The ocean's "submesoscale" consists of vortices, fronts and filaments occurring at scales well below the internal deformation radius, where stratification and rotation start to lose control. They affect Earth's climate in (at least) three major ways: (1) mixed-layer instability restratifies upper-ocean horizontal gradients; (2) the energy released through restratification generates submesoscale turbulence that strengthens the mesoscale via an inverse cascade; and (3) submesoscale processes drive intense fronts and filaments with associated vertical velocities that punch through the mixed-layer, creating conduits for oxygen, carbon and heat fluxes between the atmosphere and ocean's interior.

The radar interferometer aboard NASA's Surface Water Ocean Topography (SWOT) satellite measures sea surface height (SSH) in parallel 50 km wide swaths, resolving SSH features (in low-noise regions) at scales below 1 kilometer. This presents a landmark opportunity to quantify upper ocean submesoscale fluxes on a global scale. However, inferring submesoscale surface velocities from this data presents a challenge. While geostrophic balance provides accurate estimates of surface velocities at scales seen by traditional nadir altimetry, it is insufficient at SWOT scales for two reasons: (1) submesoscale dynamics is characterized by O(1) Rossby number and exhibit a wide range ageostrophic motions like convergent fronts and strong vortical asymmetry; and (2) inertia-gravity waves (IGWs) strongly influence SSH at these scales. Both of these types of motions are ageostrophic, but only the former achieves significant transport. The latter is confounding because their super-inertial timescales make it infeasible to remove them from the 21-day repeat cycle SWOT data using temporal filtering. Alternate methods, not relying on geostrophy or temporal filtering, need to be developed if we wish to quantify non-wave submesoscale processes from SWOT.

Along with many collaborators on the SWOT Science Team, we have focused on a multi-pronged effort to estimate the upper-ocean's "transport-active" velocity field from satellite data, using a mix of dynamical, statistical, and machine-learning approaches. In this talk, I'll touch on past work developing the use of joint PDFs of vorticity-strain-divergence to isolate convergent fronts, strong cyclones, and internal waves from flow snapshots, and their use in guiding machine learning methods to recover velocity from SSH. From current efforts I'll discuss two distinct efforts, one on the use of data-assimilating generative diffusion models to extrapolate partial observations to provide spatio-temporal maps of SSH, and in progress work using higher-order balance models to recover non-geostrophic flow from these satellite-derived 2D SSH maps. My aim is to leave you with an understanding of the complexities of the overall task, and how these seemingly-disjoint projects fit together.

Venue: MJC_2006 (Mountjoy Centre)

In this series of three talks I want to give an introduction into the wonderful world of modular forms. These are special functions with huge symmetry groups that are all-pervasive in modern mathematics and theoretical physics. It has even been said -- indeed, I may have said it myself -- that "modular forms are everywhere". The first lecture, on the many connections between modular forms and differential equations, is meant for a general audience, including applied mathematicians and mathematical physicists, while the two other talks will introduce more general types of modular objects that have emerged in recent years in contexts ranging from the string theory of black holes to the arithmetic properties of quantum invariants of knots. There will be many examples in all the talks, and no prior knowledge will be assumed.

Venue: MCS0001

Cosmological wavefunctions share many striking properties with scattering amplitudes, but their underlying structure remains less understood. I will describe a new connection between tree-level cosmological wavefunctions in scalar theories and amplitude-like graph objects. The main focus will be a new class of hidden zeros: kinematic configurations where these objects vanish, generalizing previously known cosmological zeros. I will show that these zeros imply a new factorization principle, dual in spirit to ordinary unitarity factorization, and can uniquely determine tree-level cosmological wavefunctions from locality alone. Finally, I will explain their equivalence to a new type of enhanced BCFW scaling.

Venue: zoom

Online: https://teams.microsoft.com/meet/351549099592375?p=kXI374gG4zfNSmtsWJ

Optimal transport theory has been a powerful tool in analysis on geometric spaces with curvature bounded since its introduction into geometric analysis. In this talk, I will first briefly introduce the application of the optimal transport theory on Riemannian manifolds and show an interpolation inequality. I will then focus on the recent work of mine, rigidity on curvature and measure of the Borell-Brascamp-Lieb inequality on weighted Riemannian manifolds satisfying the curvature dimension condition. I will also discuss the Brunn-Minkowski inequality and its rigidity, as well as a few open questions related.

Venue: MCS2068

In this series of three talks I want to give an introduction into the wonderful world of modular forms. These are special functions with huge symmetry groups that are all-pervasive in modern mathematics and theoretical physics. It has even been said -- indeed, I may have said it myself -- that "modular forms are everywhere". The first lecture, on the many connections between modular forms and differential equations, is meant for a general audience, including applied mathematicians and mathematical physicists, while the two other talks will introduce more general types of modular objects that have emerged in recent years in contexts ranging from the string theory of black holes to the arithmetic properties of quantum invariants of knots. There will be many examples in all the talks, and no prior knowledge will be assumed.

Venue: CLC202

The saying is ascribed to Gauss that Mathematics is the queen of the sciences and number theory is the queen of Mathematics. If so, then the theory of Diophantine equations is surely the queen of number theory. It is named after the semi-mythical Diophantus, whose works were lost for almost 1500 years but were still revolutionary when they were rediscovered, and is concerned with the simplest-sounding problems imaginable: finding solutions in whole numbers or fractions of collections of polynomial equations. Some of the oldest of the problems in this field go back to Archimedes or even earlier, long before Diophantus, while new and equally difficult elementary problems are still being discovered today. The most famous problems in this domain -- Fermat's last theorem, the problem of congruent numbers, the representability of integers as sums of two perfect cubes, and many others -- have led to some of the deepest and most difficult developments in mathematics. The lecture will attempt to describe some of the most fascinating of these problems. The high point will be the theory of elliptic curves and the conjecture of Birch and Swinnerton-Dyer, one of the seven Clay Millennium Prize problems, which I will try to explain in the simplest form possible without betraying its essence.

Venue: CG93 at/from 14:00

Link: here