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Friday, December 11, 2020 — 3:30 PM EST

**Title: **Sparse PSD approximation of the PSD cone

Speaker: | Santanu Dey |

Affiliation: |
H. Milton Stewart School of Industrial and Systems Engineering at Georgia Institute of Technology |

Zoom: | Please email Emma Watson |

**Abstract:**

While semidefinite programming (SDP) problems are polynomially solvable in theory, it is often difficult to solve large SDP instances in practice. One computational technique used to address this issue is to relax the global positive-semidefiniteness (PSD) constraint and only enforce PSD-ness on smaller k × k principal submatrices — we call this the sparse SDP relaxation.

Thursday, December 10, 2020 — 1:00 PM EST

**Title:** Chromatic symmetric functions of Dyck paths and $q$-rook theory

Speaker: | Laura Colmenarejo |

Affiliation: | UMass Amherst |

Zoom: | Contact Karen Yeats |

**Abstract:**

Given a graph and a set of colors, a coloring of the graph is a function that associates each vertex in the graph with a color. In 1995, Stanley generalized this definition to symmetric functions by looking at the number of times each color is used and extending the set of colors to $\mathbb{Z}^+$. In 2012, Shareshian and Wachs introduced a refinement of the chromatic functions for ordered graphs as $q$-analogues.

Monday, December 7, 2020 — 11:30 AM EST

**Title:** Distinct Eignvalues and Sensitivity

Speaker: | Shahla Nasserasr |

Affiliation: | Rochester Institute of Technology |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

For a graph $G$, the class of real-valued symmetric matrices whose zero-nonzero pattern of off-diagonal entries is described by the adjacencies in $G$ is denoted by $S(G)$. The inverse eigenvalue problem for the multiplicities of the eigenvalues of $G$ is to determine for which ordered list of positive integers $m_1\geq m_2\geq \cdots\geq m_k$ with $\sum_{i=1}^{k} m_i=|V(G)|$, there exists a matrix in $S(G)$ with distinct eigenvalues ${\lambda_1,\lambda_2,\cdots, \lambda_k}$ such that $\lambda_i$ has multiplicity $m_i$.

Friday, December 4, 2020 — 3:30 PM EST

**Title: **Partial orders on the symmetric group

Speaker: | Oliver Pechenik |

Affiliation: | University of Waterloo |

Zoom: | Please email Emma Watson |

**Abstract:**

The symmetric group of permutations is naturally a poset in at least 4 different ways, the (strong) Bruhat order and three flavors of weak order. Stanley showed in 1980 that the Bruhat order is Sperner, essentially meaning that the obvious large antichains are in fact the largest possible. The corresponding fact for weak orders was open until last year, when it was established by Gaetz and Gao.

Thursday, December 3, 2020 — 1:00 PM EST

**Title:**** **Twisted Hopf algebras

Speaker: | Loïc Foissy |

Affiliation: | Université du Côte d'Opale |

Zoom: | Contact Karen Yeats |

**Abstract:**

A twisted Hopf algebra is a Hopf algebra in the category of linear species. The Fock functors allow to recover "classical" Hopf algebras from twisted ones. Numerous constructions and results can be lifted to the level of twisted bialgebras, such that cofreeness, shuffle and quasi-shuffles products, etc.

Monday, November 30, 2020 — 11:30 AM EST

**Title:** Simple eigenvalues of graph

Speaker: | Krystal Guo |

Affiliation: | University of Amsterdam |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

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 vertex-transitive graph to be simple. We study cubic vertex-transitive graphs with a non-trivial simple eigenvalue, and discover remarkable connections to arc-transitivity, regular maps and Chebyshev polynomials.

Monday, November 23, 2020 — 11:30 AM EST

**Title:** Complexity Measures on the Symmetric Group and Beyond

Speaker: | Nathan Lindzey |

Affiliation: | CU Boulder |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

A classical result in complexity theory states that a degree-d 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?

Friday, November 20, 2020 — 3:30 PM EST

Title: Beyond rank

Speaker: | Jordan Ellenberg |

Affiliation: | University of Wisconsin |

Zoom: | Please email Emma Watson |

**Abstract:**

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.

Thursday, November 19, 2020 — 1:00 PM EST

**Title:** Some new lemmas about polynomials with only real roots

Speaker: | David Wagner |

Affiliation: | University of Waterloo |

Zoom: | Contact Karen Yeats |

**Abstract:**

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).

Monday, November 16, 2020 — 11:30 AM EST

**Title:** Fractional revival on graphs

Speaker: | Xiaohong Zhang |

Affiliation: | University of Waterloo |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

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.

Friday, November 13, 2020 — 3:30 PM EST

**Title: **Sampling Under Symmetry

Speaker: | Nisheeth Vishnoi |

Affiliation: | Yale University |

Zoom: | Please email Emma Watson |

**Abstract:**

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.

Thursday, November 12, 2020 — 1:00 PM EST

**Title:** Face enumeration and real-rootedness

Speaker: | Christos Athanasiadis |

Affiliation: | University of Athens |

Zoom: | Contact Karen Yeats |

**Abstract:**

About fifteen years ago F. Brenti and V. Welker showed that the face enumerating polynomial of the barycentric subdivision of any Cohen-Macaulay 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.

Monday, November 9, 2020 — 8:00 PM EST

**Title:** Scaling limits for the Gibbs states on distance-regular graphs with classical parameters

Speaker: | Hajime Tanaka |

Affiliation: | Tohoku University |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

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 distance-regular graphs, and emphasizing how the theory is related to the Terwilliger algebra.

Friday, November 6, 2020 — 3:30 PM EST

**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 Watson |

**Abstract:**

This talk is motivated by analyzing the security of the cryptographic key exchange protocol SIDH (Supersingular Isogeny Diffie-Hellman), 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.

Thursday, November 5, 2020 — 1:00 PM EST

**Title:** Filtering Grassmannian cohomology via k-Schur functions

Speakers: | Huda Ahmed and Yuanning Zhang |

Affiliation: | New York University and UC Berkeley |

Zoom: | Contact Karen Yeats |

**Abstract:**

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 k-Schur functions. Second we formulate an analogous conjecture for Lagrangian Grassmannians.

Joint work with Michael Feigen, Victor Reiner, and Ajmain Yamin.

Thursday, November 5, 2020 — 1:00 PM EST

**Title:** Packings of partial difference sets

Speaker: | Jonathan Jedwab |

Affiliation: | Simon Fraser University |

Zoom: | Contact Karen Yeats |

**Abstract:**

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.

Monday, November 2, 2020 — 11:30 AM EST

**Title:** Monogamy Violations in Perfect State Transfer

Speakers: | Sabrina Lato & Christino Tamon |

Affiliations: | University of Waterloo & Clarkson Unversity |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

Continuous-time 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.

Friday, October 30, 2020 — 3:30 PM EDT

**Title: **The Tutte Symmetric Function

Speaker: | Logan Crew |

Affiliation: | University of Waterloo |

Zoom: | Please email Emma Watson |

**Abstract:**

The Tutte polynomial is one of the most celebrated and most well-studied graph functions, in part because it specializes to every graph polynomial with a linear deletion-contraction relation, such as the chromatic polynomial. In the 1990s, Stanley generalized the Tutte polynomial to a symmetric function, but at the cost of the deletion-contraction relation.

Thursday, October 29, 2020 — 1:00 PM EDT

**Title:** qRSt: A probabilistic Robinson--Schensted correspondence for Macdonald polynomials

Speaker: | Florian Aigner |

Affiliation: | Université du Québec à Montréal |

Zoom: | Contact Karen Yeats |

**Abstract:**

The Robinson--Schensted (RS) correspondence is a bijection between permutations and pairs of standard Young tableaux which plays a central role in the theory of Schur polynomials. In this talk, I will present a (q,t)-dependent probabilistic deformation of Robinson--Schensted which is related to the Cauchy identity for Macdonald polynomials.

Monday, October 26, 2020 — 11:30 AM EDT

**Title:** Pseuodrandom Cliquefree Graphs, Finite Geometry, and Spectra

Speaker: | Ferdinand Ihringer |

Affiliation: | Ghent University, Belgium |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

A regular graph is called optimally pseudorandom if its second largest eigenvalue in absolute value is, up to a constant factor, as small as possible. Determining the largest degree of an optimally pseudorandom graph without a clique of size s is a well-known open problem in extremal graph theory.

Friday, October 23, 2020 — 3:30 PM EDT

**Title: **Semidefinite Programming Relaxations of the Traveling Salesman Problem

Speaker: | David P. Williamson |

Affiliation: | Cornell University |

Zoom: | Please email Emma Watson |

**Abstract:**

Finding a polynomial-time solvable relaxation of the traveling salesman problem whose integrality gap better matches what is seen in practice has been an outstanding open problem in combinatorial optimization for some time. We study several semidefinite programming relaxations of the traveling salesman problem proposed in the literature and show that all known relaxations have an unbounded integrality gap.

Thursday, October 22, 2020 — 1:00 PM EDT

**Title:** Coxeter combinatorics and spherical Schubert geometry

Speaker: | Reuven Hodges |

Affiliation: | University of Illinois |

Zoom: | Contact Karen Yeats |

**Abstract:**

This talk will introduce spherical elements in a finite Coxeter system. These spherical elements are a generalization of Coxeter elements, that conjecturally, for Weyl groups, index Schubert varieties in the flag variety G/B that are spherical for the action of a Levi subgroup.

Wednesday, October 21, 2020 — 4:30 PM EDT

**Title:** On the Theory of the Analytical Forms called Trees

Speaker: | Nick Olson-Harris |

Affiliation: | University of Waterloo |

Zoom: | Contact Maxwell Levit |

**Abstract:**

Trees are among the most fundamental of combinatorial structures. Nowadays they appear all over mathematics and computer science, but this has not always been the case. Trees were first introduced, at least under that name, in an 1857 paper of Cayley by the same title as this talk.

Monday, October 19, 2020 — 3:00 PM EDT

**Title:** The Hepp bound of a matroid: flags, volumes and integrals

Speaker: | Erik Panzer |

Affiliation: | University of Oxford |

Zoom: | Contact Rose McCarty |

**Abstract:**

Invariants of combinatorial structures can be very useful tools that capture some specific characteristics, and repackage them in a meaningful way. For example, the famous Tutte polynomial of a matroid or graph tracks the rank statistics of its submatroids, which has many applications, and relations like contraction-deletion establish a very close connection between the algebraic structure of the invariant (e.g. Tutte polynomials) and the actual matroid itself.

Monday, October 19, 2020 — 11:30 AM EDT

**Title:** Pretty Good State Transfer and Minimal Polynomials

Speaker: | Christopher van Bommel |

Affiliation: | University of Manitoba |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

We examine conditions for a pair of strongly cospectral vertices to have pretty good quantum state transfer in terms of minimal polynomials, and provide cases where pretty good state transfer can be ruled out.

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The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land promised to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Indigenous Initiatives Office.