Johns Hopkins Math Colloquium

Abstract: The past few years have seen numerous advances in the study of the p-adic cohomology of p-adic formal schemes, with applications in algebraic geometry, K-theory, and commutative algebra. More recently, starting with the work of Bhatt-Lurie and Drinfeld, the language of algebraic stacks has played an increasing role in the subject, and led to new cohomologies as well as new insights about old cohomologies. I will give a gentle introduction to this circle of ideas.

Abstract: By the maximum principle, the zero set of a real-valued harmonic function in the plane cannot contain a closed loop. It remains an open question whether a biharmonic function in the plane can have a nest of two zero loops, one inside the other. A positive answer (that it cannot) would not immediately imply unknown results, but it provides a useful toy model for one-dimensional problems in real algebraic geometry concerning bounds on the number of real roots of combinations of polynomial-like expressions with radicals. 

We present an elementary example of this type. Let P_i(x), i = 1, …, n, be positive quadratic polynomials on the real line, and let F(x) be a linear combination of sqrt(P_i(x)). A question of N. Alon asks whether the number of real zeros of F admits a bound linear in n. We will mention a recent positive answer to this question (joint work in progress with D. Zakharov) that uses a PDE argument. With minor tweaks, the argument extends to linear combinations of (P_i)^a with a fixed negative real exponent a. However, if one allows different exponents a_i (e.g., two distinct values), the method breaks down; this failure motivates studying zeros of linear combinations of solutions to different linear elliptic PDEs. 

An interesting question is understanding the topological complexity of zero sets of linear combinations of Laplace eigenfunctions on manifolds. The Courant–Herrmann theorem formulated in 1931 (disproved 50 years later by Arnold) states that the zero set of a linear combination of the first n Laplace eigenfunctions on a closed manifold divides the manifold into no more than n connected components. The theorem is false, because one can construct a manifold and a linear combination of just two eigenfunctions with infinitely many zero loops. However, these loops are not nested, and an open question for linear combinations of two eigenfunctions is to estimate the number of double nests in terms of the eigenvalue and the geometry of the manifold. We will discuss Gudkov’s complexity of real algebraic curves related to Hilbert’s 16th problem and its analogue for Laplace eigenfunctions. We will also review a recent result of Yoav Krause on realizing prescribed topological configurations of loops on the sphere as zero sets of Laplace eigenfunctions. Finally, we turn to a probabilistic viewpoint:  realizing a random topological configuration of loops as the zero set of a polynomial of minimal degree, and as the zero set of a Laplace eigenfunction obtained by a small perturbation of the metric on the sphere (joint work with D. Elboim).

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