Graduate Student Colloquium

Date and time:  Tuesday, April 30th from 3:30 to 4:30 pm

Location: STEM 664

Speaker: Khalil Besrour

Title: A Tale of Modular Foms and L-functions

Abstract: Modular forms are fundamental mathematical entities with applications spanning various domains, including number theory, representation theory, and mathematical physics. Their significance is notably highlighted in the proof of Fermat’s Last Theorem, attributed to the Shimura-Taniyama-Weil conjecture asserting the modularity of elliptic curves. A vital link between modular forms and arithmetic is established through L-functions, exemplified by the Riemann ζ-function. In this Colloquium, we will be introducing the attendees to the world of modular forms and their applications.

Bio: Khalil Besrour is a 3rd year PhD student working under the supervision of Abdellah Sebbar. His research area covers modular forms (classical, vector valued and p-adic), elliptic curves and equivariant functions.

Date and time:  Thursday, March 21st from 4:30 to 5:30 pm

Location: STEM 664

Speaker: Autumne Cadieux

Title: Got (too much) milk? Investigations into the treatments and indications of iron deficiency anemia in children.

Abstract: Have you ever considered if you drink way too much milk? Well for the average adult, this doesn’t usually cause too many problems, but for a young child, this can lead to a lack of iron storage within their bodies, and this can lead to iron deficiency anemia (IDA). But what do we do when a child presents with IDA? What are the common treatments, and which is most effective for each individualized case? The goal of this research work was to investigate the key indicators and treatments of IDA to support evidence-based decision making for clinicians treating IDA in the pediatric emergency department.

Bio: I am a second year MSc Statistics student, aiming to submit my thesis in the coming weeks. My work centers around applied biostatistics, particularly within the field of pediatric healthcare. I am interested in many facets of applied statistics, specifically with applications to Indigenous well-being, economic prosperity and healthcare, a field in which I would like to explore in future work.

Date and time: Tuesday, February 13th  from 4:00 to 5:00 pm.

Location: STEM 664

Speaker: Daniel Dallaire

Title: A category linking planar graph colouring to sl_2-representation theory

Abstract: The chromatic category is a diagrammatic monoidal category which encodes information about the proper colourings of planar graphs. Although it is related to graph colouring from its usual definition, it can also be constructed in a representation theoretic context as well. This yields an interesting connection between the problem of colouring planar graphs (in particular, the four colour theorem) and the representation theory of the lie algebra sl_2. In this talk, I will introduce the chromatic category and discuss how it can instead be constructed as a category of representations.

Bio: I am a 2nd year M.Sc. student in the department of Mathematics and Statistics here at the University of Ottawa with a background in math, stats, and computer science. My current research interests involve string diagrams, categorification, and representation theory, however I am interested in a variety of math and stats topics.

Date and time: Thursday, December 7th  from 5:00 to 6:00 pm.

Location: STEM 464

Speaker: Joe Ghafari

Title: Introduction to stochastic partial differential equations

Abstract: Stochastic partial differential equations (SPDE) are PDE with ”noise”. A very large class of singular SPDE remained open problems and unsolved for decades, until 2011, when Martin Hairer solved the KPZ equation using rough paths theory and later, in 2013, he introduced a new theory of ”regularity structures” allowing to give a rigorous meaning and solve many SPDE arising from physics and consequently earning a medal fields for his work. In this talk, we give a gentle introduction on SPDE. After defining them, we will outline examples and challenges of singular SPDEs and along the way we discuss formally the sub-criticality of these equations. Lastly, we present briefly three techniques allowing to solve SPDE: Martin Hairer theory of regularity structures, Paracontrolled distributions (Gubinelli, Imkeller and Perkowski), and renormalization group (Kupiainen). An effort has been made to keep the exposition concise and self-contained.

Bio: I am a second PhD student at the university of Ottawa, co-supervised by Jeremy Quastel (university of Toronto) and Aaron Smith (university of Ottawa). My research interests are: stochastic calculus and stochastic partial differential equations.

Date and time: Thursday, November 16th, from 4:00 - 5:00 PM

Location: STEM 464

Speaker: Omar Cayley

Title: How L_\infty is isomorphic to ℓ_\infty

Abstract: In functional analysis, we know that the space $L_p$ is not isomorphic to $ℓ_p$ for every $p\geq 1$ except for $p = 2$. It is somewhat a surprising fact that the Banach spaces $L_\infty$ and $ℓ_\infty$ are isomorphic. In this talk, we will prove this using a method that was first introduced by the Polish mathematician Aleksander Pełczyński in 1958. This result illustrates another consequence of the axiom of choice.

Bio: I am a first year PhD student. My research focuses on nonlocal games, or quantum information theory generally. I am glad to be serving as a committee member of the MSGSA for the first time. In my free time I like to watch the news or play chess, I also enjoy hiking and exploring new grounds.

Date and time: Friday, September 15th, from 4:00 - 5:00 PM

Location: STEM 464

Speaker: Maruša Lekše

Title: The spouse-loving variant of the Oberwolfach problem

Abstract: At the conference in Oberwolfach there was a tradition for participants to have dinner together each evening of the conference. In 1967 Gerhard Ringel asked the following question: given a room with t round tables of sizes ℓ1, . . . , ℓt , is it possible for people to sit around those tables in such a way that over an appropriate amount of meals, every person sits next to every other person exactly once? Or equivalently, given integers ℓ1, . . . , ℓt ≥ 3 , with n = ℓ1 +. . .+ ℓt being an odd integer, does there exist a decomposition of the complete graph Kn into copies of a graph F, where F is a disjoint union of t cycles of lengths ℓ1, . . . , ℓt? 

A lot of results have been obtained on this problem and its many variants, but in general it still remains unsolved. In this talk, we will focus on one of the recently studied variants - the spouse loving variant, in which the participants at the conference are couples, and every person wants to sit next to every other person exactly once, except for their spouse, next to whom they want to sit exactly twice. OP +(ℓ1, . . . , ℓt) has been previously solved for the case when all ℓi are even, the case when ℓ1 = . . . = ℓt , as well as the two-table case when one table has size 3, and we will look at some of these constructions. We will then present a new result on the two-table spouse-loving variant.

Bio: Maruša Lekše is a second year Ph.D. student at the University of Ljubljana, Slovenija. This summer she worked on the Oberwolfach problem under the supervision of Prof. Mateja Sajna. Besides mathematics, she enjoys hiking, diving and playing board games.

Date and time: Wednesday, July 26th, from 4:00 - 5:00 PM 

Location: STEM 664

Speaker: Antoine Velut

Title: Introduction to p-adic numbers and local fields

Abstract: The p-adic numbers have come to play a central role in modern number theory, with applications in class field theory, Iwasawa theory, Hodge theory and many others. Moreover, they are accessible right from the undergraduate level. Yet they are not a part of the usual mathematics courses, so they might still be unfamiliar to many.

In this talk, I will give an elementary exposition of their construction, while focusing on giving visual intuition of what this non-archimedean world looks like. I will then present some properties of the more general local fields, which encompass the p-adics, and explain some of the links with algebraic number fields.

Bio: Antoine Velut is a graduate student at the ENS de Lyon in France. He is at uOttawa for a summer internship under the supervision of Antonio Lei.

Date and time: Wednesday, June 28th, 2023, from 4:00 - 5:00 PM

Location: STEM 664

Speaker: Mihir Deo 

Title: Factorization of unbounded p-adic L-functions 

Abstract: p-adic L-functions are the power series with coefficients in the p-adic rings like Z_p and they p-adically interpolate special values of classical L-series attached to arithmetic objects such as modular forms. For example, let f = Σ a_n q^n be a modular form of weight k ≥ 2 which is p-ordinary, i.e. p does not divide its p-th Fourier coefficient a_p. Then, the p-adic L-function L_(p,f) attached to f is an element of the Iwasawa algebra Q_p ⊗ Z_p[[T]]. One could ask what happens if p | a_p. Then we get two p-adic L-functions L(p,f,α_i), i ∈ {1, 2}, where α_i are roots of X^2 − a_p X + p^(k−1). In this case, both power series have unbounded denominators and are no longer elements of Iwasawa algebra. In general, let F_α, F_β be two power series over a finite extension of the field of p-adic numbers Q_p satisfying certain interpolation formulae. Suppose further that the coefficients of the power series have unbounded denominators satisfying certain growth condition. In this talk, we will discuss the decomposition of F_α and F_β into linear combinations of two power series with integral coefficients. This is a part of my ongoing project which deals with the factorization of two variable p-adic L-function attached to a Bianchi modular form. 

Bio: Mihir Deo is a PhD student under the supervision of Prof. Antonio Lei. Mihir did his undergraduation in Electronics engineering from University of Mumbai and then M.Sc. in Mathematics from IIT Gandhinagar. His research interests are p-adic L-functions and Iwasawa theory, arithmetic aspects of automorphic forms and p-adic Hodge theory.

Date and time: Thursday, March 23rd, from 4:30 - 5:30 PM

Location: STEM 664

Speaker: Masoomeh Akbari

Title: On the generalized Honeymoon-Oberwolfach Problem

Abstract: One of the most interesting problems in graph theory is the Oberwolfach Problem (OP), which was first presented by Gerhard Ringel in 1967. The OP asks whether it is possible to seat the n participants at a conference at t round tables of sizes m_1, m_2, … , m_t, where m_1+m_2+....+m_t = n, for several consecutive nights so that each person sits beside every other person precisely once.

In my thesis, I focus on a variant of the Oberwolfach Problem called the Honeymoon Oberwolfach Problem (HOP), which was introduced by Sajna. In this variation, we have 2n participants consisting of n newlywed couples. We want to see whether it is possible to seat them at t round tables of specified sizes for several consecutive nights so that each participant sits next to their spouse every time and next to each other participant exactly once. Some cases of HOP have been solved, and results have been published recently. However, some interesting and important cases are left unsolved. Moreover, HOP has been defined with the constraint that each table size is at least 4. In this research, I aim to generalize the problem to allow for tables of size two.

Bio: Masoomeh Akbari is a third-year Ph.D. student in mathematics. She works in the area of Graph Theory under the supervision of Prof. Sajna. She is doing research on the honeymoon Oberwolfach problem as her Ph.D. thesis. Besides academia, she enjoys traveling, walking, a warm cup of tea, and socializing with friends.

Date and time: January 26th 2023, 4:30pm - 5:30pm

Location: STEM 464

Speaker: Samuel Desrochers

Title: If you’ve heard of Universal Properties, then you can understand Adjoint Functors

Abstract: Have you ever come across the notion of a “universal property”, and wondered what that really means? Or have you heard a category theorist talk about “adjoint functors”, but failed to understand what they are or why they’re interesting? In this presentation, I’ll address both of these questions! Using examples of universal properties as motivation, I’ll explain what adjoint functors are, how they come about, and why people study them. No prior knowledge of category theory is necessary, but knowing some examples of universal properties might help you relate more to the presentation. 

Date and time: December 1st 2022            

Location: STEM 664

Speaker: Lord Kavi

Title: Hamming Graphs and 3-independent sets.  

Abstract: An independent set, also known as a stable set or coclique, in a graph is a set of vertices, no two of which are adjacent. The size of a largest independent set is called the independence number. A k-independent set in a graph is a set of vertices such that any two vertices in the set are at distance at least k+1 in the graph. The k-independence number of a graph is the size of a largest k-independent set in the graph.

Using interlacing, Abiad et al generalized the Hoffman spectral bounds on k-independence, which involves taking polynomials of degree at most k. By getting the right choice of a polynomial we present a spectral bound on the 3-independence number of graphs and apply this bound to well-known families of graphs. We investigate tightness of this bound on the Hamming graph H(d; q). In particular, we give a construction of 3-independent sets in H(d,2) and show tightness of the bound for d = 2^r and d = 2^(r-1) with r in Z+.