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Please note: The University of Waterloo is closed for all events until further notice.

Thursday, March 25, 2021 — 1:00 PM EDT

**Title:** An Efficient Algorithm for Deciding the Vanishing of Schubert Polynomial Coefficients

Speaker: | Colleen Robichaux |

Affiliation: | University of Illinois at Urbana-Champaign |

Zoom: | Contact Karen Yeats |

**Abstract:**

Schubert polynomials form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. The vanishing problem for Schubert polynomials asks if a coefficient of a Schubert polynomial is zero. We give a tableau criterion to solve this problem, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the n x n grid. In contrast, we show that computing these coefficients explicitly is #P-complete. This is joint work with Anshul Adve and Alexander Yong.

Friday, March 12, 2021 — 3:30 PM EST

**Title: **An approximate solution to a 2,079,471-point traveling salesman problem

Speaker: | Bill Cook |

Affliation: | University of Waterloo |

Zoom: | Please email Emma Watson |

**Abstract:**

Together with Keld Helsguan, we have found a TSP tour through the 3D positions of 2,079,471 stars. We discuss how linear programming allows us to prove the tour is at most a factor of 0.0000074 longer than an optimal solution. The talk will focus on the use of minimum cuts and GF(2) linear systems, to drive the cutting-plane method towards strong LP relaxations.

Thursday, March 11, 2021 — 1:00 PM EST

**Title:** Equivalences of Wilson loop diagrams

Speaker: | Karen Yeats |

Affiliation: | University of Waterloo |

Zoom: | Contact Karen Yeats |

**Abstract:**

I will talk about Wilson loop diagrams, explain a bit about what they are, and some of the combinatorial questions that come out of them, with a focus on when they are equivalent. This is joint work with Susama Agarwala and Zee Fryer.

Monday, March 8, 2021 — 11:30 AM EST

**Title:** Lexicographic products, wreath products, and generalisations

Speaker: | Joy Morris |

Affiliation: | University of Lethbridge |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

I will present a history and overview of some of the work that has been done on the lexicographic product of graphs, and related generalisations. The focus of my talk will be on the automorphism groups of such graphs, and the relationship to the wreath product of permutation groups.

Friday, March 5, 2021 — 3:30 PM EST

**Title: **The embezzlement of entanglement and its applications

Speaker: | Debbie Leung |

Affliation: | University of Waterloo |

Zoom: | Please email Emma Watson |

**Abstract:**

Embezzlement of entanglement is the (impossible) task of producing an entangled state from a product state via a local change of basis, when a suitable *catalytic* entangled state is available.

The possibility to approximate this task was first observed by van Dam and Hayden in 2002. Since then, the phenomenon is found to play crucial roles in many aspects of quantum information theory. In this talk, we will discuss aspects of embezzlement and some applications (such as why quantum correlations do not form a closed set, and why there are nonlocal games that cannot be played optimally with a finite amount of entanglement, and why additive quantities cannot be more than asymptotically continuous).

Thursday, March 4, 2021 — 4:00 PM EST

**Title:** The Erdos-Szekeres theorem

Speaker: | Lukas Nabergall |

Affiliation: | University of Waterloo |

Zoom: | Contact Maxwell Levit |

**Abstract:**

What lies at the intersection of combinatorial geometry, graph theory, order theory, analysis, and statistics? Why, only one of the most beautiful theorems you may have never heard of. Let me take you on a journey from early 20th century Budapest through to the heights of modern mathematics and show you why this classic result of Erdos and Szekeres is worth adding to your mathematical repertoire. Along the way we'll even see a proof so good it must come from The Book.

Thursday, March 4, 2021 — 1:00 PM EST

**Title:** Graphs and combinatorics with a relationship to algebra, geometry and physics

Speaker: | Ralph Kaufmann |

Affiliation: | Purdue |

Zoom: | Contact Karen Yeats |

**Abstract:**

Several algebraic and geometric structures are most naturally encoded via graphs. These include restrictions, such as trees, and decorations, such as planar graphs, ribbon graphs, bi-partite graphs (aka. hypergraphs), directed versions, etc. Particularly nice properties satisfy some kind of hereditary condition. This affords a dual perspective. Either as (nested) subsets and decomposition, or as composition, gluing locally. Both views relate to category theory, algebra, and combinatorics in terms of finite sets, cospans etc. We will give examples of these phenomena and provide a general background.

Monday, March 1, 2021 — 11:30 AM EST

**Title:** Perfect Colorings of the Generalized Petersen Graphs

Speaker: | Hamed Karami |

Affiliation: | Iran University of Science and Technology |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

For a graph G and an integer m, a mapping T:V(G) -> {1,...,m} is called a perfect m-coloring with matrix A=(a_ij), i,j \in {1,...,m}, if it is surjective, and for all i,j, for every vertex of color i, the number of its neighbors of color j is equal to a_ij. There is another term for this concept in literature as "equitable partition". In this talk, we present some important results about enumerating parameter matrices of all perfect 2-colorings and perfect 3-colorings of generalized Petersen graphs GP(n,k).

Friday, February 26, 2021 — 3:30 PM EST

**Title: **Flat Littlewood Polynomials Exist

Speaker: | Robert Morris |

Affliation: | IMPA (Instituto de Matemática Pura e Aplicada) |

Zoom: | Please email Emma Watson |

**Abstract:**

In a Littlewood polynomial, all coefficients are either 1 or -1. Littlewood proved many beautiful theorems about these polynomials over his long life, and in his 1968 monograph he stated several influential conjectures about them. One of the most famous of these was inspired by a question of Erdos, who asked in 1957 whether there exist "flat" Littlewood polynomials of degree n, that is, with |P(z)| of order n^{1/2} for all (complex) z with |z| = 1.

Thursday, February 25, 2021 — 1:00 PM EST

**Title:** Chain decompositions for q,t-Catalan numbers

Speaker: | Nick Loehr |

Affiliation: | Virginia Tech |

Zoom: | Contact Karen Yeats |

**Abstract:**

The q,t-Catalan numbers Cat_n(q,t) are polynomials in q and t that reduce to the ordinary Catalan numbers when q=t=1. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Work of Garsia, Haglund, and Haiman has given us combinatorial formulas for Cat_n(q,t) as sums of Dyck vectors weighted by area and dinv. This talk narrates our ongoing quest for a bijective proof of the notorious symmetry property Cat_n(q,t)=Cat_n(t,q).

Monday, February 22, 2021 — 6:00 PM EST

*Note different start time

**Title:** Real Chromatic Roots of Graphs

Speaker: | Gordon Royle |

Affiliation: | The University of Western Australia |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

In February 1988, I arrived at C&O Waterloo for a postdoc with the late Ron Read. He handed me a paper by Beraha, Kahane and Weiss, and told me to apply it to determining the location of the complex roots of chromatic polynomials. I’ve returned to the topic every few years since then, with varying degrees of success---some positive results, but still many open problems and conjectures remain.

Friday, February 12, 2021 — 3:30 PM EST

**Title: **Mixed Integer Programming - Strength of adding integer variables

Speaker: | Robert Hildebrand |

Affliation: | Virginia Tech |

Zoom: | Please email Emma Watson |

**Abstract:**

Mixed Integer Programming is the problem of optimizing a multi-variate function over some domain constraints where some variables are required to take integer values. From a complexity-theoretic perspective, problems with fewer integer variables are easier to solve. However, this is not always the case in practice. We will discuss how performance can be improved when adding integer variables in the context of cutting planes and branch and bound. We will compare several frameworks for doing so in both the context of converting lifting integer and continuous variables to more variables. We will conclude with recent work on mixed-integer quadratic programming and mention some computational results.

Thursday, February 11, 2021 — 1:00 PM EST

**Title: **Solving Prellberg and Mortimer's conjecture - bijection(s)

between Motzkin paths and triangular walks

Speaker: | Julien Courtiel |

Affiliation: | Université de Caen |

Zoom: | Contact Karen Yeats |

**Abstract:**

In these difficult times, what we need to feel better is some colorful and elegant bijections.

This talk introduces the work we did with Andrew Elvey-Price (Tours, France) and Irène Marcovici (Nancy, France). Together we answered an open question from Mortimer and Prellberg, asking for a bijection between a family of walks inside a bounded triangular domain (think about a large equilateral triangle subdivided in several smaller equilateral triangles) and the famous Motzkin paths, but which have bounded height.

Monday, February 8, 2021 — 11:30 AM EST

**Title:** Cospectral Vertices and Isospectral Reductions

Speaker: | Mark Kempton |

Affiliation: | Brigham Young University |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

Understanding cospectral vertices in graphs is fundamental to understanding what the spectrum of the adjacency matrix can tell us about a graph. Furthermore, cospectral vertices are necessary in constructions of graphs exhibiting perfect quantum state transfer. I will talk about how to recognize cospectral vertices via a tool from network dynamics: the isospectral reduction of a graph. I will explore possible ways of getting new constructions of cospectral vertices by looking at isospectral reductions.

Friday, February 5, 2021 — 3:30 PM EST

**Title: **Probabilistic Aspects of Voting, Intransitivity, and Manipulation

Speaker: | Elchanan Mossel |

Affliation: | MIT Mathematics |

Zoom: | Please email Emma Watson |

**Abstract:**

Marquis de Condorcet, a French philosopher, mathematician, and political scientist, studied mathematical aspects of voting in the eighteenth century. Condorcet was interested in studying voting rules as procedures for aggregating noisy signals and in the paradoxical nature of ranking 3 or more alternatives. We will begin with a quick survey of some of the main mathematical models, tools, and results in this theory and discuss some recent progress in the area.

Thursday, February 4, 2021 — 1:00 PM EST

**Title:** Promotion and rowmotion – an ocean of notions

Speaker: | Jessica Striker |

Affiliation: | North Dakota State University |

Zoom: | Contact Karen Yeats |

**Abstract: **

Dynamical Algebraic Combinatorics studies objects important in algebraic combinatorics through the lens of dynamical actions. In this talk, we give a flavor of this field by investigating ever more general domains in which the actions of promotion on tableaux (or tableaux-like objects) and rowmotion on order ideals (or generalizations of order ideals) correspond. This is based on joint works with J. Bernstein, K. Dilks, O. Pechenik, C. Vorland, and N. Williams.

Friday, January 29, 2021 — 3:30 PM EST

**Title: **Finding twin smooth integers for isogeny-based cryptography

Speaker: | Michael Naehrig |

Affliation: | Microsoft Research |

Zoom: | Please email Emma Watson |

**Abstract:**

Efficient and secure instantiations of cryptographic protocols require careful parameter selection. For the isogeny-based cryptographic protocol B-SIDH, a variant of the Supersingular-Isogeny Diffie Hellman (SIDH) key exchange, one needs to find two consecutive B-smooth integers of cryptographic size such that their sum is prime. The smaller the smoothness bound B is, the more efficient the protocol becomes. This talk discusses a sieving algorithm to find such twin smooth integers that uses solutions to the Prouhet-Tarry-Escott problem.

Thursday, January 28, 2021 — 1:00 PM EST

**Title: **The multispecies TAZRP and modified Macdonald polynomials

Speaker: | Olya Mandelshtam |

Affiliation: | University of Waterloo |

Zoom: | Contact Karen Yeats |

**Abstract:**

Recently, a formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ was given in terms of objects called multiline queues, which also compute probabilities of a statistical mechanics model called the multispecies asymmetric simple exclusion process (ASEP) on a ring. It is natural to ask whether the modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$ can be obtained using a combinatorial gadget for some other statistical mechanics model.

Monday, January 25, 2021 — 11:30 AM EST

**Title:** The Matchings Polynomial

Speaker: | Chris Godsil |

Affiliation: | University of Waterloo |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

A $k$-matching in a graph is a matching of size $k$, and $p(X,k)$ denotes the number of $k$-matchings in $X$.

The matching polynomial of a graph is a form of generating function for the sequence $(p(X,k))_{k\ge0}$.

If is closely related to the characteristic polynomial of a graph. I will discuss some of the many interesting properties of this polynomial, and some of the related open problems.

Friday, January 22, 2021 — 3:30 PM EST

**Title: **Fast simulation of planar Clifford circuits

Speaker: | David Gosset |

Aflliation: | University of Waterloo |

YouTube Link: | https://youtu.be/LjmjiEPTSNo |

**Abstract:**

Clifford circuits are a special family of quantum circuits that can be simulated on a classical computer in polynomial time using linear algebra. Recent work has shown that Clifford circuits composed of nearest-neighbor gates in planar geometries can solve certain linear algebra problems provably faster --as measured by circuit depth-- than classical computers.

Thursday, January 21, 2021 — 1:30 PM EST

**Title:** The growth of groups and algebras

Speaker: | Jason Bell |

Affiliation: | University of Waterloo |

Zoom: | Contact Karen Yeats |

**Abstract:**

We give an overview of the theory of growth functions for associative algebras and explain their significance when trying to understand algebras from a combinatorial point of view. We then give a classification for which functions can occur as the growth function of a finitely generated associative algebra up to asymptotic equivalence. This is joint work with Efim Zelmanov.

Monday, January 18, 2021 — 11:30 AM EST

**Title:** Various Maximum Nullities Associated with a Graph

Speaker: | Shaun Fallat |

Affiliation: | University of Regina |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

Given a graph, we associate a collection of (typically symmetric) matrices S whose pattern of non-zero entries off of the main diagonal respects the edges in the graph. To this set, we let M denote the maximum possible nullity over all matrices in S. Depending on the choice of the set S, and the family of graphs considered, the parameter M often corresponds to an interesting combinatorial characteristic (planarity, connectivity, coverings, etc.) of the underlying graph.

Friday, January 15, 2021 — 3:30 PM EST

**Title: **Finding and Counting k-cuts in Graphs

Speaker: | Anupam Gupta |

Affiliation: |
Carnegie Mellon University |

Zoom: | Please email Emma Watson |

**Abstract:**

For an undirected graph with edge weights, a k-cut is a set of edges whose deletion breaks the graph into at least k connected components. How fast can we find a minimum-weight k-cut? And how many minimum k-cuts can a graph have? The two problems are closely linked. In 1996 Karger and Stein showed how to find a minimum k-cut in approximately n^{2k-2} time; their proof also bounded the number of minimum k-cuts by n^{2k-2}, using the probabilistic method.

Thursday, January 14, 2021 — 1:00 PM EST

**Title:** Analytic Combinatorics, Rigorous Numerics, and Uniqueness of Biomembranes

Speaker: | Steve Melczer |

Affiliation: | University of Waterloo |

Zoom: | Contact Karen Yeats |

**Abstract:**

Since the invention of the compound microscope in the early seventeenth century, scientists have marvelled over red blood cells and their surprising shape. An influential model of Canham predicts the shapes of blood cells and similar biomembranes come from a variational problem minimizing the "bending energy" of these surfaces. Because observed (healthy) cells have the same shape in humans, it is natural to ask whether the model admits a unique solution. Here, we prove solution uniqueness for the genus one Canham problem.

Monday, January 11, 2021 — 11:30 AM EST

**Title:** K-fractional revival and approximate K-fractional revival on path graphs

Speaker: | Whitney Drazen |

Affiliation: | Northeastern University |

Zoom: | Contact Soffia Arnadottir |

**Abstract:**

A continuous-time quantum walk is a process on a network of quantum particles that is governed by the transition matrix U(t) = e^{-itA}, where is A is the adjacency matrix of the graph. The two-vertex phenomenon fractional revival occurs between vertices u and v at time t if the columns of U(t) corresponding to u and v are only supported on the rows indexed by those same two vertices. The well-studied perfect state transfer is a special case of this.

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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 granted 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.