division rings) is much more complex. For example, there exist rings without zero divisors that do not admit embeddings into skew fields, and rings with several non-isomorphic "skew fields of fractions". This led Paul Moritz Cohn (1924-2006) to introduce the notion of the universal skew field of fractions to the general theory of skew fields in the 70's. However, almost all known examples of rings admitting universal skew fields of fractions belong to a relatively narrow family of Sylvester domains. One of the exceptions is the tensor product of two free algebras. After a decent introduction, we will look at the skew field of multipartite rational functions, whose construction via matrix evaluations of formal rational expressions is inspired by methods in free analysis. This skew field turns out to be the universal skew field of fractions of a tensor product of free algebras (for arbitrary finite number of factors). |
Date: | September 21, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Chun-Hung Liu, Texas A&M University |
Title: | Clustered coloring on old graph coloring conjectures |
Abstract: | The famous Four Color Theorem states that every graph that can be drawn in the plane without edge-crossing is properly 4-colorable, which means that one can color its vertices with 4 colors such that every pair of adjacent vertices receive different colors. It is equivalent to say that every graph that does not contain a subgraph contractible to K_5 or K_{3,3} is properly 4-colorable. Hadwiger in 1943 proposed a far generalization of the Four Color Theorem: every graph that does not contain a subgraph contractible to K_{t+1} is properly t-colorable. Hajos, and Gerards and Seymour, respectively, proposed two strengthening of Hadwiger's conjecture, where only special kinds of edges are allowed to be contracted. More precisely, these three conjectures state that every graph that does no contain K_{t+1} as a minor (topological minor, or odd minor, respectively) is properly t-colorable. These three conjectures are either open or false, except for some very small t. One weakening of these three conjectures is to color the vertices such that every monochromatic component has bounded size, which is called clustered coloring. In this talk we will show joint work with David Wood about a series of tight results about clustered coloring on graphs with no subgraph isomorphic to a fixed complete bipartite graph. These results have a number of applications, including nearly optimal or first linear bound for the number of colors on the clustered coloring version of the previous three conjectures, as well as results on graphs embeddable in a surface of bounded genus where edge-crossings are allowed. No background about graph theory is required for this talk. |
Date: | September 28, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Laura Matusevich, Texas A&M University |
Title: | Standard pairs |
Abstract: | Standard pairs are a useful gadget for studying monomial ideals in polynomial rings. I will recall the definition, mention some known applications, and introduce a new generalization to the context of monomial ideals in semigroup rings. |
Date: | October 5, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Li Ying, Texas A&M University |
Title: | Generalized stability of Heisenberg coefficients |
Abstract: | Stembridge introduced a new concept, Kronecker stable triple, which generalized the classical Murnaghan's stability result of Kronecker coefficients. Sam and Snowden proved a conjecture of Stembridge concerning when a Kronecker triple is stable, and they also showed an analogous result for Littlewood--Richardson coefficients. Heisenberg coefficients are Schur structure constants of the Heisenberg product which generalize Littlewood--Richardson coefficients and Kronecker coefficients. In this talk, I will recall the definition and explain some known results. I will show that any stable triple for Kronecker coefficients or Littlewood--Richardson coefficients also stabilizes Heisenberg coefficients, and I follow Vallejo's idea of using matrix additivity to generate Heisenberg stable triples. |
Date: | October 12, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Andrew Maurer, University of Georgia |
Title: | Relative Cohomology of Classical Lie Superalgebras |
Abstract: | Cohomology is a powerful technique in representation theory. In this talk, I will define Lie superalgebras and their relative cohomology theory. We will see two classical results on geometric properties of relative cohomology, and I will present original results which relate the two aforementioned results. |
Date: | October 19, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Frank Sottile, Texas A&M University |
Title: | A geometric proof of an equivariant Pieri rule for flag manifolds |
Abstract: | The Pieri formulas describes the multiplication of Schubert classes in the equivariant cohomology of any manifold of partial flags. I will describe how to use geometric and combinatorial arguments that provide a short proof of an equivariant Pieri rule in the classical flag manifold. This is joint work with Changzheng Li, Vijay Ravikumar, and Mingzhi Yang. |
Date: | October 26, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Anastasia Chavez, University of California, Davis |
Title: | Dual Equivalence Graphs and CAT(0) Combinatorics |
Abstract: | In this talk we will explore the combinatorial structure of dual equivalence graphs G_lambda. The vertices are Standard Young tableaux of fixed shape lambda that allows us to further understand the combinatorial structure of G_lambda, and the edges are given by dual Knuth equivalences. The graph G_lambda is the 1-skeleton of a cubical complex C_lambda. One can ask whether the cubical complex is CAT(0); this is a desirable metric property that allows us to describe the combinatorial structure of G_lambda very explicitly. We will discuss the CAT(0) characterization of Ardila--Owen--Sullivant. It is constructive and provides an algorithm for determining when a cubical complex is CAT(0). Using their characterization, we prove that C_lambda is CAT(0) if and only if lambda is a hook. This is joint work with John Guo. |
Date: | November 2, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Emily Witt, University of Kansas |
Title: | Connectedness and local cohomology |
Abstract: | Local cohomology modules are algebraic objects that encode important properties; e.g., the dimension of a ring, the depth of a ring on an ideal, and the number of equations needed to define a variety. The second vanishing theorem of local cohomology characterizes the connectedness of the punctured spectrum. In this talk, we seek to understand how local cohomology determines more refined connectedness properties, and introduce a "third vanishing theorem." This is joint work with Luis Núñez-Betancourt and Sandra Spiroff. |
Date: | November 9, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Westin King, Texas A&M University |
Title: | Parking on Directed Graphs |
Abstract: | Parking functions were first defined in the 1960s in order to study collision resolution on hashing tables. Since then, they have appeared nearly ubiquitously throughout combinatorics and have several generalizations. I will present a new generalization which can be thought of as drivers with preferred parking spaces searching for an available parking spot (vertices) along a series of one-way streets (directed edges). I will discuss parking functions on several families of directed graphs, their surprisingly nice enumeration, and multiple further research directions for this generalization. |
Date: | November 30, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Dmitri Nikshych, University of New Hampshire |
Title: | Quantum Manin pairs and fusion categories |
Abstract: | A quantum Manin pair consists of a non-degenerate braided fusion category and a Lagrangian algebra in it. There is a bijection between equivalence classes of such pairs and fusion categories. This allows to study the latter using categorical linear algebra techniques. We explain how various categorical structures related to a fusion category (such as braidings, subcategories, module categories, fiber functors) can be interpreted and classified by means of the corresponding Manin pair. A part of this talk is based on a joint work with Alexei Davydov, Michael Mueger, and Victor Ostrik. |
Date: | December 5, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Yeong-Nan Yeh, Institute of Mathematics, Academia Sinica |
Title: | γ-Minimal parking functions and permutation statistics |
Abstract: | In this talk, given a choice function γ, we will introduce the concept of γ-minimal parking functions, discuss properties of γ-minimal parking functions and study some permutation statistics from the point of view of γ-minimal parking functions. In particular, for some choice functions γ, we will characterize γ-minimal parking functions. We will also discuss a family of bijections from the set of γ-minimal parking functions of length n to the set of permutations of {1, 2,…, n}. We define some permutation statistics via γ-minimal parking functions, derive the recurrent relations of the generating functions for these statistics and show that several statistics are Mahonian. |
Date: | December 7, 2018 |
Time: | 3:00pm |
Location: | BLOC 628 |
Speaker: | Yang Qi, University of Chicago |
Title: | On the rank preserving property of a linear section and its applications in tensors |
Abstract: | This talk is motivated by several conjectures on tensor ranks arising from signal processing and complexity theory. In the talk, we will first translate these conjectures into the geometric language, and reduce the problems to the study of a particular property of a linear section of an irreducible nondegenerate projective variety, namely the rank preserving property. Then we will introduce several useful tools and show some results obtained via these tools. This talk is based on a joint work with Lek-Heng Lim. |