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Events - 2021

Friday, June 25, 2021 — 3:30 PM EDT

Title: From low probability to high confidence in stochastic convex optimization

Speaker: Dmitriy Drusvyatskiy
Affliliation: University of Washington
Zoom: Contact Emma Watson

Abstract:

Standard results in stochastic convex optimization bound the number of samples that an algorithm needs to generate a point with small function value in expectation. More nuanced high probability guarantees are rare, and typically either rely on “light-tail” noise assumptions or exhibit worse sample complexity. In this work, we show that a wide class of stochastic optimization algorithms can be augmented with high confidence bounds at an overhead cost that is only logarithmic in the confidence level and polylogarithmic in the condition number.

Thursday, June 24, 2021 — 1:00 PM EDT

Title: Arctic curves for groves

Speaker: Terrence George
Affiliation: University of Michigan
Zoom: Contact Stephen Melczer

Abstract:

The limit shape phenomenon is a "law of large numbers" for random surfaces: the random surface looks macroscopically like the "average surface". The first result of this kind was the celebrated arctic circle theorem for domino tilings of the aztec diamond. The limit shape has macroscopic regions with different qualitative behavior, and the arctic curve is the boundary separating these regions.

Monday, June 21, 2021 — 11:30 AM EDT

Title: Average Mixing Matrices of Trees and Stars

Speaker: Paula Kimmerling
Affiliation: Washington State University
Zoom: Contact Soffia Arnadottir

Abstract:

We define the average mixing matrix (AMM) of a continuous-time quantum walk on a graph using the orthogonal projections onto the eigenspaces of the adjacency matrix A. From there, one of the properties that has been studied is the rank of the AMM. This is easiest to do if the eigenvalues of A are simple, and we’ll review some of the results on this from Coutinho et. al. (2018).

Friday, June 18, 2021 — 3:30 PM EDT

Title: Lattice Walk Enumeration: Analytic, algebraic and geometric aspects

Speaker: Marni Mishna 
Affliliation: Simon Fraser University
Zoom: Contact Emma Watson

Abstract:

This talk will examine the rich topic of lattice path enumeration. A very classic object of combinatorics, lattice walks withstand study from a variety of perspectives. Even the simple task of classifying the two dimensional walks restricted to the first quadrant has brought into play a surprising diversity of techniques from algebra to analysis to geometry. We will consider walks under a few different lenses.

Friday, June 18, 2021 — 11:30 AM EDT

Title: The spectral radius of graphs with no odd wheels

Speaker: Dheer Noal Desai
Affiliation: University of Delaware
Zoom: Contact Soffia Arnadottir

Abstract:

The odd wheel W_{2k+1} is the graph formed by joining a vertex to a cycle of length 2k. In this talk, we will investigate the largest value of the spectral radius of the adjacency matrix of an n-vertex graph that does not contain W_{2k+1}.

Thursday, June 17, 2021 — 1:00 PM EDT

Title: Identities for ninth variation Schur Q-functions

Speaker: Angèle Hamel
Affiliation: Wilfrid Laurier University
Zoom: Contact Stephen Melczer

Abstract:

Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald's ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel's Pfaffian outside decomposition identity.

Friday, June 11, 2021 — 3:30 PM EDT

Title: The discrete logarithm problem in finite fields

Speaker: Cécile Pierrot
Affliliation: French National Institute for Computer Science Research (INRIA)
Zoom: Contact Emma Watson

Abstract:

The security of currently deployed public key protocols relies on the presumed hardness of problems often coming from number theory, such as factoring a large integer or solving the discrete logarithm problem in some group.

In this talk we focus on discrete logarithms in finite fields. We explain what is a discrete logarithm, why cryptographers need them, and we focus then on algorithms to solve the related problem, together with open questions in this area.

Thursday, June 10, 2021 — 4:00 PM EDT

Title: Counting Antichains in the Boolean Lattice

Speaker: Shayla Redlin
Affiliation: University of Waterloo
Zoom: Contact Maxwell Levit

Abstract:

How many antichains are there in the Boolean lattice P(n)? Sperner's theorem (1928) tells us that the largest antichain in P(n) has size A = (n choose n/2). A subset of an antichain is an antichain, so there are at least 2^A antichains in P(n). Interestingly, it turns out that this is close to the total, as Kleitman (1969) showed that the number of antichains is 2^(A(1+x)) where x goes to zero as n goes to infinity.

Thursday, June 10, 2021 — 1:00 PM EDT

Title: Enumerating hereditary classes of chord diagrams

Speaker: Lukas Nabergall
Affiliation: University of Waterloo
Zoom: Contact Stephen Melczer

Abstract:

A class of combinatorial structures is hereditary if membership in the class is closed under taking substructures. Hereditary classes have been extensively studied for a variety of objects, notably graphs and permutations. A central problem is to determine the number of objects of size n in a given hereditary class. We discuss this problem for chord diagrams, perfect matchings of [2n].

Friday, June 4, 2021 — 3:30 PM EDT

Title: On the approximability of the maximum cardinality stable matching problem with ties  

Speaker: Kanstantsin Pashkovich
Affliliation: University of Waterloo
Zoom: Contact Emma Watson

Abstract:

The maximum cardinality stable matching problem is central in game theory. When participants are allowed to have ties in their preferences, the problem of finding a stable matching of maximum cardinality is NP-hard, even when the ties are of size two. Moreover, in this setting it is UGC-hard to provide an approximation for the maximum cardinality stable matching problem with a constant factor smaller than 4/3.

Thursday, June 3, 2021 — 1:00 PM EDT

Title: What is the degree of a Grothendieck polynomial?

Speaker: Oliver Pechenik
Affiliation: University of Waterloo
Zoom: Contact Stephen Melczer

Abstract:

Jenna Rajchgot observed that the Castelnuovo-Mumford regularity of matrix Schubert varieties is computed by the degrees of the corresponding Grothendieck polynomials. We give a formula for these degrees.

Monday, May 31, 2021 — 11:30 AM EDT

Title: An algebraic framework for twualities of embedded graphs

Speaker: Jo Ellis-Monaghan
Affiliation: Korteweg-de Vries Instituut voor Wiskunde, Universiteit van Amsterdam
Zoom: Contact Soffia Arnadottir

Abstract:

We develop algebraic tools to identify and generate new surface embeddings of graphs with various forms of self-twuality including geometric duality, Petrie duality, Wilson duality, and both forms of triality (which is like duality, but of order three instead of two).  These operations are of particular interest because of their interplay with graph symmetries and graph polynomials.

Friday, May 28, 2021 — 3:30 PM EDT

Title: Algebraic formulations of Zauner's conjecture

Speaker: Jon Yard
Affliliation: University of Waterloo
Zoom: Contact Emma Watson

Abstract:

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).  They are conjectured by Zauner to exist in every dimension, even as specific group orbits. 

Friday, May 21, 2021 — 3:30 PM EDT

Title: Positivity Problems for Linear Recurrences

Speaker: Steve Melczer
Affliliation: University of Waterloo
Zoom: Contact Emma Watson

Abstract:

Although sequences satisfying linear recurrence relations have been studied for centuries, and appear as some of the first examples of combinatorial sequences encountered in an introductory combinatorics class, there are natural examples of simply stated problems related to their basic behaviour whose decidability is unknown. In this talk we survey some open computability and complexity questions related to the positivity of linearly recurrent sequences, before examining a new approach to proving positivity using rigorous numerical methods for functions satisfying linear differential equations.

Thursday, May 20, 2021 — 1:00 PM EDT

Title: q-Whittaker functions, finite fields, and Jordan forms

Speaker: Steven Karp
Affiliation: UQAM
Zoom: Contact Steve Melczer

Abstract:

The q-Whittaker symmetric function associated to an integer partition is a q-analogue of the Schur symmetric function. We give a new formula for the q-Whittaker function in terms of partial flags compatible with a nilpotent endomorphism over the finite field of size 1/q.

Monday, May 17, 2021 — 11:30 PM EDT

Title: Minimum eigenvalue of nonbipartite graphs

Speaker: Bojan Mohar
Affiliation: Simon Fraser University
Zoom: Contact Soffia Arnadottir

Abstract:

Let \rho and \lambda be the largest and the smallest eigenvalue of a connected graph G. It is well-known that \rho + \lambda \geq 0 and that equality occurs if and only if G is bipartite. The speaker will discuss what else can we say when G is not bipartite.

Friday, May 14, 2021 — 3:30 PM EDT

Title: Interlacing methods in Extremal Combinatorics

Speaker: Hao Huang
Affliliation: Emory University
Zoom: Contact Emma Watson

Abstract:

Extremal Combinatorics studies how large or how small a collection of finite objects could be, if it must satisfy certain restrictions. In this talk, we will discuss applications of spectral graph theory, more specifically eigenvalue interlacing, to prove various interesting results in Extremal Combinatorics. We will discuss the Erdos-Ko-Rado Theorem and its degree version, an isodiametric inequality for discrete cubes, and the resolution of a thirty-year-old open problem in Theoretical Computer Science, the Sensitivity Conjecture of Nisan and Szegedy. Several open problems will also be mentioned during this talk.

Monday, May 10, 2021 — 11:30 AM EDT

Title: Quantum independence number

Speaker: Mariia Sobchuk
Affiliation: University of Waterloo
Zoom: Contact Soffia Arnadottir

Abstract:

From this talk you will be able to learn what quantum independence number is and how it is different from the classical independence number. I will provide both known individual and infinite families of the graphs where classical and quantum independent numbers are different, as well as some of our generalisations of these examples.  

Monday, April 26, 2021 — 11:30 AM EDT

Title: A Spectral Moore Bound for Bipartite Semiregular Graphs

Speaker: Sabrina Lato
Affiliation: University of Waterloo
Zoom: Contact Soffia Arnadottir

Abstract:

The Moore bound provides an upper bound on the number of vertices of a regular graph with a given degree and diameter, though there are disappointingly few graphs that achieve this bound. Thus, it is interesting to ask what additional information can be used to give Moore-type bounds that are tight for a larger number of graphs. Cioaba, Koolen, Nozaki, and Vermette considered regular graphs with a given second-largest eigenvalue, and found an upper bound for such graphs.

Friday, April 23, 2021 — 3:30 PM EDT

Title: Robust Interior Point Methods for Key Rate Computation in Quantum Key Distribution

Speaker: Hao Hu
Affliliation: University of Waterloo
Zoom: Contact Emma Watson

Abstract:

We study semidefinite programs for computing the key rate in finite dimensional quantum key distribution (QKD) problems. Through facial reduction, we derive a semidefinite program which is robust and stable in the numerical computation. Our program avoids the difficulties for current algorithms from singularities that arise due to loss of positive definiteness. This allows for the derivation of an efficient Gauss-Newton interior point approach. We provide provable lower and upper bounds for the hard nonlinear semidefinite programming problem.

Monday, April 19, 2021 — 11:03 AM EDT

Title: Quantum walks on Cayley graphs

Speaker: Julien Sorci
Affiliation: University of Florida
Zoom: Contact Soffia Arnadottir

Abstract:

In this talk we will look at the continuous-time quantum walk on Cayley graphs of finite groups. We will show that normal Cayley graphs enjoy several nice algebraic properties, and then look at state transfer phenomena in Cayley graphs of certain non-abelian p-groups called the extraspecial p-groups. Some of the results we present are part of joint work with Peter Sin.

Friday, April 16, 2021 — 3:30 PM EDT

Title: A proof of the Erdős–Faber–Lovász conjecture

Speaker: Tom Kelly
Affliliation: University of Birmingham
Zoom: Contact Emma Watson

Abstract:

The Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$.  We prove this conjecture for every sufficiently large $n$.  This is joint work with Dong Yeap Kang, Daniela Kühn, Abhishek Methuku, and Deryk Osthus.

Thursday, April 15, 2021 — 1:00 PM EDT

Title: Algebraic structure of the Hopf algebra of double posets

Speaker: Yannic Vargas
Affiliation: Potsdam University
Zoom: Contact Karen Yeats

Abstract:

A Hopf algebra of double posets was introduced by Claudia Malvenuto and Christophe Reutenauer in 2011, motivated by the study of pictures of tableaux as defined by Zelevinsky. Starting from the correspondence between top-cones in the braid arrangement and partial orders, we investigate several properties of the Hopf algebra of double posets as the image of a Hopf monoid (via the Fock functor). In particular, we obtain a non-cancellative formula for the antipode. A description of the primitive space is also discussed.

Monday, April 12, 2021 — 11:30 AM EDT

Title: Feynman integrals as algebraic graph theory

Speaker: Karen Yeats
Affiliation: University of Waterloo
Zoom: Contact Soffia Arnadottir

Abstract:

I will overview how Feynman integrals should feel very familiar to algebraic graph theorists, and then say a few words about current directions of interest to me, particularly the c_2 invariant.

Friday, April 9, 2021 — 3:30 PM EDT

Title: Circuit QED Lattices: Synthetic Quantum Systems on Line Graphs

Speaker: Alicia Kollár
Affliliation: University of Maryland
Zoom: Contact Emma Watson

Abstract:

After two decades of development, superconducting circuits have emerged as a rich platform for quantum computation and simulation. Lattices of coplanar waveguide (CPW) resonators realize artificial photonic materials or photon-mediated spin models. Here I will highlight the special property that these lattice sites are deformable and allow for the implementation of devices with graph-like configurational flexibility. In particular, I will show that it is possible to create synthetic materials in which microwave photons experience negative curvature, which is impossible in conventional electronic materials [1].

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