Ph.D. Candidacy Exam Lecture

Speaker: Vladimir Gorchakov (Western)

"Three-Dimensional Small Covers and Links"

Time: 14:00 - 15:00

Room: MC 107

We study certain orientation-preserving involutions on three-dimensional small covers. We prove that the quotient space of an orientable three-dimensional small cover by such an involution belonging to the 2-torus is homeomorphic to a connected sum of copies of $S^2 \times S^1$.

Geometry and Topology

Speaker: Kensuke Arakawa (Kyoto University)

"On Pavlov's conjecture on presentably symmetric monoidal $\infty$-categories"

Time: 15:30 - 16:30

Room: MC 107

A classical result of Dugger and Lurie says that presentable $\infty$-categories are precisely the $\infty$-categories that underlie combinatorial model categories.

There are (at least) two generalizations of this result in the literature:

- A symmetric monoidal version of Dugger-Lurie's theorem: Presentably symmetric monoidal $\infty$-categories are exactly those that underlie combinatorial symmetric monoidal model categories (Nikolaus-Sagave, 2017).

- Lifting Dugger-Lurie's theorem to the level of entire homotopy theories: The homotopy theory of combinatorial model categories is equivalent to that of presentable $\infty$-categories (Pavlov, 2025).

As a natural meeting point of these directions, Pavlov conjectured that the homotopy theory of combinatorial symmetric monoidal model categories is equivalent to that of presentably symmetric monoidal $\infty$-categories. This would, for example, ensure the existence and uniqueness of (combinatorial) model-categorical presentations of important objects like spectra (with smash product) and operads (with the Boardman-Vogt tensor product).

In this talk, we will give an affirmative answer to this conjecture (and its monoidal version), explain the main ideas behind the proof, and sketch some applications to the theory of $\infty$-operads.

Transformation Groups Seminar

Speaker: Kumar Shukla (Western)

"Hochster's Decomposition"

Time: 09:30 - 11:00

Room: MC 108

The cohomology of a moment-angle complex is given by the Koszul homology of the Stanley-Reisner ring of the underlying simplicial complex. The Hochster decomposition gives an easy way to compute this Koszul homology from the combinatorics of the simplicial complex. In this talk, we will discuss Hochster's proof of the decomposition. Then, we will look at some 'colorful' generalizations of the decomposition which correspond to cohomology of certain partial quotients of moment-angle complexes.

Ph.D. Candidacy Exam Lecture

Speaker: Thomas Thorbjornsen (Western)

"Resolution-free derived functors in HoTT"

Time: 10:00 - 11:00

Room: MC 107

Classically, derived functors are computed by projective or injective resolutions. For any ring R, the category of R-modules has enough projective and injective objects by the axiom of choice. However, this axiom is not assumed in Homotopy Type Theory (HoTT), preventing us from using the standard resolutions. Building upon earlier work by Yoneda and Buchsbaum, we present a constructive resolution-free approach to derived functors. Using this framework, we give a constructive derivation of Tor in HoTT.

Ph.D. Candidacy Exam Lecture

Speaker: Ben Connors (Western)

"Formalizing the Small Object Argument"

Time: 13:00 - 14:00

Room: MC 107

A perennial question in type theory is how to do type theory within type theory. Rather than interpreting the syntax, which has proven very difficult even for simple type theories, one can attempt to internalize categorical models of type theory instead. We give a roadmap for formalizing the simplicial model of homotopy type theory (HoTT) inside HoTT, with particular emphasis on the first step: weak factorization systems and the small object argument.

Ph.D. Candidacy Exam Lecture

Speaker: Mac Martin (Western)

"Strong Approximation for the Classifying Stack of G-Torsors"

Time: 10:00 - 11:00

Room: MC 107

Strong approximation is a technique in arithmetic geometry to find a k-point of a variety over a number field which is v-adically close to a finite number of local points. It is desirable to understand what the obstructions to strong approximation are and when they vanish. In this talk we will review several important concepts of strong approximation before then showing how one is able to extend these ideas from varieties to algebraic stacks. We will then apply these concepts to BG, the classifying stack of G-torsors, and see that the etale-Brauer-Manin obstruction is the only obstruction in this case.

Ph.D. Candidacy Exam Lecture

Speaker: Deepak Sadanandan (Western)

"Monodromy solving and fibre-preserving maps"

Time: 10:30 - 11:30

Room: MC 107

Monodromy solving is the idea that for a parametrized polynomial system with a single known solution at a given parameter p, the other solutions can be found by tracking the solution along loops based at p. We extend this concept to the concept of fibre-preserving maps, and explore how adding in the option of tracking the solutions using fibre-preserving maps could help us make the algorithm faster.

M.Sc. Public Lecture

Speaker: Xiangyuan Liang (Western)

"Improved Criteria for Analyzing Slow-Fast Oscillations in Nonlinear Systems"

Time: 13:00 - 14:00

Room: MC 204

Recurrent oscillations in biological systems, particularly in models of persistent infections, often indicate underlying slow-fast dynamics. Classical approaches such as Geometric Singular Perturbation Methods (GSPM) rely on the existence of critical manifolds and are limited to systems with small perturbation parameters. However, these conditions are not always met in complex or higher-dimensional models. This thesis introduces a set of refined analytical criteria for detecting slow-fast oscillations beyond the scope of traditional perturbation theory. Applying this framework to a three-dimensional SIR model with secondary transmission and intrinsic population growth, we use bifurcation analysis, center manifold theory, and normal form analysis to identify conditions under which Hopf bifurcations and periodic solutions arise, even without asymptotically small growth rates. Numerical simulations support the theoretical findings, demonstrating the robustness and broader applicability of the proposed method in capturing oscillatory behavior across a wide range of parameter regimes.

M.Sc. Public Lecture

Speaker: Michelle Hatzel (Western)

"Homotopy Methods and Pseudozeros"

Time: 09:00 - 10:00

Room: zoom

In this work we study numerical homotopy continuation, a method used for solving nonlinear equations and nonlinear systems of equations. We focus on detecting and resolving error in a fixed step predictor-correct method for numerically solving univariate polynomials. We verify the homotopy paths with a Lipschitz condition to detect path jumping that leads to false multiplicities of roots and missed roots; we show that mutual path jumping cannot occur. At path ends we may obtain spurious roots, which are easy to detect for univariate polynomials when we check for non-zero residuals at the final path points. We show that pseudozeros---roots of nearby polynomials---provide another means of understanding error at path end points and give insight on how to optimize a homotopy continuation algorithm for a specific problem. We demonstrate that a change of basis can improve homotopy continuation results when finding ill-conditioned roots.

M.Sc. Public Lecture

Speaker: Victoria Quance (Western)

"Modeling the Effect of Hunting on Female Wild Turkey Behaviour during Incubation"

Time: 10:00 - 11:00

Room: MC 108

Wild turkeys (Meleagris gallopavo) were over-hunted and faced near extinction in the United States in the 1930s. Hunting practices have since adapted, but the US still experiences unexplained turkey population declines. Nest success is a primary driver of annual population change. Female turkeys (hens) are the sole caretakers of the nest, so factors affecting their behaviour influence population dynamics. Little is known about the effect of hunting on hens, as most hunting-related studies of turkeys focus on males. To investigate the effect of hunting on hens during their incubation period, we analyze GPS movement data from over 600 hens in sites across the southeastern US. Hunting occurs in most sites and is prohibited in one. Hen behaviour can be inferred from location data. We are interested in the ways in which hunting affects both the behaviour and physiological state of the hens. We develop a hidden Markov model, treating physiological state as the hidden variable and behaviour as the observed variable. The model assumes that behaviour varies with physiological state and hunting presence. We found that hunting has an effect on hen behaviour, and this effect changes depending on their physiological state. Further, the model predicts that, on average, hens in hunted sites leave their nest for longer consecutive periods than hens in non-hunted sites. Results suggest that hens in hunted sites place their nest at greater risk and prioritize their own survival. These results should be considered when developing hunting policies.

M.Sc. Public Lecture

Speaker: Jiayin Lyu (Western)

"APPLICATION OF COMPLEX-VALUED NEURAL NETWORKS FOR IMAGE RECONSTRUCTION AND INTERPRETABLE STATE SPACE MODELS"

Time: 12:00 - 13:00

Room: MC 107

Complex-valued neural networks (cv-NNs) have shown strong potential for tasks that rely on phase-based representations, short-term memory, and dynamic pattern selection — echoing ideas found in oscillator networks and wave patterns phenomena [3, 12]. Building on the previous work, Chapter 2 of this thesis demonstrates how a dynamic cv-NN can reconstruct partially occluded digit images by evolving toward one of several stable states, guided by the system’s oscillatory dynamics and eigenvalue structure [3]. This approach provides an interpretable mechanism for recovering missing information and highlights how complex eigenmodes can drive pattern completion in neural systems. This line of work naturally invites the question: how can such a framework be generalized to operate beyond specific matrix classes such as the HiPPO matrix [11] or other structured normal matrices? To address this, Chapter 3 establishes a new factorization result for diagonaliz-able complex orthogonal matrices [25], making it possible to extend the solvable cv-NN approach to arbitrary real matrices. This factorization bridges synchronization-inspired models with structured state-space representations, offering fresh insights into the spectral properties of HiPPO matrices and suggesting a unified view of oscillatory neural dynamics and modern sequential modeling.

Geometry and Topology

Speaker: Felix Cherubini (University of Gothenburg and Chalmers University of Technology, Sweden)

"Projective space in synthetic algebraic geometry"

Time: 15:30 - 16:30

Room: MC 107

The central objects of algebraic geometry, schemes, are an intricate notion which takes some effort to define - in synthetic algebraic geometry, schemes are just sets with an additional property, but the rules for manipulating sets are different than what is assumed in the classical theory.

More precisely, the language of synthetic algebraic geometry is homotopy type theory together with three axioms. We will explain these axioms and some essential definitions needed to describe schemes and the example of projective space. To show non-trivial facts about projective space, like the classification of line bundles, we will make use of the higher types available in homotopy type theory.