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No seminar this week.

Title: Distribution of primes and approximation on weighted Dirichlet spaces Speaker: Daniel Seco University Carlos III de Madrid Time: 4:00pm–5:00pm Place: CMC 130

We study zero-free regions of the Riemann zeta function \(\zeta\) related to an approximation problem in the weighted Dirichlet space \(D_{-2}\) which is known to be equivalent to the Riemann Hypothesis since the work of Báez–Duarte. We prove, indeed, that analogous approximation problems for the standard weighted Dirichlet spaces \(D_{\alpha}\) when \(\alpha\in (-3,-2)\) give conditions so that the half-plane \(\{s\in\mathbf{C}:\Re (s) > -\frac{\alpha+1}{2}\}\) is also zero-free for \(\zeta\). Moreover, we extend such results to a large family of weighted \(\ell^p\)-spaces of analytic functions. As a particular instance, in the limit case \(p=1\) and \(\alpha=-2\), we provide a new proof of the Prime Number Theorem. This is a joint work with Eva Gallardo–Gutiérrez.

Title: Some musings on inner functions, Part II Speaker: Catherine Bénéteau Time: 4:00pm–5:00pm Place: CMC 130

Title: Some musings on inner functions Speaker: Catherine Bénéteau Time: 4:00pm–5:00pm Place: CMC 130

In this talk, I will discuss inner functions in different contexts, in particular how they arise naturally in connection with analytic functions of one variable and how they are related to shift-invariant subspaces and certain extremal problems. I will examine the notion of weakly inner and see how it compares, and will discuss some connections with optimal polynomial approximants. Along the way, I might pose some questions about what can happen in analytic function spaces of two variables.

Title: Mandrekar's theorem using reproducing kernels, Part II Speaker: Linus Bergqvist Stockholm University Time: 4:00pm–5:00pm Place: CMC 130

Title: Mandrekar's theorem using reproducing kernels Speaker: Linus Bergqvist Stockholm University Time: 4:00pm–5:00pm Place: CMC 130

A classical theorem by Beurling states that all invariant subspaces of the shift operator acting on the Hardy space on the disc will be of the form \(I(z)\,H^2(D)\) for some inner function \(I(z)\), and in particular all invariant subspaces are generated by one function. For Hardy spaces on polydiscs the situation is a lot more complicated, and in fact there are invariant subspaces that are not finitely generated. However, a theorem by Mandrekar shows that an invariant subspace of \(H^2\left(D^2\right)\) is generated by an inner function if and only if the shift operators acting on the invariant subspace are doubly commuting. In this talk we will discuss function spaces on polydiscs, shift invariant subspaces of such spaces, and give an alternative proof of Mandrekar's theorem using reproducing kernels.

No seminar this week — Spring Break.

Title: Zeta-function regularization of functional determinants, the Polyakov Alvarez formula, and everything Razvan talked about last week, Part III Speaker: Nathan Hayford Time: 4:00pm–5:00pm Place: CMC 130

Title: Zeta-function regularization of functional determinants, the Polyakov Alvarez formula, and everything Razvan talked about last week, Part II Speaker: Nathan Hayford Time: 4:00pm–5:00pm Place: CMC 130

Title: Choreography in Nature (towards theory of dancing curves, superintegrability) Speaker: Alexander Turbiner ICN-UNAM, Mexico/Stony Brook University Time: 4:00pm–5:00pm Place: CMC 130

By definition, the choreographic curve (dancing curve) is a closed trajectory on which \(n\) classical bodies move chasing each other without collisions. The first choreography — the Remarkable Figure Eight — at zero angular momentum was discovered unexpectedly in physics by C. Moore (Santa Fe Institute) in 1993 for 3 equal masses in R3 Newtonian gravity numerically and independently in mathematics by Chenciner (Paris)-Montgomery (Santa Cruz) in 2000. At the moment, about 6,000 choreographies in R3 Newtonian gravity are found, all numerically, for different \(n > 2\). All of them are represented by transcendental curves. It manifests the major discovery in celestial mechanics, next after H. Poincaré's proof of chaotic nature of \(n\) body problem.

Does there exist (non)-Newtonian gravity for which dancing curve is known analytically? — Yes, a single example is known — it is the algebraic lemniscate by Jacob Bernoulli (1694), unusually parametrized - and it will be the subject of the talk. Astonishingly, the Figure Eight trajectory in R3 Newtonian gravity coincides with algebraic lemniscate with \(\chi^2\) deviation \(\sim 10-7\). Both choreographies admit any odd numbers of bodies on them. Both 3-body choreographies define maximally superintegrable trajectories with 7 constants of motion and Liouville integrals. The talk will be accompanied by numerous animations.

Title: Zeta-function regularization of functional determinants, the Polyakov Alvarez formula, and everything Razvan talked about last week Speaker: Nathan Hayford Time: 4:00pm–5:00pm Place: CMC 130

I will try to explain what Razvan talked about last week. Zeta function regularization of functional determinants will be discussed, and write a variational formula for the log of the determinant of the Dirichlet Laplace operator under a perturbation of the boundary. If time permits (and I can find a formula for it), I will also derive a variational formula for the log of the determinant of the Laplacian under a suitably chosen perturbation of the operator itself.

Title: Variations on Hadamard's variational formula, Part II Speaker: Razvan Teodorescu Time: 4:00pm–5:00pm Place: CMC 130

Title: Variations on Hadamard's variational formula Speaker: Razvan Teodorescu Time: 4:00pm–5:00pm Place: CMC 130

Starting from Hadamard's first variation formula for the Dirichlet Green's function of a domain with finite connectivity, we consider its extensions and derivation within optimization theory. The formula's implications for theoretical physics are inevitable, but ignorable 🙂.