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.

Transformation Groups Seminar

Speaker: Mathieu Vallée (Université Libre, Brussels)

"Classification of toric manifold of small Picard number"

Time: 14:30

Room: MC 108

Toric varieties form a specific class of algebraic varieties equipped with a well-behaved action of an algebraic torus. They provide a useful setting for testing conjectures, as they admit a particularly explicit and combinatorial description. The fundamental theorem of toric geometry states that toric varieties correspond to fans, that is, sets of strongly convex polyhedral cones in $\mathbb{R}^n$ that are closed under taking faces and whose relative interiors are pairwise disjoint. Properties of the fan translate directly into geometric properties of the associated toric variety. In particular, a toric variety is complete if and only if the cones of the fan cover the whole space $\mathbb{R}^n$, and it is non-singular if and only if each cone is generated by part of a basis of the integer lattice $\mathbb{Z}^n$. We focus here on characterizing complete non-singular toric varieties, also called toric manifolds. The Picard number of a toric manifold is the rank of its Picard group; this equals the number of the 1-dimensional cones minus the dimension of its associated fan. There are two major directions of research toward this characterization: studying toric manifolds of fixed (small) dimension, or studying those with fixed (small) Picard number. In dimension 2, toric manifolds are completely understood: they are obtained by a sequence of toric blow-ups startying either from the complex projective plane or from a Hirzebruch surface. In any dimension $n$, the unique toric manifold of Picard number 1 is the complex projective space $\mathbb{C}P^n$, whose fan corresponds to the normal fan of a unimodular $n$-simplex. Kleinschmidt (1988) and Batyrev (1991) classified toric manifolds of Picard number 2 and 3, respectively. In this talk, I will present a sequence of joint works with Suyoung Choi and Hyeontae Jang leading to the classification of toric manifolds of Picard number 4 in terms of fans, relying mainly on a combinatorial construction known as the (simplicial) wedge operation; which was used for instance by F. Santos in his construction for disproving the Hirsch conjecture. I will also discuss recent advances toward the case of Picard number 5.

Transformation Groups Seminar

Speaker: Xin Fu (SIMIS, Shanghai)

"Szczarba's twisted shuffle map and equivariant path homology of directed graphs"

Time: 14:30

Room: MC 108

Inspired by the GLMY path homology theory of directed graphs and its generalisations to quivers and marked categories, we associate a path chain complex to a marked simplicial set and define its path homology. In this talk, I will introduce a Borel-type construction for marked simplicial sets equipped with simplicial group actions and twisting functions. This construction is given by a marked version of the twisted Cartesian product using the box product.

A classical theorem of Szczarba states that the twisted shuffle map induces a quasi-isomorphism between the chain complex of a twisted Cartesian product and an associated twisted tensor product. In the marked setting, we prove that this map restricts to an isomorphism of path chain complexes. As an application, digraphs with group actions admit a natural Borel construction as a special case of our framework. This leads to a notion of equivariant path homology, which can be computed via an explicit twisted tensor product. This is joint work with Shing-Tung Yau.

PhD Thesis Defence

Speaker: Tao Gong (Western)

"Toric varieties and Weyl groups"

Time: 12:30

Room: MC 204

We study toric varieties associated with $W$-permutohedra and their quotients by parabolic subgroups. For a Weyl group $W$ and a $W$-permutohedron $P$, the quotient $P/W_K$ by a parabolic subgroup $W_K$ can be identified with a polytope inside $P$, giving rise to toric varieties $X_P$ and $X_{P/W_K}$.

We construct an explicit algebra isomorphism between the rational cohomology ring of $X_{P/W_K}$ and the invariant of the rational cohomology ring of $X_P$$$H^*(X_{P/W_K};\mathbb{Q})\cong H^*(X_P;\mathbb{Q})^{W_K},$$ and generalize this result to non-degenerate $W$-symmetric polytopes, to intermediate lattices, and to finite Coxeter groups via a polytopal algebra model.

From a topological perspective, we prove that $X_P/W_K$ is homotopy equivalent to $X_{P/W_K}$, and extend this result to non-degenerate $W$-symmetric polytopes. In joint work with Crowley and Simpson, we further establish a variety isomorphism between $X_P/W_K$ and $X_{P/W_K}$, and generalize this isomorphism to $W$-symmetric polytopes.

These results give affirmative answers to questions of Horiguchi--Masuda--Shareshian--Song concerning equivalences between $X_P/W_K$ and $X_{P/W_K}$.

Finally, we study the real locus $X_P^{\mathbb{R}}$ of the toric variety $X_P$. We show that the quotient $X_P^{\mathbb{R}}/W$ is contractible, and describe the homotopy type of $X_P^{\R}/W_K$ in low-dimensional cases.

PhD Thesis Defence

Speaker: Yucen Jin (Western)

"Delay-Induced Bifurcations and Complex Dynamics in a Host–Competitor–Parasite Model"

Time: 13:00

Room: MC 204

This thesis investigates the dynamics of a host–parasite–competitor model with biologically motivated time delays. By incorporating a parasite maturation delay, the delayed system exhibits dynamical behaviours that are absent in the corresponding ordinary differential equation model.

Using linear stability and bifurcation analysis, conditions for Hopf, generalized Hopf (Bautin), and Hopf–zero bifurcations are established, providing analytical criteria for the stability and local bifurcation structure of the delayed system. Numerical simulations, together with bifurcation diagrams, Lyapunov exponent estimates, and return maps, are used to investigate the post-Hopf dynamics of the delayed system.

Overall, this thesis presents a theoretical and numerical investigation of delay-induced dynamics in host–parasite–competitor systems. The results demonstrate how biologically motivated time delays influence the stability and bifurcation structure of the model, providing new insights into the dynamics of ecological systems with delayed interactions.

PhD Thesis Defence

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.

MSc Thesis Defense

Speaker: Meagan James (Western)

"Automorphisms of the fine separating curve graph"

Time: 14:00

Room: MC 107

Inspired by the work of Long, Margalit, Pham, Verberne, and Yao, we prove that the fine separating curve graph of a surface of genus at least 3 has automorphism group isomorphic to the group of self-homeomorphisms of the surface. To obtain this result, we prove that the natural map from the group of homeomorphisms to the group of automorphisms of the fine inessential curve graph is an isomorphism. The first result is analogous to that of Brendle and Margalit on the separating curve graph and its relationship to the mapping class group. The main results of this project serve as evidence in partial support of an Ivanov-type metaconjecture in the context of the homeomorphism group.

In this presentation, we will introduce surfaces and curves, the extended mapping class group and its action on curve graphs, Ivanov's metaconjecture, analogous results on fine curve graphs, and the Ivanov-type metaconjecture for the group of homeomorphisms. We will then encode points in a surface using convergent sequences and configurations of separating curves and show that these are preserved by automorphisms of the fine separating curve graph. In other words, we will show that, despite being restricted to separating curves, the graph-theoretic data of the fine separating curve graph is sufficient to encode the topological properties of the surface and is preserved by automorphisms of the graph, allowing us to prove the desired group isomorphism.

Geometry and Topology

Speaker: Fredrik Bakke (Norwegian University of Science and Technology)

"Some results in constructive cardinal theory"

Time: 15:30

Room: MC 107

While the basic study of cardinals are undeniably important in mathematics, they are surprisingly ill-behaved in the constructive setting, where they are currently poorly understood. In this seminar, I will present a series of novel constructive results on cardinals, including a constructivization of the Cantor-Schröder-Bernstein theorem,a predicative proof of the uncountability of the (MacNeille) reals, and, if time permits, a constructivization of König’s theorem. Along the way, I will highlight aspects and challenges in the study of constructive cardinals, as well as some ways to tackle them.

MSc Thesis Defense

Speaker: Mahajabin Mohsin (Western)

"Restricted Horizontal Transmission Reverses the Effect of Host Mixing on Virulence Evolution"

Time: 09:30

Room: MC 204

Understanding how host mixing influences virulence evolution is important for predicting disease emergence and spread. Previous theoretical studies have shown that increased host mixing favors higher pathogen virulence, whereas vertical transmission tends to select for more prudent host exploitation. In this thesis, we investigate whether this prediction remains valid using a model with different pathogen transmission scales. Specifically, we assume that horizontal transmission occurs at a local scale within patches, while vertical transmission occurs at a global scale across the population. Our model predicts that the relationship between host mixing and virulence can be reversed when horizontal transmission is locally restricted. We propose that this reversal occurs because lower virulence allows pathogens to exploit local host populations for longer periods. Overall, our key assumptions emphasize that virulence evolution with host mixing depends not only on the pathogens' transmission mode, but also on the spatial structure of the host population.

MSc Thesis Defense

Speaker: Juhee Kim (Western)

"Recovering Permutation Groups from an Error-Prone Sampler"

Time: 13:00

Room: MC 107

This thesis studies the recovery of an unknown permutation group $G \leq S_n$ from independent observations of an error-prone sampling process that samples uniformly from $G$ with probability $1-p$, and samples uniformly from $S_n$ with probability $p$. This model is motivated by the numerical computation of monodromy groups using homotopy continuation, where path-tracking errors may result in faulty monodromy permutation computations. We study the effects of these errors under the simplifying assumption that the distribution induced by path-tracking errors is the uniform distribution on $S_n$. We first establish the fragility of the standard approach for computing $G$ and, applying results from probabilistic group theory, present our own algorithms which can reliably recover $G$ even when the error rate $p$ is high.

MSc Thesis Defense

Speaker: Aditya Rao (Western)

"Control and computation in nonlinear oscillator networks"

Time: 13:30

Room: ZOOM

Nonlinear oscillator networks are of widespread interest in physics, chemistry, biology, engineering, and machine learning. Here, we study the dynamics of nonlinear oscillator networks in two settings. First, we develop a mathematically exact method for controlling the Kuramoto-Sakaguchi model of coupled oscillator networks, capable of steering such networks to a rich variety of dynamical states, such as synchrony, desynchrony, chimera states, and phase-locked attractors of any configuration. Then, we formulate a recurrent neural network for machine learning, related to the Kuramoto-Sakaguchi model, and analyze its input-driven dynamics in terms of a spectral decomposition as the network performs an image sequence prediction task. These results contribute to the theory of nonlinear oscillator networks and to their potential for mathematically grounded applications.

PhD Thesis Defence

Speaker: Anif Shikder (Western)

"TBA"

Time: 13:00

Room: MC 204