Events

Carrie Knoll, Department of Pure Mathematics, University of Waterloo

“Idempotent operations in reflexive digraphs”

Every projection map is idempotent, but the converse is not necessarily true. In a general structure A, it is possible that the only binary idempotent operations on A are the binary projection maps, but we can find a 3-ary idempotent operation that is not a projection. This cannot happen for finite reflexive digraphs.

Timothy Caley, Department of Pure Mathematics, University of Waterloo

"A new algorithm for the Prouhet-Tarry-Escott problem"

Abstract: Given natural numbers n and k, the Prouhet-Tarry-Escott (PTE)
asks for integers x_1,..,x_n and y_1,...,y_n such that the sums of the
first k powers are equal. This problem has connections to combinatorics
and theoretical computer science, as well as to other areas of number

Kai-Cheong Chan, University of Waterloo

“On tensor products of digraph algebras over pre-ordered groups ”

Two natural operator algebra structures, A⊗maxB and A⊗minB, exist on the tensor product of two given unital operator algebras A and B. Because of the different properties enjoyed by the two tensor products in connection to dilation theory, it is of interest to know when they coincide.

Jeffrey Shallit, School of Computer Science, University of Waterloo

"Rational numbers and automata"

In this talk, I will describe a new model for describing certain sets S of rational numbers using finite automata. We will see that it is decidable if every element of S is an integer, and that sup S is computable. However, closely related questions are still open. There are applications to combinatorics on words.

Refreshments will be served in MC 5046 at 3:30pm. All are welcome.

Cassie Naymie, Pure Mathematics, University of Waterloo

“Roth’s theorem on ﬁnite abelian groups”

Roth’s theorem, proved by Roth in 1953, states that when A ⊆ [1, N ] with A dense enough, A has a three term arithmetic progression (3-AP). Since then the bound originally given by Roth has been improved upon by number theorists several times. The theorem can also be generalised to ﬁnite abelian groups. In 1994 Meshulam worked on ﬁnding an upper bound for subsets containing only trivial 3-APs based on the number of components in a ﬁnite abelian group.

Ross Willard, Pure Mathematics, University of Waterloo

"Larose's theorem"

Putting together some of the machinery developed this term, I will prove Larose’s Theorem: if X is a ﬁnite, connected reﬂexive digraph and X admits a Taylor operation, then for every k ≥ 1, the k-th homotopy group of X is trivial.

Jerry Wang, Harvard University

"Pencils of quadrics and 2-Selmer groups of Jacobians of hyperelliptic curves"

Since Bhargava and Shankar's new method of counting orbits, average orders of the 2,3,4,5-Selmer groups of elliptic curves over Q have been obtained. In this talk we will look at a construction of torsors of Jacobians of hyperelliptic curves using pencils of quadrics and see how they are used to compute the average order of the 2-Selmer groups of Jacobians of hyperelliptic curves over Q with a rational (non-)Weierstrass point.

Ian Payne, Pure Mathematics, University of Waterloo

"I Can't Get No (Constraint) Satisfaction"

The objective of this talk is to introduce the audience to the constraint satisfaction problem, or CSP. This broad class of decision problems was motivated by theoretical computer science, but is of interest to mathematicians since universal algebra is the main tool used in the study of their complexity. I will define relational structures, which give the framework for the CSP.

Chun Kit Lai, McMaster University

"Fourier frames on general measure spaces"

Motivated from the Fuglede conjecture and the discovery of exponential
orthonormal basis on the one-fourth Cantor measure, but not for the
one-third one, there has been interest in understanding the kind of
measures that admit some exponential type bases and their relatives such
as Fourier frames and Riesz bases. By decomposing the measure into

Malabika Pramanik, University of British Columbia

"Needles, bushes, hairbrushes and polynomials"

Points, lines and circles are among the most primitive and fundamental of mathematical concepts, yet few geometric objects have generated more beautiful and nontrivial mathematics. Deceptively simple in their formulation, many classical problems involving sets of lines or circles remain open to this day.

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Learning seminar on Finite Relational Structures