Schedule

The above talks are 50 minutes (except as noted).

Titles and Abstracts

Monday

Karen Meagher.

Erdős-Ko-Rado Theorems for Groups: Two permutations are intersecting if they both map some $i$ to the same point, so permutations $\sigma$ and $\pi$ are intersecting if and only if $\pi^{-1}\sigma$ has a fixed point. For any permutation group we can ask what is the size of the largest set of intersecting permutations. For example, in the symmetric group, the largest such set is the stabilizer of a point (or the coset of a stabilizer of a point). This can be considered to be a version of the Erdős-Ko-Rado theorem for permutations. In this talk I will go over common techniques used to find bounds on the size of intersecting sets of permutations in groups and also how to determine the structure of such sets. I will give an overview of what is known for different groups and which questions are still open.

Ada Chan.

Instantaneous uniform mixing: Let $A$ be the adjacency matrix of a graph $X$ on $n$ vertices. The transition matrix of the continuous-time quantum walk on $X$ is $e^{-i t A}$. We are interested state transfer and mixing in quantum walks. While we have characterizations for perfect state transfer, pretty good state transfer and fractional revival, much less is known for mixing. In this talk, we focus on instantaneous uniform mixing which happens when all entries of the transition matrix have the same absolute value, or equivalently, when $\sqrt{n} e^{-i t A}$ is a complex Hadamard matrix. We discuss instantaneous uniform mixing on graphs in association schemes, and related problems.

Jon Yard.

Tight 2-designs in complex projective spaces: Tight complex projective 2-designs are simultaneously maximal sets of equiangular lines and minimal complex projective 2-designs. In quantum information theory, they define optimal measurements known as SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures). In this talk, I will discuss the existence of explicit constructions from group orbits, focusing on algebraic and geometric properties, highlighting connections to some open questions in number theory and topology.

Tuesday

Joey Iverson.

Equiangular lines over finite fields: We discuss equiangular lines in classical geometries over finite fields, and explore connections with various open problems in algebraic combinatorics and discrete geometry. Joint work with Gary Greaves, John Jasper, and Dustin Mixon.

Nathan Lindzey.

Algebraic Aspects of t-Intersecting Families: We outline a general algebraic method for bounding the size of t-intersecting families of various combinatorial objects. We discuss some results that have been obtained using this method and identify other domains that seem amenable to this approach.

Hadi Kharaghani.

Balancedly splittable Orthogonal Designs: The concept of balancedly splittable orthogonal designs is introduced along with a recursive construction. As an application, equiangular tight frames over the real, complex, and quaternions meeting the Delsarte-Goethals-Seidel upper bound are obtained. Connections to unbiased orthogonal designs will be discussed in details.

Wednesday

David Roberson.

Quantum Isomorphisms: Results and Open Questions: I will introduce the notion of quantum isomorphisms and give an overview of some of the known results. In particular I will discuss how to construct non-isomorphic but quantum isomorphic graphs, and connections to both quantum groups and graph homomorphism counts. If time permits I will also discuss some new work showing that homomorphism counts from graphs of bounded degree do not determine a graph up to isomorphism.

Ferdinand Ihringer.

Variants of the Erdős-Ko-Rado problem: We survey some generalizations of the Erdős–Ko–Rado (EKR) problem which the speaker considers interesting. Currently, the plan is to focus on EKR with regularity/symmetry conditions, Erdős' Matching Conjecture, and low degree Boolean functions.

Andriaherimanana Sarobidy Razafimahatratra.

Intersection density of transitive groups: A set of permutations $\mathcal{F}$ of a finite transitive group $G\leq \mathrm{Sym}(\Omega)$ is intersecting if any two permutations in $\mathcal{F}$ agree on an element of $\Omega$. The intersection density of the intersecting set $\mathcal{F}\subset G$ is the rational number $\rho(\mathcal{F}) : =\frac{|\mathcal{F}|}{|G_\omega|}$, where $\omega\in \Omega$. The intersection density of the group $G$ is the number $\rho(G) := \max \{\rho(\mathcal{F}) : \mathcal{F}\subset G \mbox{ is intersecting}\}$. The group $G$ is said to have the Erdős-Ko-Rado (EKR) property if $\rho(G)=1$. The standard tool used to study the EKR property of the transitive group $G$ is its derangement graph $\Gamma_G$. This graph is the Cayley graph of $G$ with connection set equal to the set of all derangements of $G$ (i.e., the fixed-point-free elements). I will talk about some recent progress on the construction of transitive groups that do not have the EKR property. My main focus will be on the transitive groups with complete multipartite derangement graphs. I will also present some open problems on the intersection density of transitive groups of certain degrees.

Thursday

Venkata Raghu Tej Pantangi.

The EKR-Module Property: Let $G$ be a finite group acting transitively on $X$. We say $g,h \in G$ are intersecting if $gh^{-1}$ fixes a point in $X$. A subset $S$ of $G$ is said to be an intersecting set if every pair of elements in $S$ intersect. Cosets of point stabilizers are canonical examples of intersecting sets. The group action version of the classical Erdos-Ko-Rado problem asks about the size and characterization of intersecting sets of maximum possible size. A group action is said to satisfy the EKR property if the size of every intersecting set is bounded above by the size of a point stabilizer. A group action is said to satisfy the strict-EKR property if every maximum intersecting set is a coset of a point stabilizer. It is an active line of research to find group actions satisfying these properties. It was shown that all $2$-transitive satisfy the EKR property. While some $2$-transitive groups satisfy the strict-EKR property, not all of them do. However a recent result shows that all $2$-transitive groups satisfy the slightly weaker "EKR-module property"(EKRM), that is, the characteristic vector of a maximum intersecting set is a linear span of characteristic vectors of cosets of point stabilizers. We will discuss about a few more infinite classes of group actions that satisfy the EKRM property. I will also provide a few non-examples and a characterization of the EKRM property using characters of $G$.

Sebi Cioaba.

Some open problems related to eigenvalues of graphs: The eigenvalues of a graph can reveal important information regarding its structure. For example, its second largest eigenvalue is related to the graph’s connectivity and expansion properties while the smallest eigenvalue is closely connected to the independence and chromatic number of the graph. In this talk, I will present describe several open problems related to these eigenvalues.

Peter Sin.

Quantum walks on Cayley graphs: I'll discuss continuous (and maybe some discrete) time quantum walks on normal Cayley graphs, and formulate various phenomena (strong cospectrality, perfect state transfer, uniform mixing) from the point of view of character theory of groups. Worked examples will include extraspecial 2-groups and finite Heisenberg groups.

Christino Tamon.

Quantum mixing through the algorithmic lens: We explore the algorithmic side of mixing in continuous-time quantum walks. After a brief survey of early works in the area, we focus on natural applications of mixing for quantum computational tasks. Along the way, we mention some related open problems and may slightly perturb the normal notions of mixing.

Friday

Xiwang Cao.

Some results on uniform mixing on abelian Cayley graphs: I will present some results about uniform mixing on abelian Cayley graphs, including a complete classification of integral abelian Cayley graphs having uniform mixing.

Chris Godsil.

Uniform mixing on oriented graphs: To quote Lueneburg, the role of theory is to explain the examples. One reason uniform mixing is difficult is that our supply of examples is very limited, and this in part because we do not have an algorithm for deciding if a given graph admits uniform mixing. (For example, does $C_9$?)

However if $S$ is the skew symmetric adjacency matrix of an oriented graph, then the matrices $U(t)=\exp(tS)$ determine a quantum walk, and here there is a useful necessary condition for uniform mixing. (Any walk on a bipartite graph can be realized as a walk on an oriented graph, so have a necessary condition for ordinary walks on biartite graphs, at least.) I will report on joint work on this topic with Xiaohong Zhang. One result of this is a characterization of the oriented Cayley graphs for abelian groups with all eigenvalues integer multiples of $i$.

Contact

For further information, contact Chris Godsil.