Video Content by Date
Speaker: David Chan
Higher algebraic K-theory is a powerful invariant which was originally defined for rings, but has since grown far beyond its initial scope to encompass increasingly rich and intricate settings. There are many different constructions of algebraic K-theory, reflecting the range of uses and perspectives encompassed by the theory. In this talk, I will describe a comparison between Waldhausen’s K-theory construction and Segal’s K-theory of symmetric monoidal categories. Precisely, given a symmetric monoidal category, we construct a Waldhausen category with an equivalent K-theory spectrum.
Speaker: Roberto Budzinski
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.
Speaker: Ben Ashby
Host–parasite interactions often resemble an evolutionary arms race, where each side must continually adapt just to keep up. These dynamics can produce oscillations in allele frequencies—often called Red Queen dynamics—that are a hallmark of host–parasite coevolution. Despite their prominence, we still have an incomplete understanding of how they influence broader evolutionary outcomes. Much of the existing theory has focused on their role in the evolution of sex and recombination, leaving their consequences for other life-history traits largely unexplored. In this talk, I will explore the mechanisms that generate coevolutionary cycles and the role of eco-evolutionary feedbacks in shaping them. I will then discuss how these cycles influence the evolution of parasite virulence.
Speaker: Cemeron Franc
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.
Speaker: David Hernandez-Aristizabal
The basement membrane (BM) is a nanoporous extracellular matrix that surrounds most tissues and blocks the passage of cells, primarily composed of covalently cross-linked collagen IV fibres and laminins. While BM breaching has traditionally been thought to be mediated by protease-mediated degradation, the failure of protease-targeting clinical trials to reduce metastasis suggests the existence of alternative protease-independent mechanisms. Recent studies indicate that invasive cells extend filopodia capable of remodelling plastic extracellular matrices. However, the covalent cross-links in collagen IV fibres are very strong, prompting the question of how filopodia might facilitate BM invasion. Collagen IV fibres undergo turnover---a dynamic process of protein renewal that may create transient weak spots in the BM. We hypothesise that filopodia exploit these weak spots during turnover to initiate and expand pores, enabling protease-independent invasion. We propose a mathematical biophysical model to test the plausibility of this mechanism using biologically relevant protruding and turnover conditions obtained from experimental observations in the literature. Invasive cells are represented as energetic biomembranes using geometric-surface partial differential equations, allowing the formation of filopodial protrusions, while the BM is modelled as a barrier with collagen IV cross-links that stochastically transition between active and inactive states. The results of the model contrasted with experimental observations identify two subpopulations of filopodia in invasive cells: thin, short-lived filopodia that contribute to global BM degradation, and long-lived, widening filopodia that locally stabilise and enlarge pores where invasion can eventually occur. Under suitable conditions, the model predicts that random turnover and filopodia can synchronise, leading to progressive pore enlargement. Further, pore enlargement arise from the collaboration of several filopodia entering and leaving the same region of the BM at different times. Although our results cannot demonstrate that this mechanism occurs in vivo, they place turnover as a plausible contributor to protease-independent invasion.
Speaker: Yang Hu
Enumerating vector bundles of a fixed rank over a given manifold is a classical question in topology. While vector bundles are stably computable via K-theory, in the unstable range they become much harder to detect.
In this talk, we will demonstrate how the orthogonal/unitary calculus of Weiss—a version of functor calculi—can be applied to enumerate unstable topological vector bundles. We will present counting results for complex vector bundles over complex projective spaces in the metastable range and, time permitting, introduce an equivariant version of the calculus theory along with some potential applications in equivariant geometry. The talk includes joint work with Hood Chatham and Morgan Opie, and with Prasit Bhattacharya.
Speaker: Francesca Ballatore
Understanding how cells migrate through confined environments is crucial for elucidating fundamental biological processes, including cancer invasion, immune surveillance, and tissue morphogenesis. The nucleus, as the largest and stiffest cellular organelle, often limits cellular deformability, making it a key factor in migration through narrow pores or highly constrained spaces. In this work, we introduce a geometric surface partial differential equation (GS-PDE) model in which the cell plasma membrane and nuclear envelope are described as evolving energetic closed surfaces governed by force-balance equations. We replicate the results of a biophysical experiment, where a microfluidic device is used to impose compressive stresses on cells by driving them through narrow microchannels under a controlled pressure gradient. The model is validated by reproducing cell entry into the microchannels. A parametric sensitivity analysis highlights the dominant influence of specific parameters, whose accurate estimation is essential for faithfully capturing the experimental setup. We found that surface tension and confinement geometry emerge as key determinants of translocation efficiency. Although tailored to this specific setup for validation purposes, the framework is sufficiently general to be applied to a broad range of cell mechanics scenarios, providing a robust and flexible tool for investigating the interplay between cell mechanics and confinement. It also offers a solid foundation for future extensions integrating more complex biochemical processes such as active confined migration.
Speaker: William Hornslien
Atiyah and Rees proved that the Chern classes and a mod 2 invariant, named the alpha invariant, classifies all rank 2 topological vector bundles on P3. They also showed that a construction by Horrocks provides algebraic representatives for all topological rank 2 bundles. However, Horrocks’ construction is non-explicit. The goal of this talk is to construct an algebraic rank 2 bundle on P3 with trivial Chern classes and non-trivial alpha invariant. We will use methods from motivic homotopy theory to construct an explicit algebraic description of the bundle. This is joint work with Jean Fasel.
Speaker: Joy Morris
Although we often introduce group theory to students using groups of symmetries, we tend to move quickly away from these intuitive representations into the realm of axioms and deductions. There are plenty of good reasons for this, not least of which is that the objects or pictures we study the symmetries of, often fail to give us effective intuition about basic properties of the symmetry group.
I will discuss research on representing groups as groups of symmetries of some basic combinatorial structures (graphs, directed graphs, graphs with special properties, posets, etc.) and how this approach can provide improved intuitive understanding of the group.
I will also discuss research on asymptotic results about symmetries of combinatorial objects. The overall lesson of this aspect of the research is that not only is symmetry rare, but even when some amount of symmetry is required or imposed, it is rare for additional symmetry to arise spontaneously.
Speaker: Paweł Morzywolek
We provide an inferential framework to assess variable importance for heterogeneous treatment effects. This assessment is especially useful in high-risk domains such as medicine, where decision makers hesitate to rely on black-box treatment recommendation algorithms. The variable importance measures we consider are local in that they may differ across individuals, while the inference is global in that it tests whether a given variable is important for any individual. Our approach builds on recent developments in semiparametric theory for function-valued parameters, and is valid even when statistical machine learning algorithms are employed to quantify treatment effect heterogeneity. We demonstrate the applicability of our method to infectious disease prevention strategies.
Speaker: Vinod Vaikuntanathan
Integer lattices play a central role in mathematics and computer science, with applications ranging from number theory and coding theory to combinatorial optimization. Over the past three decades, they have also become a cornerstone of modern cryptography.
In this talk, I will describe the evolution of lattices in cryptography: from the early use of lattices to break classical cryptosystems; to their application in designing new encryption and digital signature schemes with (conjectured) post-quantum security; and to their role in achieving long-standing cryptographic goals such as fully homomorphic encryption that allow us to compute directly on encrypted data.
The talk will not assume any prior background in cryptography.
Speaker: Trisha Lawrence
The global population with access to electricity is constantly increasing from 84 to 92 percent. However, as the world continues to advance towards sustainable energy targets, there still exist 900 million people living without access to electricity.
In this talk, we provide a modeling framework for analyzing mini‑grid project performance and evaluating the economic impact of battery energy storage in competitive electricity markets. Using a dataset of 104 rural mini‑grid installations, we estimate the probability of project success through both a Probit regression model and a Bayesian hierarchical model. Community ownership and the presence of storage systems emerge as statistically significant predictors, with Bayesian posterior estimates closely aligning with frequentist results while providing improved predictive stability.
Furthermore, we analyze the bidding behavior of the Alberta electricity market and construct a mixed‑integer self‑scheduling model that determines optimal charging and discharging strategies. Through numerical experiments we demonstrate how storage can enhance arbitrage profitability, influence market clearing prices, and support system reliability.
Our results highlight the value of combining statistical inference with optimization‑based modeling to guide investment decisions.
Speaker: Puneet Velidi
Advances in protein folding and structure prediction models have enabled new computational approaches to immunotherapeutic research by providing access to high-quality structural information at scale. In this talk, we present three core application areas. (1) Antigen structure prediction, where folding models are used to characterize the three-dimensional structure of viral, tumor-associated, and neoantigen targets in the absence of experimental data. (2) Antibody–antigen complex prediction, where multimeric and joint modeling approaches are leveraged to infer binding modes, paratope–epitope interactions, and structural determinants of specificity. (3) Immunogenicity prediction, where predicted structures are analyzed to assess surface accessibility, conformational stability, and geometric features that influence immune recognition. Together, these applications illustrate how protein folding models function not only as structure predictors, but as foundational components in quantitative pipelines for immunotherapeutic discovery and design.
Speaker: Kostya Druzhkov
Differential equations can be studied from a purely geometric point of view, translating many constructions from finite-dimensional differential geometry into their language. This approach helps to clarify such notions as symmetries, conservation laws, presymplectic structures, and others. However, a number of questions arise in this framework whose answers are either incomplete or currently unknown. In particular, the problem of defining the cotangent equation in terms of the intrinsic geometry of PDEs remains open. This problem is directly related to the Hamiltonian formalism for differential equations.
From an applied perspective, methods for constructing exact solutions of differential equations are of particular interest. One of the most powerful approaches is based on the study of solutions invariant under certain symmetries of the given equation. A question of practical importance in this context is how the systems describing such invariant solutions inherit geometric structures from the original system.
In this talk, I will explain how these two topics are brought together within a reduction mechanism, which in particular clarifies how Hamiltonian operators are inherited by systems describing solutions of a given equation that are invariant under some of its symmetries. To fully implement this mechanism, an interpretation of cotangent equations in intrinsic geometric terms is also required. This can be achieved in the case where the reduced system turns out to be finite-dimensional.
Speaker: Valentina Zapata Castro
Model categories provide a powerful framework for abstract homotopy theory, but their complexity often makes them difficult to classify. By focusing on finite categories, especially grids, we gain a combinatorial setting where the problem becomes explicit. In this talk, we explore model structures through weak factorization systems (WFS) on posets, which are in one-to-one correspondence with transfer systems and their duals, both introduced here. This perspective leads to a method for constructing model structures and a characterization theorem for finding weak equivalence sets in posets. Our approach offers a pathway towards classifying model structures in a controlled setting.
This is joint work with Kristen Mazur, Angélica Osorno, Constanze Roitzheim, Rekha Santhanam and Danika Van Niel.
