| Date: | September 3, 2021 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Title: | Organization meeting |
|
| Date: | September 17, 2021 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Chun-Hung Liu, TAMU |
| Title: | Homomorphism counts in robustly sparse graphs |
| Abstract: | For a fixed graph H and for arbitrarily large host graphs G, the number of homomorphisms from H to G and the number of subgraphs isomorphic to H contained in G have been extensively studied in extremal graph theory and graph limits theory when the host graphs are allowed to be dense. This talk addresses the case when the host graphs are robustly sparse and proves a general theorem that solves a number of open questions proposed since 1990s and strengthens a number of results in the literature. In particular, our result determines, up to a constant multiplicative error, the maximum number of subgraphs isomorphic to H of an n-vertex graph in any fixed class of graphs with bounded expansion, which applies to any (topological) minor-closed family and many graph classes with certain geometric properties. |
|
| Date: | October 8, 2021 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Roberto Palomares, TAMU |
| Title: | Q-systems and higher unitary idempotent completion for C* 2-categories |
| Abstract: | A Q-system is a unitary version of a Frobenius algebra object in a tensor category or a C* 2-category. Q-systems were introduced by Longo to characterize the canonical endomorphism of a finite index inclusion of infinite von Neumann factors. Following work of Douglass-Reutter, a Q-system is also a unitary version of a higher idempotent. We will define a higher unitary idempotent completion for C* 2-categories called Q-system completion, and describe some of its properties and examples.
We will show that C*Alg, the C* 2-category of right correspondences of unital C*-algebras is Q-system complete by adapting a technique from subfactors theory called realization. This result allows for the straightforward adaptation of subfactor results to C*-algebras, characterizing finite Jones-Watatani-index extensions of unital C*-algebras $A \subset B$ equipped with a faithful conditional expectation $E:B \to A$ in terms of the Q-systems in C*Alg. |
|
| Date: | October 15, 2021 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Eric Rowell, TAMU |
| Title: | Zesting Braided Fusion Categories |
| Abstract: | I will describe a construction of new fusion categories from a given G-graded braided fusion category known as zesting. Zesting fits into the general theory of G-graded extensions and so are subject to the cohomological yin and yang of obstructions and parameterization torsors. On the other hand, zesting has several computational advantages. To name a few: the fusion rules are immediately available, we may easily explore braiding and pivotal structures, and when the resulting category is modular the data can be expressed succinctly in terms of the original data. These ideas will be illustrated with examples. |
|
| Date: | October 29, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Elizabeth Grimm, Illinois State University |
| Title: | Hamiltonicity of 3-tough (P2 ∪3P1)-free graphs |
| Abstract: | Chv ́atal conjectured in 1973 the existence of some constant t
such that all t-tough graphs with at least three vertices are Hamiltonian.
While the conjecture has been proven for some special classes of graphs, it
remains open in general. We say that a graph is (P2 ∪3P1)-free if it contains
no induced subgraph isomorphic to P2 ∪3P1, where P2 ∪3P1 is the disjoint
union of an edge and three isolated vertices. In this talk, we show that every
3-tough (P2 ∪3P1)-free graph with at least three vertices is Hamiltonian. |
|
| Date: | November 5, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Guoli Ding, Louisiana State University |
| Title: | New results on unavoidable large graphs |
| Abstract: | Ramsey theorem states that a large clique or coclique is unavoidable in every sufficiently large graph. This result can be extended to graphs of other connectivity. For instance, a big W_n- or K_{3,n}-minor is unavoidable in every sufficiently large 3-connected graph. In this talk we present similar results including results for rooted graphs. |
|
| Date: | November 19, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Anna Pun, University of Virginia |
| Title: | A Note on the Higher order Tur\'{a}n inequalities for k-regular partitions |
| Abstract: | Nicolas and DeSalvo and Pak proved that the partition function $p(n)$ is log concave for $n \geq 25$. Chen, Jia and Wang proved that $p(n)$ satisfies the third order Tur\'{a}n inequality, and that the associated degree 3 Jensen polynomials are hyperbolic for $n \geq 94$. Recently, Griffin, Ono, Rolen and Zagier proved more generally that for all $d$, the degree $d$ Jensen polynomials associated to $p(n)$ are hyperbolic for sufficiently large $n$.
In this talk, we will see that the same result holds for the $k$-regular partition function $p_k(n)$ for $k \geq 2$. In particular, for any positive integers $d$ and $k$, the order $d$ Tur\'{a}n inequalities hold for $p_k(n)$ for sufficiently large $n$. The case when $d = k = 2$ proves a conjecture by Neil Sloane that $p_2(n)$ is log concave.
This is a joint work with William Craig.
|
|
| Date: | December 3, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Zhanar Berikkyzy, Fairfield University |
| Title: | Long cycles in Balanced Tripartite Graphs |
| Abstract: | In this talk, we will survey the relevant literature, namely degree and edge conditions for Hamiltonicity and long cycles in graphs, including bipartite and $k$-partite results. We will then prove that if $G$ is a balanced tripartite graph on $3n$ vertices, $G$ must contain a cycle of length at least $3n-1$, provided that $e(G) \geq 3n^2-4n+5$ and $n\geq 14$. The result will be generalized to long cycles for 2-connected graphs when the minimum degree is large enough. Joint work with G. Araujo-Pardo, J. Faudree, K. Hogenson, R. Kirsch, L. Lesniak, and J. McDonald. |
|
Menu