Monday 30 September 2024, 15:00-16:00
Arts, Lecture Room 5
The Gowers uniformity norms have become well-used tools in additive combinatorics, ergodic theory and analytic number theory. We discuss an effective version of the inverse theorem for the Gowers U3-norm for functions supported on high-rank quadratic level sets in finite vector spaces. This enables one to run density increment arguments with respect to quadratic level sets, which are analogues of Bohr sets in the context of quadratic Fourier analysis on finite vector spaces. For instance, one can derive a polyexponential bound on the Ramsey number of three-term progressions which are the same colour as their common difference ("Brauer quadruples"), a result it seems difficult to obtain by other means.
Monday 21 October 2024, 15:00-16:00
Arts, Lecture Room 5
It is well-known that when sets have the same Fourier and Hausdorff dimensions there are no exceptions to Marstrand's projection theorem. In this talk we will show how we can use the method of dimension interpolation to improve the state-of-the-art estimates for the Hausdorff dimension of the exceptional set of projections in higher dimensions. Joint work with Jonathan Fraser.
Monday 28 October 2024, 15:00-16:00
Arts, Lecture Room 5
I will discuss our recent work on constructing singular Gibbs measures, using the radial Φ44 measure (measure supported on radial functions) as an example. Utilising Barashkov-Gubinelli's variational approach, alongside new ideas by exploiting the Markov property of the Gaussian free field and Green function estimates, allows us to overcome the singularities appearing at the origin.
This is joint work with Tadahiro Oh (Edinburgh), Leonardo Tolomeo (Edinburgh), and Nikolay Tzvetkov (Lyon).
Monday 11 November 2024, 15:00-16:00
Arts, Lecture Room 5
In Fourier restriction theory, we bound exponential sums whose frequencies lie in sets with special properties, e.g., random sets or curved sets. Bourgain, Demeter, and Guth proved sharp decoupling inequalities, which show that many such functions enjoy square root cancellation behavior. This theory lies in the larger context of bounding matrix p to q norms, which is well-studied in the CS literature. We will discuss a new polynomial-time algorithm of myself, Guth, and Urschel that is inspired by Fourier restriction theory and which reduces the multiplicative error of computing matrix 2 to q norms from roughly n1/q to n1/2q, where n is the size of the matrix.
Monday 18 November 2024, 15:00-16:00
Arts, Lecture Room 5
How many rational points with denominator of a given size lie within a certain distance from a compact, 'non-degenerate' manifold? This talk is about some recent progress towards answering this question. We shall describe how the geometric and analytic properties of the manifold play a key role in determining this count, and present a heuristic for the same. We shall then discuss how Fourier analytic techniques can be exploited to establish the desired asymptotic for rational points near manifolds satisfying certain strong geometric conditions. The key proof ingredient is a bootstrapping argument relying on Poisson summation, projective duality and oscillatory integral techniques.
Monday 25 November 2024, 15:00-16:00
Arts, Lecture Room 5
The main mathematical manifestation of the Stark Effect in Quantum Mechanics is the shift and the formation of clusters of eigenvalues when a spherical Hamiltonian is perturbed by lower order terms. Understanding this mechanism turned out to be fundamental in the description of the large-time asymptotics of the associated Schrödinger groups and can be responsible for the lack of dispersion. In this talk, we will first explain the construction of a family of spectrally projected intertwining operators in the case of constant perturbations on the sphere (inverse-square potential), reminiscent of Kato's Wave Operators, and also prove their boundedness in Lp. Next, we extend this framework to define intertwining operators for non constant spherical perturbations in space dimensions two and higher. In addition, we investigate the mapping properties between Lp-spaces of these operators.
In 2D, we prove a complete result, for the Schrödinger Hamiltonian with a (fixed) magnetic potential and an electric potential, both scaling critical, allowing us to prove dispersive estimates, uniform resolvent estimates, and Lp-bounds of Bochner-Riesz means. In higher dimensions, apart from recovering the example of inverse-square potential, we can conjecture a complete result in the presence of some symmetries (zonal potentials), and open some interesting spectral problems concerning the asymptotics of eigenfunctions.
Monday 2 December 2024, 15:00-16:00
Arts, Lecture Room 5
The study of boundary value problems in the upper half-space for divergence-form elliptic equations with block structure and complex coefficients independent of the transversal direction to the boundary has recently been settled by P. Auscher and M. Egert, who streamlined the methods developed over the last two decades to solve such problems for many boundary data spaces, including those of Lp-type, in their recent monograph.
In this talk we shall describe how this machinery can be adapted to the study of Dirichlet and (Dirichlet) regularity boundary value problems for a class of singular Schrödinger-type elliptic equations on the upper half-space, by relying on some essential L2 estimates recently obtained by A. Morris and A. Turner for a class of Schrödinger operators with singular potentials.
This talk is based on joint work with Andrew Morris.
Monday 9 December 2024, 15:00-16:00
Arts, Lecture Room 5
In this talk I will present gradient estimates of Hamilton-Souplet-Zhang and Li-Yau types for a class of non-linear diffusion equations on smooth metric measure spaces. The Laplace-Beltrami operator gives its place to the Witten Laplacian and the Riemannian metric tensor and potential evolve with time (a geometric flow). The estimates are established under different curvature conditions and lower bounds on the Bakry-Emery Ricci tensor and are then used to prove a number of important results such as Harnack inequalities, spectral bounds, sharp Logarithmic Sobolev inequalities (LSI) and general Liouville and global constancy results. If time allows, I will present applications of the above to entropy dissipation inequalities and characterisation of ancient/eternal solutions.