Video Lectures

 I shall present joint work with Maxim Kontsevich (arXiv:2105.10161) describing an interesting domain of complex metrics on a smooth manifold. It is a complexification of the space of ordinary Riemannian metrics, and has the Lorentzian metrics (but...

We explain our proof of the unbounded denominators conjecture. This talk will require the main theorem of the lecture on Nov. 17, 2021, as a “black box” but otherwise be logically independent of that talk.

In this talk, we discuss the Diophantine study of relative SL2-character varieties of surfaces. In particular, we prove that the integral points on these varieties are effectively finitely generated in a precise sense, and in particular their...

I will explain a proof of a joint with D. Alvarez-Gavela and D. Nadler theorem which allows to make a Lagrangian submanifold transverse to a given Lagrangian distribution in exchange for making them piecewise smooth with canonical singularities...

What governs whether large-scale planetary habitability is sustained through time? Lakes, valley networks, groundwaters, hydrothermal systems, and chemically open-system weathering once existed on Mars. The suite of aqueous environments preserved in...

Abstract 1: In recent years, weak lensing of the cosmic microwave background (CMB) has emerged as a powerful tool to probe fundamental physics. The prime target of CMB lensing surveys is the lensing potential, which is reconstructed from observed...

We describe two extensions, called the virtual Morse-Bott index and circle-equivariant virtual Morse-Bott index, of the classical Morse-Bott index of a Morse-Bott function on a smooth manifold to the setting of (a) suitably defined analytic...

The regularity theory for the Navier-Stokes equation will be reviewed. Motivations from Kolmogorov’s phenomenological theory of turbulence will be discussed. Rigorous mathematical results are obtained to confirm some of the phenomenologies.

I will discuss joint work with McLean and Smith, lifting the results of Seidel, Lalonde, McDuff, and Polterovich concerning the topology of Hamiltonian fibrations over the 2-sphere from rational cohomology to complex cobordism. In addition to the...