This talk concerns a twenty-thousand-year old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to a cohomological interpretation of the Hasse-Weil zeta function by regularized determinants, as envisioned by Deninger.

In this talk, based on joint work with Plamen Stefanov and Gunther Uhlmann, I discuss the boundary rigidity problem on manifolds with boundary (for instance, a domain in Euclidean space with a perturbed metric), i.e. determining a Riemannian metric from the restriction of its distance  function to the boundary. This corresponds to travel time tomography, i.e. finding the Riemannian metric from the time it takes for solutions of the corresponding wave equation to travel between boundary points. A version of this relates to finding the speed of seismic waves inside the Earth from travel time data, which in turn permits a study of the structure of the inside of the Earth.

This non-linear problem in turn builds on the geodesic X-ray transform on such a Riemannian manifold with boundary. The geodesic X-ray transform on functions associates to a function its integral along geodesic curves, so for instance in domains in Euclidean space along straight lines. The X-ray transform on symmetric tensors is similar, but one integrates the tensor contracted with the tangent vector of the geodesics. I will explain how, under suitable convexity assumptions, one can invert the geodesic X-ray transform on functions, i.e. determine the function from its X-ray transform, in a stable manner, as well as the analogous tensor result, and the connection to the full boundary rigidity problem.

(Distinguished Lecture)

Non-self-adjoint operators appear in many settings, from kinetic theory 
and quantum mechanics to linearizations of equations of mathematical 
physics. The spectral analysis of such operators, while often notoriously 
difficult, reveals a wealth of new phenomena, compared with their 
self-adjoint counterparts. Spectra for non-self-adjoint operators display 
fascinating features, such as lattices of eigenvalues for operators of 
Kramers-Fokker-Planck type, say, and eigenvalues for operators with 
analytic coefficients in dimension one, concentrated to unions of curves 
in the complex spectral plane. In this talk, after a general introduction, 
we shall discuss spectra for non-self-adjoint perturbations of 
self-adjoint operators in dimension two, under the assumption that the 
classical flow of the unperturbed part is completely integrable.
The role played by the flow-invariant Lagrangian tori of the completely 
integrable system, both Diophantine and rational, in the spectral analysis 
of the non-self-adjoint operators will be described. In particular, we 
shall discuss the spectral contributions of rational tori, leading to 
eigenvalues having the form of the "legs in a spectral centipede". This 
talk is based on joint work with Johannes Sj\"ostrand.

Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications. This talk is designed for a general audience in mathematics and related fields.