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3 PhD Thesis Defence
PhD Thesis Defence Speaker: Siyuan Yu (Western) "Symplectic embeddings of five balls into the complex projective plane" Time: 13:00 - 14:00 Room: MC 107 In 1985, Gromov proved the remarkable Non-Squeezing Theorem: a symplectic embedding of a closed ball of radius r into the symplectic cylinder $Z^{2n}(R)=B^2(R)\times \mathbb{R}^{(2n-2)}$ exists if and only if $r\le R$. This show that in general, symplectic embeddings have more obstructions than volume constraints. One of the obstructions is known as the Gromov’s capacity. In this thesis, we study the problem of symplectic embeddings of five disjoint closed balls of Gromov’s capacities $c_1,…,c_5$ into the complex projective space $\mathbb{C}P^2$. By investigation of the action of the group of symplectomorphisms of $\mathbb{C}P^2$ on the space of symplectic embeddings, the homotopy type of the space of symplectic embeddings can be computed: it is homotopy equivalent to a union of strata in the configuration space of five points on $\mathbb{C}P^2$, with the precise strata determined by the chosen capacities. Moreover, the homotopy type of the space of symplectic embeddings of five balls remains constant as the capacities vary within any given stability chamber of capacities. The complete set of stability chambers is also determined. Mathematics Departmental Presentation 2026
Mathematics Departmental Presentation 2026 Speaker: Thomas Thorbjørnsen (Western) "Finitely Adequate Modules in Synthetic Algebraic Geometry" Time: 15:30 - 16:30 Room: MC 108 Synthetic algebraic geometry (SAG) is an extension of homotopy type theory that provides a language for internal reasoning about the big Zariski topos. In SAG, we postulate the existence of a generic local ring R with some additional properties. Schemes over R are not defined by giving the underlying space a structure sheaf; rather, they are defined by a property of the space itself. Sheaves on a scheme are then expressed as bundles over the scheme, and on the sheaves themselves we have many of the usual operations, such as taking cohomology. However, algebraic geometry often looks different from this internal point of view, compared to the classical external one. For instance, we can show that the generic local ring R is not Noetherian, and so the category of finitely presented R-modules is not abelian. In particular, the cohomology groups of sheaves of finitely presented R-modules may no longer be finitely presented. In this talk, we shall study the abelian closure of the finitely presented R-modules in the category of all R-modules, which we call the finitely adequate R-modules. We will characterize the finitely adequate R-modules which are injective and projective in this subcategory. Then, we prove that finitely adequate R-modules are closed under extensions. We hope that the category of finitely adequate R-modules gives us a suitable replacement for the category of finitely presented modules, so that the cohomology groups of finitely adequate sheaves are finitely adequate. |
4 Ph.D. Candidacy Exam Lecture
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Theo Chatzidiamantis Christoforidis (Western) Fixed Point Properties in Synthetic Homotopy Theory
Ph.D. Candidacy Exam Lecture Speaker: Theo Chatzidiamantis Christoforidis (Western) "Fixed Point Properties in Synthetic Homotopy Theory" Time: 10:00 - 11:00 Room: MC 107 In topology, results such as Brouwer's fixed point theorem are often not accessible from the homotopy-theoretic point of view, since they usually depend on more than just the homotopy type of a given space, and many are also not constructively provable. By working in the setting of homotopy type theory, we will see that studying the property "every self-map has a fixed point" provides a different, homotopy-invariant notion. After constructing counterexamples, we show that classifying spaces of non-Abelian finite simple groups satisfy this property. Along the way, we compute the homotopy groups of the space of maps into a classifying space in homotopy type theory. In particular, if G and H are finite groups, we will show that we can completely describe the space of maps between their classifying spaces constructively. Our results are formalised in the Rocq proof assistant.This talk is based on joint work with Dan Christensen.Ph.D. Candidacy Exam Lecture
Ph.D. Candidacy Exam Lecture Speaker: Zack Dooley (Western) "The Internal Language Conjecture" Time: 14:00 - 15:00 Room: MC 107 Dependent type theory is an alternative foundation to mathematics that has provided the basis for proof assistant software such as Rocq or Lean. This is enabled by the nice computational properties of type theory, but another valuable aspect of type theory is its syntax/semantics relationship with category theory. This relationship allows us to not just study our foundations using category theory, but also to study (higher) category theory using type theory. While we understand this relationship well in certain cases, such as with the lambda calculus and Cartesian closed categories, for many type theories, including most of those used in proof assistants, our understanding of this relationship is still incomplete. In this talk I will discuss the nature of this relationship and explain part of the internal language conjectures which posit an equivalence between certain type theories and higher categories. |
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11 Dept Oral Exam
Dept Oral Exam Speaker: Priya Bucha Jain (Western) "Analytical Spectral Methods for Structure and Dynamics in Complex Networks" Time: 13:00 - 14:00 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
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 - 15:30 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
Information Session: MITACS Opportunities Speaker: Xiang Liang, Advisor, Business Development (Mitacs) "Information Session: Mitacs Opportunities for Math, Statistics & Actuarial Students" Time: 15:30 - 16:30 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. |
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15 Geometry and Topology
Geometry and Topology Speaker: Kensuke Arakawa (RIMS, Kyoto University) "Towards the discrete homotopy hypothesis" Time: 13:30 - 14:30 Room: MC 107 The A-groups of graphs, named after Atkin, are the key invariants in discrete homotopy theory. A natural problem is to identify the $\infty$-category obtained by localizing the category of graphs at maps inducing isomorphisms on A-groups. The discrete homotopy hypothesis (DHH), a major conjecture in the homotopy theory of graphs, asserts that the resulting $\infty$-category is equivalent to that of spaces. What makes DHH interesting is that this conjecture is that it appears within reach, but not quite. On the one hand, there is substantial positive evidence, including Carranza-Kapulkin's recent work on a version of DHH for homotopy n-types. On the other hand, some basic questions, such as the existence of homotopy (co)limits of graphs, remain open.In this talk, I will explain what additional ingredients are needed to deduce DHH from the evidence currently available. Remarkably, the existence of countable homotopy products would already suffice. In addition, I will sketch a possible approach for constructing such products.Geometry and Combinatorics
Geometry and Combinatorics Speaker: Colin Crowley (University of Oregon) "Toric varieties modulo reflections" Time: 15:30 - 16:30 Room: MC 108 We show that the quotient of a complex projective toric variety by a finite group, generated by reflections of the associated lattice polytope, is another toric variety. This result generalizes results of Blume and of Gong on permutohedra, and positively answers a question of Horiguchi, Masuda, Shareshian, and Song. In a similar direction, we also study the topological quotient of the real points of a projective toric variety, and prove that the quotient is contractible in the permutohedron case. Joint with Tao Gong and Connor Simpson. |
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M.Sc. Public Lecture
M.Sc. Public Lecture Speaker: Jacob Ender (Western) "TBA" Time: 13:00 - 14:00 Room: MC 107 |
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