Dept Oral Exam

Speaker: Priya Bucha Jain (Western)

"Analytical Spectral Methods for Structure and Dynamics in Complex Networks"

Time: 13:00

Room: ZOOM

Networks provide a mathematical framework for studying complex systems across biology, physics, ecology, neuroscience, and many other fields. In this talk, I will present two projects from my thesis that use spectral graph-theoretic methods to understand both the structure and dynamics of complex networks. One part of the talk focuses on community detection, where communities represent groups of nodes that are more strongly connected to each other than to the rest of the network. I will present a new analytical approach for detecting community structure in directed weighted networks. Unlike many computational methods, this approach relies only on the eigenspectrum of the network adjacency matrix and does not require free parameters to be tuned or trained. I will describe the method, explain how the number of communities can be estimated from the spectral information, prove the validity of the approach in an ideal setting, and demonstrate its performance through numerical simulations on weighted graphs with random edge perturbations. The other part of the talk focuses on dynamics in multilayer networks of Kuramoto oscillators. The Kuramoto model describes synchronization processes in many natural systems, including interacting neural populations. I will show how the dynamics of a large multilayer Kuramoto system can be related to two smaller systems: one describing intra-layer interactions and the other describing inter-layer interactions. This decomposition makes it possible to construct solutions for the full multilayer system and study their linear stability. Together, these projects show how spectral methods can provide analytical insight into both the organization of networks and the collective dynamics they support.

Noncommutative Geometry

Speaker: Joshua Y. L. Jones (School of Theoretical Physics, Dublin Institute for Advanced Studies)

"A Conjecture for Lorentzian Spectral Geometry"

Time: 14:30

Room: MC 107

Lorentzian spectral geometry, as a field, has enjoyed much less progress than its Riemannian counterpart. I will suggest that the causal propagator (the difference between the retarded and advanced Green functions) is the appropriate operator to be spectrally considered on Lorentzian manifolds. I will present a conjecture that connects null geodesic lengths (in a sense that will be explained), and the eigenvalues of the causal propagator. This gives the leading term in the asymptotic scaling of the spectral density, in analogy with Weyl's law for the Laplace-Beltrami operator. This opens many avenues for work in Lorentzian spectral geometry. My talk will be based on: https://arxiv.org/abs/2606.00311.

Information Session: MITACS Opportunities

Speaker: Xiang Liang, Advisor, Business Development (Mitacs)

"Information Session: Mitacs Opportunities for Math, Statistics & Actuarial Students"

Time: 15:30

Room: MC 105B

This session will introduce paid, non competitive internship opportunities available to current students and recent graduates with strong backgrounds in mathematics, statistics, and actuarial science. The presentation will outline: •Opportunities open to graduate students and recent grads •Examples of how mathematical and statistical skills are applied in industry and research settings •Guidance on positioning academic training for early career roles This session is intended for students exploring practical career pathways beyond academia and seeking accessible entry points into internships and early career opportunities.