Zoom: Meeting ID: 961 2738 3116 Passcode: the last four digits above in the reverse order
We consider the structured Gaussian matrix G_A=(a_{ij}g_{ij}), where g_{ij}'s are independent standard Gaussian variables. The exact behavior of the spectral norm of the structured Gaussian matrix is known due to the result of Latala, van Handel, and Youssef from 2018. We are interested in two-sided bounds for the expected value of the norm of G_A treated as an operator from l_p^n to l_q^m. We conjecture the sharp estimates expressed only in the terms of the coefficients a_{ij}'s. We confirm the conjectured lower bound up to the constant depending only on p and q, and the upper bound up to the multiplicative constant depending linearly on a certain (small) power of ln(mn). This is joint work with Radoslaw Adamczak, Joscha Prochno, and Michal Strzelecki.
Zoom ID 989 6529 0738. Passcode: the last four digits of the zoom ID in the reverse order
Take a uniformly random graph with a fixed edge density e. Its triangle density will typically be about e^3, and we are interested in the large deviations behavior: what's the probability that the triangle density is about e^3 - delta? The general theory for this sort of problem was studied by Chatterjee-Varadhan and Dembo-Lubetzky, who showed that the solution can be written in terms of an optimization over certain integral kernels. This optimization is difficult to solve explicitly, but Kenyon, Radin, Ren and Sadun used numerics to come up with a fascinating and intricate set of conjectures regarding both the probabilities and the structures of the conditioned random graphs. We prove these conjectures in a small region of the parameter space.
Asymptotic freeness of independent Haar distributed unitary matrices was discovered by Voiculescu. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random permutations acting on the orthogonal of the Perron-Frobenius eigenvector. In this talk, we consider a new matrix unitary model appearing naturally from representation theory of compact groups. We fix a non-trivial signature, i.e. two finite sequences of non-increasing natural numbers, and for n large enough, consider the irreducible representation of Un associated to this signature. We show that strong asymptotic freeness holds in this generalized context when drawing independent copies of the Haar measure. We also obtain the orthogonal variant of this result. This is a joint work with Benoît Collins.
The restriction problem in harmonic analysis asks for L^p bounds on the Fourier transform of functions defined on curved surfaces. In this talk, we will present improved restriction estimates for hyperbolic paraboloids, that depend on the signature of the paraboloids. These estimates still hold, and are sharp, in the variable coefficient regime. This is joint work with Jonathan Hickman