Ph.D. Candidacy Exam Lecture

Speaker: Saleh Ahmed (Western)

"Failure of Strong Approximation for the Moduli Stack of Elliptic Curves"

Time: 13:00

Room: MC 108

A classical problem in arithmetic geometry is determining when the global properties of a space over a number field are determined by its local behavior at all completions. This local-to-global relation can be topologically formulated via strong approximation, which asks whether the rational points of a space are dense in its adelic points away from a specified finite set of places. To make this topological notion of density meaningful, one requires a functorial topology on adelic points that behaves compatibly with the geometric nature of the objects of study. Conrad provides a construction of such a topology on the adelic points of separated and finite type algebraic spaces. In a recent work, Dhillon extended this framework to certain algebraic stacks, enabling the study of strong approximation in the stack-theoretic setting. In this talk, we investigate strong approximation for the moduli stack of elliptic curves $\mathcal{M}_{1,1}$. Since its coarse moduli space is the affine line $\mathbb{A}^1$, a space which satisfies strong approximation, one might expect $\mathcal{M}_{1,1}$ to inherit this property. We show that this intuition is false. Using a quotient stack presentation of $\mathcal{M}_{1,1}$ and analyzing the image of the $j$-invariant, we construct a non-empty open subset of the adelic points of $\mathcal{M}_{1,1}$ containing no rational points, hence proving that $\mathcal{M}_{1,1}$ fails to satisfy strong approximation away from the places $\{2, 3, \infty\}$.

PhD Thesis Defence

Speaker: Jeremy Gamble (Western)

"Random Matrices and Schwinger-Dyson Equations for Dirac Ensembles with Fermions"

Time: 10:00

Room: MC 107

Dirac ensembles lie at the intersection of random matrix theory and noncommutative geometry. In particular, we look at the (0,1) fuzzy Dirac ensemble with fermion in the case where the fermion mass is positive. We give an introduction to random matrix models, both analytic and formal, including the relationship between matrix integrals and generating functions of maps in the formal case. We also motivate and introduce the concept of a spectral triple from noncommutative geometry. One application of spectral triples is Dirac ensembles, consisting of a certain finite spectral triple with a probability distribution on the space of Dirac operators. Dirac ensembles describe a toy model of quantum gravity. We describe Dirac ensembles and their underlying hermitian matrix models, giving special attention to the (0,1) fuzzy Dirac ensemble with a fermion. The Schwinger-Dyson equations for this model can be found through complex analytic techniques, and allows for a perturbative solution of the moments. In the case where the potential is Gaussian, an exact solution for moments can be obtained using complex analytic techniques and elliptic integrals.