at 12:15pm
in M1060

(University of Lethbridge)

Please bring your favourite (math) problems. Anyone with a problem to share will be given about 5 minutes to present it. We will also choose most of the speakers for the rest of the semester.

at 12:15pm
in M1060

This talk focuses on the inverse eigenvalue problem for graphs (IEPG), which seeks to determine the possible spectra of symmetric matrices associated with a given graph $G$. These matrices have off-diagonal non-zero entries corresponding to the edges of $G$, while diagonal entries are unrestricted. A key parameter in IEPG is $q(G)$, the minimum number of distinct eigenvalues among such matrices. The Johnson and Hamming graphs are well-studied families of graphs with many interesting combinatorial and algebraic properties. We prove that every Johnson graph admits a signed adjacency matrix with exactly two distinct eigenvalues, establishing that its $q$-value is two. Additionally, we explore the behavior of $q(G)$ for Hamming graphs. This is a joint work with Shaun Fallat, Allen Herman, and Johnna Parenteau.

at 12:15pm
in M1060

The study of the cohomology of line bundles on (partial) flag varieties is an important problem at the intersection of algebraic geometry, commutative algebra, and representation theory. Over fields of characteristic zero, this is well-understood thanks to the Borel-Weil-Bott theorem, but in positive characteristics, it remains largely open.

In this talk, I will focus on the incidence correspondence, the partial flag variety parameterizing pairs consisting of a point in projective space and a hyperplane containing it. I will describe joint work with C. Raicu, A. Kyomuhangi, and E. Reed, where we establish a recursive formula for the characters of the cohomology of line bundles on the incidence correspondence in positive characteristic.

Finally, I will highlight how this problem is unexpectedly connected to other open questions in positive characteristic. In particular, I will explain how our work leads to a better understanding of the Han-Monsky representation ring, the ring of isomorphism classes of finite-length graded $k[T]$ modules.

at 12:15pm
in M1060

(University of Lethbridge)

The prime number theorem proven independently by de la Vallée Poussin and Hadamard (1896) is an asymptotic statement about prime counting functions. It holds for sufficiently large numbers $x$. In 1941 Rosser authored an article giving explicit versions of the prime number theorem which holds for $x$ in various ranges. This work was later updated in 1962 and 1975, in joint work of Rosser and Schoenfeld. In recent years there has been a flurry of activity on this subject with contributions made by Dusart, Faber-Kadiri, Büthe, Fiori-Kadiri-Swidinsky, Johnston-Yang, and Chirre-Helfgott. Very recently, Tao has initiated the Integrated Explicit Analytic Number Theory network which has the goal to formalize many results in this field. Some of the key ideas that are used include a partial verification of the Riemann Hypothesis, explicit zero-free regions, explicit zero-density, explicit zero-counting formulae, and optimal functions. In this talk, I will provide a survey and history of results in explicit prime number theory. I will also present recent new bounds on Mertens sums and products which is joint work with Broadbent, Fiori, Kadiri, and Wilk.

at 12:15pm
in M1060

Vertex operator algebras (VOAs) are algebraic objects that arose in the study of infinite dimensional lie algebras, mathematical physics, and in the classification of finite simple groups. These days they are understood to give rise to vector bundles on moduli spaces of algebraic curves that are useful in a variety of areas of mathematics and physics. In number theory one frequently encounters them via their incarnation on modular curves. In this talk we will recall background on VOAs and modular forms, and we will give a concrete description of the corresponding VOA bundles in terms of modular forms. We will also describe their connection with quasi-modular forms, which arises naturally from the VOA structure.

at 12:15pm
in M1060

(University of Lethbridge)

Understanding how network structure gives rise to spatiotemporal dynamics and computation is a central challenge in computational neuroscience and artificial intelligence. Despite increasingly detailed connectomic data in neuroscience and large-scale datasets in machine learning, establishing principled links between connectivity, dynamics, and function in nonlinear neural systems remains difficult. In this talk, I will present a mathematical framework that directly relates network architecture to emergent dynamical patterns and computational capabilities in analytically tractable models. Our approach focuses on networks of coupled oscillators, which are widely used to model interacting neural populations and have recently gained interest as computational substrates in artificial neural networks. With this approach, we can show how key structural features of these networks — including connectivity patterns and transmission delays — determine the emergence and stability of spatiotemporal activity, enabling analytical predictions of collective phenomena such as traveling waves. When applied to empirically derived brain networks, the framework provides a rigorous connection between large-scale anatomy, distance-dependent delays, and wave dynamics observed at mesoscopic and whole-brain scales. Building on these results, we introduce a new class of neural networks that leverage structured spatiotemporal dynamics for computation while remaining exactly solvable. Together, these results outline a general strategy for linking network structure, emergent dynamics, and computation, with implications for understanding neural activity and for developing interpretable dynamical models for neural computation.

at 12:15pm
in M1060

L-functions appear as generating functions encapsulating information about various objects such as Galois representations, elliptic curves, arithmetic functions, modular forms, Maass forms, etc. Studying L-functions is therefore of utmost importance in number theory at large. Two of their attached data carry critical information: their zeros, which govern the distributional behavior of underlying objects; and their central values, which are related to invariants such as the class number of a field extension. We will discuss the important conjectures, one concerning the distribution of the zeros and one concerning the distribution of the central values, and explain a general principle that any restricted result towards the first conjecture can be refined to show that most corresponding central values have the typical distribution predicted by the second conjecture. We will instantiate this general principle for a wide class of L-functions, and provide a more detailed discussion in the case of L-functions attached to modular forms.

at 12:15pm
in M1060

A $c$-colouring of a combinatorial design is an assignment of colours, chosen from a set of size $c$, to the points of the design. A $c$-colouring is equitable if every block of size $k$ contains $\lfloor k/c \rfloor$ or $\lceil k/c \rceil$ points of each colour. In 2016, Luther and Pike characterized $c$-colourable balanced incomplete block designs (BIBDs), proving that nontrivial equitable $c$-colourings exist only for a family of highly restricted parameter sets. By contrast, equitable colourings of cycle designs seem to exist more widely. In this talk, we review known results on equitable colourings and discuss some recent results on equitable colourings of various classes of design. Motivated by the scarcity of equitably-colourable BIBDs, we introduce the concept of a semi-equitable $c$-colouring, where in each block of size $k$, one colour which does not appear at all but each other colour appears on $\lfloor k/(c-1) \rfloor$ or $\lceil k/(c-1) \rceil$ points. Reporting on joint work with William Kellough and David Pike, we give necessary conditions for the existence of semi-equitably colourable BIBDs and present construction methods for such designs using Hadamard matrices, affine planes and twin prime powers.

at 12:15pm
in M1060

One of my favorite quotations in mathematics is due to Titchmarsh, who remarked: “The finer theory of the partial sums of the Möbius function is extremely obscure, and the results are not nearly so precise as the corresponding ones in the prime number problem.” In this talk, we show how certain extremal functions in Fourier analysis can be used to obtain strong bounds for the partial sums of the Möbius function and von Mangoldt function. This is joint work with Harald A. Helfgott (CNRS).