L6

Classical finite element methods discretize scalar functions using piecewise polynomials. Vector finite elements, such as those developed by Raviart-Thomas, Nédélec, and Brezzi-Douglas-Marini in the 1970s and 1980s, have since undergone significant theoretical advancements and found wide-ranging applications. Subsequently, Bossavit recognized that these finite element spaces are specific instances of Whitney’s discrete differential forms, which inspired the systematic development of Finite Element Exterior Calculus (FEEC). These discrete topological structures and patterns also emerge in fields like Topological Data Analysis.

In this talk, we present an overview of discrete and finite element differential forms motivated by applications from topological hydrodynamics, alongside recent advancements in tensorial finite elements. The Bernstein-Gelfand-Gelfand (BGG) sequences encode the algebraic and differential structures of tensorial problems, such as those encountered in solid mechanics, differential geometry, and general relativity. Discretization of the BGG sequences extends the periodic table of finite elements, originally developed for Whitney forms, to include Christiansen’s finite element interpretation of Regge calculus and various distributional finite elements for fluids and solids as special cases. This approach further illuminates connections between algebraic and geometric structures, generalized continuum models, finite elements, and discrete differential geometry.

I will report on recent progress on influential conjectures from the 1970s and 1980s (Berry-Tabor, Bohigas-Giannoni-Schmit), which suggest that the spectral statistics of the Laplace-Beltrami operator on a given compact Riemannian manifold should be described either by a Poisson point process or by a random matrix ensemble, depending on whether the  geodesic flow is integrable or “chaotic”. This talk will straddle aspects of analysis, geometry, probability, number theory and ergodic theory, and should be accessible to a broad audience. The two most recent results presented in this lecture were obtained in collaboration with Laura Monk and with Wooyeon Kim and Matthew Welsh. 

Studying the topology of zero sets of maps is a central topic in many areas of mathematics. Classical homological invariants, such as Betti numbers, are not always suitable for this purpose due to the fact that they do not distinguish between topological features of different sizes. Topological data analysis provides a way to study topology coarsely by ignoring small-scale features. This approach yields generalizations of a number of classical theorems, such as Bézout's theorem and Courant’s nodal domain theorem, to a wider class of maps. We will explain this circle of ideas and discuss potential directions for future research. The talk is partially based on joint works with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich and E. Shelukhin.

I will report on recent progress on influential conjectures from the 1970s and 1980s (Berry-Tabor, Bohigas-Giannoni-Schmit), which suggest that the spectral statistics of the Laplace-Beltrami operator on a given compact Riemannian manifold should be described either by a Poisson point process or by a random matrix ensemble, depending on whether the  geodesic flow is integrable or “chaotic”. This talk will straddle aspects of analysis, geometry, probability, number theory and ergodic theory, and should be accessible to a broad audience. The two most recent results presented in this lecture were obtained in collaboration with Laura Monk and with Wooyeon Kim and Matthew Welsh. 

Renormalization group (RG) flow is a central aspect of our modern understanding of QFT. We may wonder about the relationship of renormalization to some of the other properties of a QFT, and if we can reconstruct RG flow from these properties. It has recently been proposed by Chavda, McLoughlin, Mizera and Staunton in [2510.25822] and [2511.10613] that unitarity can give us at least a part of RG flow, which is known as the Unitarity Flow Conjecture. In this talk, I will summarize the central ideas of this conjecture, and provide some evidence for it.

Two different, almost orthogonal approaches to QFT are: (1) the study of von Neumann algebras of local observables in flat space, and (2) the study of extended and topological defects in general spacetime manifolds. While naively the two focus on different aspects, it has been recently pointed out that some of the axioms of approach (1) clash with certain expectations from approach (2). In this JC talk, I’ll give a brief introduction to both approaches and review the recent discussion in [2008.11748], [2503.20863], and [2509.03589], explaining (i) what the tensions are, (ii) a recent proposal to solve them, and (iii) why it can be useful.

Let g be the Lie algebra of a simple algebraic group over an algebraically closed field of characteristic p. When p=0 the celebrated Jacobson-Morozov Theorem promises that every non-zero nilpotent element of g is contained in a simple 3-dimensional subalgebra of g (an sl2). This has been extended to odd primes but what about p=2? There is still a unique 3-dimensional simple Lie algebra, known colloquially as fake sl2, but there are other very sensible candidates like sl2 and pgl2. In this talk, Adam Thomas from the University of Warwick will discuss recent joint work with David Stewart (Manchester) determining which nilpotent elements of g live in subalgebras isomorphic to one of these three Lie algebras. There will be an abundance of concrete examples, calculations with small matrices and even some combinatorics.