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

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.

Monday, January 4, 2021 — 11:30 AM EST

**Title: **Complex Hadamard diagonalizable graphs

Speaker: | Ada Chan |

Affiliation: | York University |

Zoom Contact: | Soffia Arnadottir |

**Abstract:**

A graph is complex Hadamard diagonalizable if its Laplacian matrix is diagonalizable by a complex Hadamard matrix.

This is a natural generalization of the Hadamard diagonalizable graphs introduced by Barik, Fallat and Kirkland.

My interest in these graphs is two-fold:

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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.