Venue: zoom

The winds of Jupiter have motivated many studies across the years, from the iconic equatorial bands to the Great Red Spot. While the dynamics of these systems are mostly understood, recent imagery from NASA's Juno spacecraft has discovered spectacular cyclones on the Jovian polar regions. Currently, there exists eight circumpolar vortices at north pole and five at the south pole, all "orbiting" a central vortex. The circumpolar vortices have largely remained a mystery until their discovery, evading global atmospheric models of Jupiter. They have largely not changed since the first flyby of 2016, aside from the appearance of a reoccurring intruding vortex at the south pole in 2019 and 2021.

This presentation will be on two subjects. The first is discussing a new regime we discovered by modelling rapidly rotating, incompressible, low Prandtl Rayleigh-Benard convections. At low Prandtl numbers, we produce large scale oscillations of Large Scale Vortices (LSVs), a phenomenon we have dubbed bursting. While bursting has been observed in the past for convection, this has not been documented for vortices. The second subject will be on applying a magnetic friction to a saturated low Prandtl case in place of a full MHD simulation to estimate the conductivity required to halt any LSV generation. This can then be used to provide a rough estimate to the depth of Jupiter's polar vortices.

Venue: MJC2004

Hilbert–Maass cusp forms are non-holomorphic automorphic forms over totally real number fields, arising as joint eigenfunctions of the Laplace operators on products of upper half-planes. This talk explores how their arithmetic structure can be used to compute explicit examples.

For fields of narrow class number one, we develop a classical Hecke theory, including reflection symmetries, normalization of eigenforms, and explicit bounds for Fourier coefficients. We then generalize Hejhal’s automorphy method to this setting, with explicit bounds for truncation and aliasing errors. Given an exact spectral parameter and assuming a one-dimensional eigenspace, verified linear algebra yields certified coefficient and function approximations.

We illustrate the method with tentative spectral data over Q(\sqrt2) and \ Q(\sqrt5), checked using unit, reflection, and Hecke relations. An example over Q(\sqrt3) highlights the challenges beyond narrow class number one. This is joint work with Sujeet Singh.

Venue: MCS2068

TBA

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

TBA

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

TBA

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MJC2012

The STACK API allows STACK questions to be viewed and marked (and tested) outside of a Moodle environment. One valuable use-case for this is the embedding of STACK questions as examples directly within course notes (in html format). This enables students to better practise questions relative to the specific part of the course notes that they are currently studying. In this seminar, we demonstrate a simple method for converting course materials to html (using Pandoc) and embedding stack questions within the resulting html using the STACK API.

Venue: MCS3070 (MAGIC)

Venue: MCS3070

Venue: MCS3070

\( \mathfrak{sl}_N \) homology is a quantum invariant of knots and links, generalising Khovanov homology. In this talk we will provide a brief introduction to how this is defined via objects called foams. We will also introduce s-invariants, which have applications in 4-manifold topology. We will then show that for the case of N a prime p, and working over \( \mathbb{F}_p \), the chain complex that this theory associates to a link diagram has a remarkably simple decomposition. From this, we can deduce new results and reprove existing results regarding s-invariants and the structure of \( \mathfrak{sl}_p \) homology.

Venue: MCS2068

Square-tiled surfaces (aptly named) are surfaces obtained by gluing together a collection of unit squares along their sides (the square torus being the simplest example). These surfaces are special cases of translation surfaces, whose moduli space carries a natural action of the group SL(2,R). The famous works of Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi were concerned with understanding the orbits of this action.

This SL(2,R)-action restricts to an action of SL(2,Z) on square-tiled surfaces, and the orbits under this restricted action can be transformed into finite regular graphs. It is a long-standing conjecture of McMullen that a specific family of these orbit graphs in genus two forms a family of expander graphs. I will discuss the history of this conjecture and recent progress towards it obtained in joint work with Carlos Matheus.

Venue: MCS2068

We will show that the eigenvalues of the discrete Hodge Laplacian of a finite simplicial complex can be estimated by the eigenvalues in lower dimensions and certain geometric constants. As a result, we obtain new criteria for vanishing cohomology and prove a recent conjecture of O on the dimensional monotonicity of the largest eigenvalues. If time permits, we will also apply these results to the Alexander dual and obtain analogous estimates in terms of the eigenvalues in higher dimensions.

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

In the usual Euclidean path-integral picture, higher symmetries are represented by invertible topological defects, and ’t Hooft anomalies are described by anomaly inflow from a topological action in one dimension higher. Is there a corresponding construction that starts directly from the algebra of local operators and its symmetries? To address this question, I will describe a mathematically precise framework based on localized symmetry transformations and their commutators. To each spatial region U, one associates a group of transformations localized in U. Locality implies that the commutator of transformations localized in regions U and V is localized in their intersection. From these data one can extract a system of higher groups, one per cover of space. Both anomalies and higher symmetries are encoded in this system of higher groups. The framework is designed to treat algebraic QFT and quantum lattice systems in a common language. The guiding idea is that topology emerges from the interplay of symmetry and locality.

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

I will review some of the progress on the directed polymer model in dimension d=2, whose partition functions can be seen as a discretised version of the stochastic heat equation. Recent results have shown that, in a suitable (and subtle) critical window of parameters, the model converges to a disordered limit, namely the Critical 2D Stochastic Heat Flow (SHF) introduced by Caravenna, Sun and Zygouras. This SHF is a non-Gaussian stochastic process of measures on R^2, and a lot of its properties have been studied over the last years. In this talk, I will first take some time to explain the (idea of the) construction of the SHF and then review some of the recent advances on its relation to the KPZ equation.

Venue: MC0001

We study differential equations with time-dependent drifts that are merely integrable in space and time, driven by additive fractional Brownian motion with Hurst parameter H<1/2​. Although the corresponding deterministic equations may be highly non-unique, the irregularity of the noise restores strong well-posedness. Under a natural subcriticality condition on the space-time integrability of the drift, together with conditions required for a change-of-measure argument, we prove strong existence and path-by-path uniqueness. We also obtain quantitative stability with respect to the initial condition and the drift, both pathwise and in moment norms, and construct a Hölder-continuous semiflow. As H approaches 1/2, the admissible regime recovers, up to an endpoint, the classical Brownian scaling condition of Krylov and Röckner (2005). I will explain the main ideas of the proof, which combine stochastic sewing, Lyons' rough-path estimates, and a quantitative John-Nirenberg inequality for stochastic processes with vanishing oscillations. Base on a joint work with Oleg Butkovsky and Toyomu Matsuda.

Venue: MCS2068

Venue: MCS2068

Solitons are localized waves displaying stunning particle-like behavior. They appear in numerous integrable PDEs like the KdV equation, the NLS equation, and, on the discrete side, the Toda lattice. Large ensembles of solitons give rise to a so-called `soliton gas' which is described by Zakharov's kinetic theory, later extended by El and Kamchatnov. Soliton gases of maximal density are called soliton condensates.     In joint work with Guido Mazzuca (arXiv:2611.12748) we study a class of Toda soliton condensates using the Deift-Zhou nonlinear steepest descent method. The characteristics of the individual solitons, i.e. their amplitudes and phases, are assumed to be independent random variables. Despite this randomness we show that in the soliton condensate limit, the solution converges deterministically to a modulated elliptic (genus $1$) or hyperelliptic (genus $2$) wave. The precise genus depends on the exact location in the space-time diagram, which exhibits slow modulation in space and time. Contrary to previous studies of soliton gases using Riemann-Hilbert methods, we observe genuinely nonlinear boundaries between different phases. As a corollary we prove that with high probability the soliton gas kinetic equation, the Whitham modulation equations and the nonlinear dispersion relations hold.     This work is supported by the European Union’s Horizon Europe research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101272799 (Aztec RHP).

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Venue: MCS2068

Formal accounts of statistical inference are usually based around the notion of closed inference. By this we mean all of the ingredients (model form, parametric choices, probabilistic specification, etc) are fixed a priori. When new information is received, inferences (for example, about likely values of the model parameters) are made according to pre-specified rules established by the formal system (for example, by use of Bayes theorem).

However, real world statistical inferences are rarely as tidy as this. Modelling, data exploration, diagnostic testing and inference are intertwined in ways which do not align well with such formal accounts of statistical inference. Further, there are general areas of uncertain inference which rely on aspects of statistical reasoning but are not well described by this kind of formalism. (For example, most forms of human reasoning, problems of Uncertainty Quantification for complex computer models, and much of reasoning by AI.)

To be able to make principled inferential statements corresponding to our actual procedures, we need the notion of open statistical inference. This describes problems for which the specification of all of the elements need not be fully described a priori, involving models which are not constrained to remain unchanged in form over time. Further, we may not be able to pre-specify the precise inferential process that we intend to follow.

We will derive, under very weak temporal constraints, certain conditions that any open inference should obey. These conditions provide a natural framework for analysing open inferences, supported constructively by new methodological tools for this purpose. In particular, they allow us to give a meaningful logical framework for closed inferences within this more general formalism.

Venue: MCS2068