Cancelled.

I will review connections between random matrix theory and statistical properties of the Riemann zeta-function, including recent developments relating to extreme values of the zeta function on the critical line.

The modified diagonal on the triple product of a curve was first introduced by Gross and Schoen in the 90’s. This simply defined object holds fundamental importance in the study of the geometry and arithmetic of curves. One basic question is whether the modified diagonal vanishes under “deformation”. I will introduce the origin of this type of question and provide a brief history of the study of the modified diagonal. Subsequently, I will discuss my collaborative works with W. Zhang, where we demonstrated that such vanishing can be dictated by the symmetry of the curve. As an application in number theory, we proved a case of the notorious Beilinson—Bloch conjecture, a generalization of the Millennium Birch—Swinnerton-Dyer conjecture. Finally, I want to propose some new questions.

The growth of an infinite-dimensional algebra is a fundamental tool to measure its "size." The growth of noncommutative algebras plays an important role in noncommutative geometry, representation theory, differential algebraic geometry, symbolic dynamics, homological stability results, and more. 

We analyze the space of growth functions of algebras, answering a question of Zelmanov on the existence of certain 'holes' in this space, and provide evidence for the ampleness of the possible growth rates of algebras with prescribed properties; we conclude a strong quantitative solution of the Kurosh Problem on algebraic algebras.

Utilizing new layers of the interplay between noncommutative algebra and symbolic dynamics, we exhibit surprising pathologies in the prime spectrum and tensor product structure of algebras with polynomial growth, thereby providing counterexamples to questions of Bergman, Krause, Lenagan, and others; applying our methods to algebras of faster growth types, we resolve a conjecture of Bartholdi on amenable representations in exponential growth.

Finally, the largest objects (groups, algebras, Lie algebras) are, in many contexts, those containing free substructures. We discuss the coexistence of this phenomenon with finiteness properties -- in particular, "almost algebraicity" of algebras and "almost periodicity" of groups -- from algebraic, geometric, and probabilistic perspectives.

This talk is partially based on joint works with Bell, Goffer, and Zelmanov.

We present a new, recent doubling method for establishing a priori estimates, then classical solvability and regularity for solutions to fully nonlinear PDEs.  The method produces the missing estimates for the quadratic Hessian and prescribed hypersurface scalar curvature PDEs in dimension four.  It also gives new proofs of the estimates and regularity for Monge–Ampère and special Lagrangian equations and provides prospects for classical solvability of alternative Dirichlet problems.