Programme

Abstracts

• Shing-Tung Yau: Complete Calabi-Yau Metric on Noncompact Manifolds

  Thanks to the pioneering work of Chern and Calabi, we see a burgeoning of complex geometry since 1940s. Nowadays, a very important theme in complex geometry is the Kähler-Einstein metric, which is the complex analogue of Einstein field equation. The Schwarzschild solution models the black hole – the most mysterious object in our universe. Actually, it is my fascination with finding such Ricci-flat metrics in general relativity that drove me to solve the Calabi conjecture in 1976. I proposed a program to solve the noncompact version of the Calabi conjecture in the 1978 Helsinki International Congress of Mathematicians. In string theory, the extra dimensions of spacetime takes the form of a Calabi-Yau 3-fold. The noncompact version of Calabi-Yau is starting to show its significance in mirror symmetry. In the talk, I will survey Kähler-Einstein metric and especially the complete Ricci-flat Kähler metric on noncompact manifolds. The Yau-Tian-Zelditch expansion is one of the most important tools in its development.

• Jian Song: Nakai-Moishezon criterions for complex Hessian equations

  The J-equation proposed by Donaldson is a complex Hessian quotient equation on Kähler manifolds. The solvability of the J-equation is proved by Song-Weinkove to be equivalent to the existence of a subsolution. It is also conjectured by Lejmi-Szekelyhidi to be equivalent to a stability condition in terms of holomorphic intersection numbers as an analogue of the Nakai-Moishezon criterion in algebraic geometry. The conjecture is recently proved by Chen under a stronger uniform stability condition. In this paper, we establish a Nakai-Moishezon type criterion for pairs of Kähler classes on analytic Kähler varieties. Consequently, we prove Lejmi-Szekelyhidi's original conjecture for the J-equation. We also apply such a criterion to obtain a family of constant scalar curvature Kähler metrics on smooth minimal models.

• Duong H. Phong: Geometric flows in symplectic geometry

  The search for supersymmetric solutions of string theories has led to new geometric partial differential equations which are interesting in their own right. In particular, the Type IIA string leads to a new geometric flow, introduced by T. Fei, S. Picard, X.W. Zhang, and the speaker, and called the Type IIA flow, which is a flow of closed and primitive 3-forms on a symplectic six-dimensional manifold. Remarkably, the Type IIA flow can also be viewed as a flow of a special class of almost-complex structures, or as a coupling of the Ricci flow with a scalar field. We survey what is now known about this flow, with emphasis on the many open questions.

• András Vasy: The Feynman propagator and self-adjointness

  In this talk I will discuss the Feynman and anti-Feynman inverses for wave operators on certain Lorentzian manifolds; these are two inverses which from a microlocal analysis perspective are more natural than the standard causal (advanced/retarded) ones. For instance, for the spectral family of the wave operator, these are the natural inverses when the spectral parameter is non-real. Indeed, I will explain that these connect to the self-adjointness of the wave operator, and the positivity properties that follow.

• Gunther Uhlmann: Some Geometric Inverse Problems

  We will survey results on some geometric inverse problems, arising in different contexts.

• Christopher Sogge: Microlocal Kakeya-Nikodym estimates and improved eigenfunction estimates

  We discuss the problem of trying to obtain improved eigenfunction estimates under geometric assumptions, such as the presence of negative sectional curvatures. In particular, we discuss how the use microlocal Kakeya-Nikodym estimates (championed by Zelditch) can be used to simplify an argument of Blair and I to obtain improvements for the critical Lebesgue exponent and hence all exponents q>2. These are used along with a combination of local and global harmonic analysis that further exploit geometric assumptions.

• John Toth: Quantitative unique continuation for L²-restrictions of eigenfunction sequences

  Let (M, g) be a C^∞ compact, Riemannian manifold and u_h ∈ C^∞(M) be a sequence of L²-normalized Laplace eigenfunctions with (−h²Δ_g−1) u_h = 0. Let H ⊂ M be a smooth hypersurface. In the terminology of Zelditch and Toth, the hypersurface H is good relative to the sequence u_h if there exist constants h₀ (H),  C_H > 0 such that for all h ∈ (0, h₀ (H)] ,

  ∫_H |u_h|² ds_H  ≥  e ^−C_H/h
  In the talk, I will describe some recent results on goodness of hypersurfaces relative to eigenfunction sequences and give some applications to nodal sets of eigenfunction sequences.

• Jeffrey Galkowski: Weyl laws and closed geodesics on typical manifolds

  We discuss the typical behavior of two important quantities on compact Riemannian manifolds: the number of primitive closed geodesics of a certain length and the error in the Weyl law. For Baire generic metrics, the qualitative behavior of both of these quantities has been well understood since the 1970's and 1980's. Nevertheless, their quantitative behavior for typical manifolds has remained mysterious. In fact, only recently, Contreras proved an exponential lower bound for the number of closed geodesics on a Baire generic manifold. Until now, this was the only quantitative estimate on the number of geodesics for typical metrics, and no such estimate existed for the remainder in the Weyl law. In this talk, we give stretched exponential upper bounds on the number of primitive closed geodesics for typical metrics. Furthermore, using recent results on the remainder in the Weyl law, we will use our dynamical estimates to show that logarithmic improvements in the remainder in the Weyl law hold for typical manifolds. The notion of typicality used in this talk is a new analog of full Lebesgue measure in infinite dimensions called predominance. Based on joint work with Y. Canzani.

• Eugenia Mallinnikova: Local version of the Courant nodal domain theorem

  Let M be a closed n-dimensional smooth Riemannian manifold. We consider the eigenfunctions of the Laplace operator on M and estimate the number of nodal domains of kth eigenfunction that intersect a given geodesic ball. The problem of local bounds for the volume and for the number of nodal domains was raised by Donnelly and Fefferman, who also proposed an idea how one can prove such bounds. We combine their idea with two new ingredients: the recent sharp Remez type inequality for eigenfunctions and the Landis type growth lemma in narrow domains. The talk is based on a joint work with Sagun Chanillo, Alexander Logunov, and Dan Mangoubi.

• Andrew Hassell: Semiclassical scarring for KAM systems

  A hallmark of Steve’s approach to mathematics is his generosity with ideas. I will describe a few occasions on which a conversation with Steve was the impetus for results which I obtained sometimes a decade or more later. I will concentrate on discussing a recently published result with Sean Gomes on scarring for KAM systems.

• Michael Douglas: Numerical methods in higher dimensional geometry

  We discuss the prospects for combining numerical and analytic methods to get rigorous results in higher dimensional differential geometry.

• Yanir Rubinstein: On Nazarov's proof of the Bourgain-Milman inequality

  Over 85 years ago, Mahler conjectured several sharp inequalities concerning the volume of convex bodies in two cases: symmetric bodies and general bodies, with neither implying the other. In the 80's, Bourgain-Milman proved a breakthrough sub-optimal version of these inequalities and since then many researchers have gradually improved the best known constant in these inequalities. All proofs so far have derived the general case from the symmetric case via the standard symmetrization trick. A decade ago Nazarov devised a beautiful Bergman kernel proof of the Bourgain-Milman in the symmetric case and suggested that this approach should yield a direct (though still sub-optimal) proof of the general case, without using symmetrization. In joint work with V. Mastrantonis, we follow many of Nazarov's ideas, some ideas of Berndtsson and Hultgren, as well as some new ingredients, to confirm Nazarov's prediction.

• Vincent Vargas: Quantum gravity and Mabuchi path integral-I

  In the first part of these talks, we will review recent beautiful ideas of Steve Zelditch (with Ferrari and Klevtsov) on 2d quantum gravity using the path integral formalism. In particular, they advocate the use of the Mabuchi functional using a Kähler parametrization of the metrics to construct quantum gravity.

• Rémi Rhodes: Quantum gravity and Mabuchi path integral-II

  In the second part of these talks, we will explain a recent alternative construction of Zelditch's (and co-authors) Mabuchi quantum gravity, which leads to a path integral whose anomaly presents the standard Liouville anomaly plus an additional K-energy term. Motivations come from models for fluctuating metrics arising when coupling (small) massive perturbations of conformal matter fields to quantum gravity. Based on a work with Hubert Lacoin.

• Bernard Shiffman: Linear statistics of random zeros on complex manifolds and Bergman kernel asymptotics

  Zeros of random complex polynomials, or of random sections of holomorphic line bundles on a Riemann surface, are self-averaging in the sense that as their degree increases, the fluctuations of the zeros are of smaller order than their density. In 2004, Sodin and Tsirelson showed that these fluctuations as measured by the variance of the "linear statistics" of the zeros (the sum of the values of a smooth test function at the zeros) of a random polynomial of increasing degree k decay at a rate proportional to 1/k. In joint work with Zelditch in 2010, we generalized this result to holomorphic line bundles on Riemann surfaces and in higher dimensions to compact Kähler manifolds. Our results depend on the off-diagonal asymptotic expansion of the Bergman kernel. In this talk, I shall apply Bergman kernel asymptotics to show that the variance of the linear statistics also has an asymptotic expansion.

• Peter Sarnak: Quantum Ergodicity and beyond

  We trace some recent developments by Zelditch and others, which have their roots in his pioneering Quantum Ergodicity Theorem.

• Semyon Dyatlov: Semiclassical measures for quantum cat maps

  Semiclassical measures are a standard object studied in quantum chaos, capturing macroscopic behavior of sequences of eigenfunctions in the high energy limit. In previous work with Jin and Nonnenmacher we showed that for Laplacian eigenfunctions on negatively curved surfaces, semiclassical measures have full support. This result is not available in higher dimensions because the key new ingredient, the fractal uncertainty principle (proved by Bourgain and the speaker), is only known for subsets of the real line. In this talk I will present joint work with Jézéquel which focuses on eigenfunctions of quantum cat maps in higher dimensions. We show that, if the quantized matrix has a unique largest eigenvalue and its characteristic polynomial is irreducible over the rationals, then all semiclassical measures have full support. The proof uses the one-dimensional fractal uncertainty principle and a careful choice of propagation time.

• Stéphane Nonnenmacher: Random eigenstates of the Quantum Cat Map

  The quantized hyperbolic automorphisms of the 2-torus, also known as the "Quantum Cat Map", is a popular toy model of quantized chaotic system. It can be obtained through various quantization procedures, including the Toeplitz quantization (by S. Zelditch in 1997). A peculiarity of this system is the presence, in the semiclassical limit, of "maximally large" spectral multiplicities. It is known that these "maximal" multiplicities allow to construct non-equidistributed eigenstates, thus contradicting Quantum Unique Ergodicity in general. In the present work we consider random eigenbases of the quantum cat map, in this context of large multiplicities. Inspired by previous works of Zelditch and Han, we prove that, almost surely in the semiclassical limit, random eigenbases satisfy QUE, including at (algebraic) small scales. We also show that the local statistics of these random eigenstates converges to that of standard Gaussian random states (in particular, their value distribution is asymptotically Gaussian).

• Nalini Anantharaman: The bottom of the spectrum of a random hyperbolic surfaces, recent developments

  I will report on joint work with Laura Monk, aiming to show that the lowest eigenvalue of a random compact hyperbolic surfaces (for the Weil-Petersson model) is typically greater than 1/4-ε, in the large genus asymptotic. We introduce generalized volume functions (associated to non-simple closed geodesics) and we show this functions belong to what we call the "Friedman-Ramanujan class" (a notion imported from J. Friedman's work on random regular graphs).

• Maciej Zworski: The scattering phase: seen at last

  The scattering phase is the analogue of the counting function for eigenvalues when dealing with exterior domains. It is closely related to Krein's shift function in spectral theory and to phase shifts in physics. We revisit classical results on asymptotics of the scattering phase (including the result of Hassell-Zelditch which was the first asymptotic formula at 0 in 2D) and point out that it is never monotone in the case of strong trapping of waves. Perhaps more interestingly, we provide the first numerical calculations of scattering phases for non-radial scatterers. They show that the asymptotic Weyl law is accurate even at low frequencies and reveal effects of trapping such as lack of monotonicity. This is achieved by using the recent high level multiphysics finite element software FreeFEM. The talk is based on joint work with J. Galkowski, P. Marchand and J. Wang.