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Welcome to Pure Mathematics

We are home to 30 faculty, four staff, approximately 60 graduate students, several research visitors, and numerous undergraduate students. We offer exciting and challenging programs leading to BMath, MMath and PhD degrees. We nurture a very active research environment and are intensely devoted to both ground-breaking research and excellent teaching.


News

Friday, September 29, 2023

Spring 2023 Graduands

Congratulations to Clement Wan, MMath and Eric Boulter, PhD, who convocated in Spring 2023. Best of luck in your future endeavours!

Events

Monday, February 24, 2025 2:30 pm - 3:30 pm EST (GMT -05:00)

Pure Math Department Colloquium

Carlo Pagano, Concordia University

Hilbert 10 via additive combinatorics

In 1900 Hilbert proposed a list of problems that have been very influential throughout the last century. In 1970 Matiyasevich, building on earlier work of Davis—Putnam—Robinson, proved that Hilbert's 10th problem is undecidable for Z. The problem of extending this result to any ring that is finitely generated over Z (eg ring of integers in number fields) has attracted significant attention since 1970 and, thanks to the efforts of many mathematicians, the task has been reduced to an arithmetic problem about elliptic curves. This problem so far had been solved only conditional on the BSD conjecture (one of the Millenium problems) by Mazur—Rubin.

In joint work with Peter Koymans we have combined additive combinatorics (Green—Tao’s celebrated theorem) with 2-descent (an old technique dating back to Fermat) to solve this problem about elliptic curves unconditionally. This shows that Hilbert 10 is undecidable over any finitely generated infinite commutative ring.

In this colloquium I will provide a gentle introduction to this undecidability result, giving a glimpse of how mathematical logic, number theory and additive combinatorics meet into one story.

MC 5501

Tuesday, February 25, 2025 10:00 am - 10:50 am EST (GMT -05:00)

Number Theory Seminar

Miao Gu, University of Michigan

On Triple Product L-functions

The Poisson summation conjecture of Braverman-Kazhdan, Lafforgue, Ngo and Sakellaridis is an ambitious proposal to prove analytic properties of quite general Langlands L-functions using vast generalizations of the Poisson summation formula. In this talk, we present the construction of a generalized Whittaker induction such that the associated L-function is the product of the triple product L-function and L-functions whose analytic properties are understood. We then formulate an extension of the Poisson summation conjecture and prove that it implies the expected analytic properties of triple product L-functions. Finally, we use the fiber bundle method to reduce this extended Poisson summation conjecture to a case of the Poisson summation conjecture in which spectral methods can be employed together with certain local compatibility statements. This is joint work with Jayce Getz, Chun-Hsien Hsu, and Spencer Leslie.

MC 5479

Tuesday, February 25, 2025 2:00 pm - 3:00 pm EST (GMT -05:00)

Logic Seminar

Clement Yung, University of Toronto

Weak A2 spaces, the Kastanas game and strategically Ramsey sets

In the past century, the insight behind the original Ramsey's theorem proved to be applicable to a wide range of mathematics, such as number theory, functional analysis and topology. This spurred two particular directions of Ramsey theory: The first one is known as topological Ramsey theory, a general procedure developed by Todorcevic to prove many seemingly unrelated Ramsey's theorem-like results. The second one is the Ramsey theory of Banach spaces, kickstarted by Gowers' shocking application of Ramsey theory to resolve a long-standing open problem in Banach space theory. In this talk, I introduce the theory of weak A2 spaces, which serves as a possible intersection between these two Ramsey theories and discuss how several infinite games that appeared in these Ramsey theories (the Kastanas game, the Gowers game and the asymptotic game) are closely related.

MC 5479