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Title: Monogamy Violations in Perfect State Transfer
Speakers: Sabrina Lato & Christino Tamon Affiliations: University of Waterloo & Clarkson Unversity Zoom: Contact Soffia ArnadottirAbstract:
Continuoustime quantum walks on a graph are defined using a Hermitian matrix associated to a graph. For a quantum walk on a graph using either the adjacency matrix or the Laplacian, there can be perfect state transfer from a vertex to at most one other vertex in the graph.
Title: Filtering Grassmannian cohomology via kSchur functions
Speakers: Huda Ahmed and Yuanning Zhang Affiliation: New York University and UC Berkeley Zoom: Contact Karen YeatsAbstract:
This talk concerns the cohomology rings of complex Grassmannians. In 2003, Reiner and Tudose conjectured the form of the Hilbert series for certain subalgebras of these cohomology rings. We build on their work in two ways. First, we conjecture two natural bases for these subalgebras that would imply their conjecture using notions from the theory of kSchur functions. Second we formulate an analogous conjecture for Lagrangian Grassmannians.
Joint work with Michael Feigen, Victor Reiner, and Ajmain Yamin.
Title: Packings of partial difference sets
Speaker: Jonathan Jedwab Affiliation: Simon Fraser University Zoom: Contact Karen YeatsAbstract:
Partial difference sets are highly structured group subsets that occur in various guises throughout design theory, finite geometry, coding theory, and graph theory. They admit only two possible nontrivial character sums and so are often studied using character theory.
Title: Constructing broken SIDH parameters: a tale of De Feo, Jao, and Plut's serendipity
Speaker: Chloe Martindale Affiliation: University of Bristol Zoom: Please email Emma WatsonAbstract:
This talk is motivated by analyzing the security of the cryptographic key exchange protocol SIDH (Supersingular Isogeny DiffieHellman), introduced by 2011 by De Feo, Jao, and Plut. We will first recall some mathematical background as well as the protocol itself. The 'keys' in this protocol are elliptic curves, which are typically described by equations in x and y of the form y^2 = x^3 + ax + b.
Title: Scaling limits for the Gibbs states on distanceregular graphs with classical parameters
Speaker: Hajime Tanaka Affiliation: Tohoku University Zoom: Contact Soffia ArnadottirAbstract:
Limits of the normalized spectral distributions and other related probability distributions of families of graphs have been studied in the context of quantum probability theory as analogues of the central limit theorem. First I will review some of the previous work by Hora, Obata, and others, focusing on the case of distanceregular graphs, and emphasizing how the theory is related to the Terwilliger algebra.
Title: Face enumeration and realrootedness
Speaker: Christos Athanasiadis Affiliation: University of Athens Zoom: Contact Karen YeatsAbstract:
About fifteen years ago F. Brenti and V. Welker showed that the face enumerating polynomial of the barycentric subdivision of any CohenMacaulay simplicial complex has only real roots. It is natural to ask whether similar results hold when barycentric subdivision is replaced by more general types of triangulations, or when simplicial complexes are replaced by more general cell complexes.
Title: Sampling Under Symmetry
Speaker: Nisheeth Vishnoi Affiliation: Yale University Zoom: Please email Emma WatsonAbstract:
Exponential densities on orbits of Lie groups such as the unitary group are endowed with surprisingly rich mathematical structure and. traditionally, arise in diverse areas of physics, random matrix theory, and statistics.
In this talk, we will discuss the computational properties of such distributions and also present new applications to quantum inference and differential privacy.
Title: Fractional revival on graphs
Speaker: Xiaohong Zhang Affiliation: University of Waterloo Zoom: Contact Soffia ArnadottirAbstract:
Let A be the adjacency matrix of a weighted graph, and let U(t)=exp(itA). If there is a time t such that U(t)e_a=\alpha e_a+\beta e_b, then we say there is fractional revival (FR) between a and b. For the special case when \alpha=0, we say there is perfect state transfer (PST) between vertices a and b. It is known that PST is monogamous (PST from a to b and PST from a to c implies b=c) and vertices a b are cospectral in this case. If \alpha\beta\neq 0, then there is proper fractional revival.
Title: Some new lemmas about polynomials with only real roots
Speaker: David Wagner Affiliation: University of Waterloo Zoom: Contact Karen YeatsAbstract:
Recent investigations in Ehrhart theory suggested some conjectures involving interlacing relations among polynomials with only real roots, and Veronese sections of them. Revisiting some old theorems, we find as corollaries some new lemmas which have been overlooked for a long time. One of these lemmas directly implies a strong form of the motivating conjecture. Similar applications of the other lemmas are anticipated. This is ongoing joint work with Christos Athanasiadis (U. Athens).
Title: Beyond rank
Speaker: Jordan Ellenberg Affiliation: University of Wisconsin Zoom: Please email Emma WatsonAbstract:
The notion of the rank of a matrix is one of the most fundamental in linear algebra. The analogues of this notion in multilinear algebra — e.g., what is the “rank” of an m x n x p array of numbers? — is much more mysterious, but it also has proven to be useful in a wide array of contexts. I will talk about some questions and answers in “higher rank” coming from complexity theory, data science, geometric combinatorics, additive number theory, and commutative algebra.
Title: Complexity Measures on the Symmetric Group and Beyond
Speaker: Nathan Lindzey Affiliation: CU Boulder Zoom: Contact Soffia ArnadottirAbstract:
A classical result in complexity theory states that a degreed Boolean function on the hypercube can be computed using a decision tree of depth poly(d). Conversely, a Boolean function computed by a decision tree of depth d has degree at most d. Thus degree and decision tree complexity are polynomially related. Many other complexity measures of Boolean functions on the hypercube are polynomially related to the degree (e.g., approximate degree, certificate complexity, block sensitivity), and last year Huang famously added sensitivity to the list. Can we prove similar results for Boolean functions on other combinatorial domains?
Title: Simple eigenvalues of graph
Speaker: Krystal Guo Affiliation: University of Amsterdam Zoom: Contact Soffia ArnadottirAbstract:
If v is an eigenvector for eigenvalue λ of a graph X and α is an automorphism of X, then α(v) is also an eigenvector for λ. Thus it is rather exceptional for an eigenvalue of a vertextransitive graph to be simple. We study cubic vertextransitive graphs with a nontrivial simple eigenvalue, and discover remarkable connections to arctransitivity, regular maps and Chebyshev polynomials.
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