Past Seminars

Abstract: In this talk, I will introduce a new family of cohomology classes on the moduli space of curves, known as Theta-classes. These classes are indexed by two integers (r,s) and arise as Euler classes of certain vector bundles. They generalise several known examples, including Norbury’s class. We show that the descendant integrals of the Theta-classes are computed by topological recursion. As a consequence, we establish that their partition function is a tau function of the r-KdV hierarchy, generalising the Brézin–Gross–Witten tau function. Furthermore, we derive differential constraints satisfied by these tau functions: they are annihilated by a set of differential operators forming a representation of a W-algebra at the self-dual level. Based on joint works with V. Bouchard, N. K. Chidambaram, E. Garcia-Failde, and S. Shadrin. 

Abstract: By the Riemann-Hilbert correspondence, connections with regular singularities on complex algebraic curves are characterized by their monodromy data. This admits an generalization to the case of irregular singularities, involving generalized monodromy data known as Stokes data.

On the other hand, there is a notion of Fourier transform for irregular connections on the projective line: it acts in a very nontrivial way, typically changing the rank, number of singularities and pole orders of the connections.

In this talk, I will present a topological way to compute the Stokes data of the Fourier transform of a connection in terms of its Stokes data in a new class of cases, relying on work of T. Mochizuki. In particular, this gives explicit isomorphisms between the corresponding wild character varieties. This is joint work with Andreas Hohl. 

Abstract: For a wide class of ODEs, we provide sufficient conditions of existence and uniqueness of a fundamental system of solutions, with specified asymptotic behaviour, in wide sectors centered at an isolated singularity of the coefficients. These singularity can be general, not just poles. So, we are dealing with systems with not necessarily meromorphic coefficients. This is a joint work with G. Cotti and D. Masoero. 

References  G. Cotti, D. Guzzetti, D. Masoero: Asymptotic solutions for linear ODEs with not-necessarily meromorphic coefficients: A Levinson type theorem on complex domains, and applications. Journal of Differential Equations 428 (2025), 1-58. 

Abstract: The signature of a path, first introduced by Chen, is the holonomy of a universal translation invariant connection on Euclidean space and it provides a transform whereby paths are represented by non-commutative power series. Chen proved that the signature characterizes a path up to thin homotopy—an equivalence relation that essentially consists of reparametrization and cancellation of retracings. Kapranov generalized the signature by introducing a universal higher connection whose holonomy represents surfaces (or higher dimensional membranes) through formal series of tensors. I will report on joint work with Darrick Lee in which we study the signature of piecewise linear surfaces and characterize the kernel in terms of thin homotopy. 

Abstract: First described by Eisenbud back in 1980 in the context of Cohen-Macaulay modules, matrix factorizations have proven to be very interesting and useful mathematical objects, appearing in algebraic geometry, quantum algebra and mathematical physics. In this talk, I will present some recent results relating them to representations of a vertex operator algebra under what is called the Landau-Ginzburg/conformal field theory correspondence. Our work generalizes some existing results by Davydov-Runkel-RC and were obtained via the description of module tensor categories of matrix factorisations. This structure was first introduced by Henriques-Penneys-Tenner, and our case is a first out-of-their-usual-arena instance. Based on joint work and work in progress with Thomas Wasserman (University of Oxford). 

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Abstract: Almost four hundred years ago, at the other end of the Mediterranean, Galileo observed that the fastest descent curve for a little ball is not the straight line! However to solve this problem rationally,  he foresaw that a ”higher” mathematics would be needed. Six decades later, J. Bernoulli posed this challenge to the mathematical world, prompting Newton’s overnight solution. The higher mathematics that emerged to solve such problems—and, ultimately, all of classical mechanics—was calculus.  

This journey through mechanics and geometry continued with Lagrange’s formulation of mechanics, Noether’s theorem on symmetries and conservation laws, and the rise of modern symplectic geometry. In this setting, Alan Weinstein’s insight—”everything is Lagrangian”—has shaped our understanding of phase space, reduction, and quantization. However, just as “higher mathematics” was needed for Galileo’s challenge, contemporary problems in symplectic reduction and singularities call for a new, more powerful framework.

In this talk, I will introduce higher and derived structures in differential geometry, inspired by Grothendieck’s derived algebraic geometry, which provide a natural language to resolve singularities and allow shifts of symplectic structure. The physical nature behind these higher structures, I believe is the dimension of our universe being 3+1 dimensional. In another word, higher and derived geometry may serve as a mathematical language to describe topological quantum field theories (TQFTs) and sigma models with greater clarity.

To illustrate this approach, I will introduce key ideas in higher and derived differential geometry and discuss their application to Marsden-Weinstein symplectic reduction, demonstrating how this modern perspective refines classical techniques and opens new directions in symplectic geometry and mathematical physics, based on work joint with Cueca, Dorsch and Sjamaar. 

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Abstract: Derived Lie (higher) groupoids provide a nice smooth model to handle singularities in symplectic reduction and Lagrangian intersection. They form an iCFO (incomplete category of fibrant objects) — not as perfect as a model category, but still gives rise to an infinity category. 

However, if one just inverts the weak equivalences in this iCFO, just like what we do for usual Lie groupoids, we can NOT obtain the correct stacks behind the groupoids — this already exhibits itself in the example, if we want to understand symplectic reductions as Lagrangian intersections. This puzzled us for the whole winter. 

We solve it by going to a larger world of homotopy n-groupoid in C^infty Schemes inspired by Pridham, Steffen, Carchedi-Roytenberg and Toen-Vezzosi’s work. Despite the distance of these more abstract objects to differential geometers, they do provide a more tolerant world so that correct Morita equivalence naturally arises. 

In this talk, I’ll try to explain this in details. Joint work with Cueca, Dorsch and Sjamaar.

Abstract: I will explain through examples my recent proof of the longstanding Quantum Resurgence Conjecture of Voros and Écalle. The proof uses purely geometric techniques involving spectral curves, Lie groupoids, and constructing global complex flows of meromorphic vector fields. 

Link: http://www.fields.utoronto.ca/talks/Resurgence-WKB-Solutions-Schr%C3%B6dinger-equations

Abstract: There are two prominent sources of groups in mathematics: symmetries and topological spaces, and they are related through the theory of differential equations, or more generally, gauge theory. Namely, by solving a differential equation, a path in a topological space can be represented as a symmetry of a vector space. In this way, the concatenation structure of paths is reflected in the composition of symmetries. The universal example of this correspondence is the path signature, a construction, originally due to Chen, that assigns a non-commutative power series to a path in Euclidean space. Remarkably, the signature completely characterizes a path up to translation, reparametrization, and cancellation, and can be used to construct the solutions to a family of differential equations. In this talk, I will give an overview of thin homotopy and the path signature. If time permits, I will discuss ongoing work, joint with Camilo Arias Abad and Darrick Lee, which aims to extend the signature so that it can be used to encode two-dimensional surfaces.

Abstract: In contact geometry, a systolic inequality aims to give a uniform upper bound on the shortest period of a periodic Reeb orbit for contact forms with fixed volume on a given manifold. This generalizes a well-studied notion in Riemannian geometry. It is known that there is no systolic inequality valid for all contact forms on any given contact manifold. In this talk, I will state a systolic inequality for contact forms that are invariant under a circle action in dimension three. 

Abstract: The Donaldson polynomials are powerful invariants in Low Dimensional Topology: they are able to distinguish exotic smooth 4-manifolds. They are rooted in theoretical physics (Yang-Mills Gauge Theory) and are difficult to compute, as they involve solving a PDE (the Anti-Self-Dual equation). Their 3D counterpart, Instanton Floer homology groups, are equally important invariants for 3-manifolds or links.

I will report on a program aimed at computing these by "cut-and-paste" operations. Specifically I will explain how to construct an "extended Topological Field Theory" that should recover them. This will involve going from gauge theory to symplectic geometry, Hamiltonian actions and some new homotopical algebra.

This will be based on arXiv:1903.10686, arXiv:2404.17393 and arXiv:2410.16225, with A. Hock and T. Mazuir. 

Abstract:  I'll give an introduction to the notion of isomonodromy in algebraic geometry, and explain how it plays the role of a "non-abelian" analogue of a Picard-Fuchs equation. I'll discuss some topological and arithmetic applications of these ideas, and a number of open questions around how to understand analogies between the cohomology of algebraic varieties and the moduli of local systems on algebraic varieties.

Abstract: L_\infty algebras, i.e. Lie algebras up to homotopy coherent homotopy, appear in a variety of contexts, including string theory and deformation theory. Over the last several decades, the outlines of a Lie theory for such objects has appeared in work of Sullivan, Getzler, Henriques and others. In this talk, we'll present joint work with Chris Rogers (UNR) establishing Lie's second and third theorems for connective L_\infty-algebras, with a focus on Lie's third theorem as an interplay of homotopical algebra, differential topology and Lie theory. 

Link: http://www.fields.utoronto.ca/activities/24-25/geostructures  

Abstract: Factorization homology 'integrates' (higher) categorical structures, such as representation categories of Hopf algebras, VOAs, or quantum groups, over manifolds. In my talk, I will discuss an approach for constructing local-to-global deformation quantizations of symplectic manifolds, such as moduli spaces of flat bundles (character varieties), based on factorization homology. In this approach, local quantizations are governed by E₂-deformations of symmetric monoidal categories, which, when integrated over 2-dimensional manifolds, lead to Poisson structures and deformation quantizations. Applying the general framework to the Drinfeld category reproduces deformations previously introduced by Li-Bland and Ševera. As a direct consequence, we can conclude a precise relation between their quantization and those introduced by Alekseev, Grosse, and Schomerus. No prior knowledge of factorization homology will be assumed. The talk is based on joint work with Eilind Karlsson, Corina Keller, and Ján Pulmann. 

Abstract: Moduli spaces of connections on principal bundles over surfaces have rich geometric structures. For example holomorphic connections over closed Riemann surfaces yield complex symplectic algebraic varieties, which are (transcendentally) mapped to character varieties by taking the monodromy data of the associated local system. This has been subsequently extended to the case of meromorphic connections by several authors, starting by allowing regular singularities (= simple poles): the generic examples involve semisimple coadjoint orbits of reductive algebraic groups, which can be deformation-quantised using Verma modules for the corresponding Lie algebra, and are then intimately related with conformal blocks in the 

Wess--Zumino--Novikov--Witten model (WZNW).

In this talk we will aim at a review of part of this story, and then describe extensions in the context of irregular singularities (= higher-order poles). In particular there are Verma modules for (nonreductive) truncated current Lie algebras which lead to deformation quantisation in this setting, and if time allows we will also sketch their relation with `irregular' WZNW conformal blocks.

[This talk is based on past work with D. Calaque, G. Felder, and R. Wentworth, as well as work in progress with M. Chaffe, L. Topley, and D. Yamakawa; but also on past work with P. Boalch, J. Douçot, and M. Tamiozzo]

Abstract: I will recall the WKB method for Schrödinger equations and explain how to make sense of it in more invariant and geometric terms. I will then explain why Schrödinger equations on compact Riemann surfaces have the so-called quantum resurgence property: the Borel transform of a formal WKB solution is is a holomorphic germ which defines an infinite Riemann surface with very interesting geometry completely reflected in the geometry of the spectral curve associated with the Schrödinger operator. Based on arXiv:2410.17224.

Abstract: In my talk I consider the generating function for the reciprocals N-th power of factorials. Using explicit constructions, I show a connection of product formulas for such series with the periods for certain families of algebraic hypersurfaces, defined via Landau-Ginzburg type models. Then I will discuss geometry of such hypersurfaces and connection with Buchstaber-Rees polynomials, which define N -valued group laws. I will also recover some integrality properties of appeared objects. Joint work with V. Rubtsov and D. van Straten.