Résumés singularities 2024

The classical umkehr map of Hopf assigns to a map of oriented manifolds, f:M \to N, `wrong-way’ homomorphisms in homology f_!: H_*(N) \to H_*(M) and in cohomology f^!:H^*(M) \to H^*(N), the latter a version of `integration over the fibers’. Similar wrong-way maps, sometimes known as transfer maps or Gysin maps, are defined for other generalized (co)homology theories as long as the manifolds are suitably oriented and have had many applications. While these maps are defined only for manifolds there has long been interest in extending them to singular spaces. I’ll discuss joint work with Markus Banagl and Paolo Piazza in which we capitalize on recent work on the index theory of signature operators to give analytic definitions of transfer maps in K-homology for stratified spaces and relate them to topological orientations.

Geometric (pre)quantization can be performed only for integral symplectic manifolds. In 1989 Alan Weinstein conjectured that using the notion of Diffeology, as well as Noncommutative Geometry methods, one might obtain the representations required to « quantise the unquantisable » from a torus bundle rather than a line bundle. In joint work with P. Antonini, we showed that the obstruction to integrality can be lifted by adding extra dimensions and passing to the diffeological category. In fact, the added dimensions force the C*-algebra associated with this construction to be nothing else than the crossed product algebra associated with a torus action.

In this talk I’ll discuss recent and ongoing work concerning two geometric flows. The first result concerns « convergence stability » for normalized Ricci flow near rotationally symmetric asymptotically hyperbolic metrics. The second result concerns a stability result for a higher-order geometric flow modeled on the Bach tensor of conformal geometry. Our analytic methods exploit semigroup techniques, and I’ll discuss a criterion for a geometric elliptic operator to generate an analytic semigroup on weighted Hölder spaces on manifolds of bounded geometry. This talk is based on joint work with Guenther, Isenberg and Mazzeo.

This is a joint work with Ilaria Mondello and David Tewodrose. In 1985, Bakry and Emery have introduced a curvature condition for Weighted Riemannian manifold generalizing the lower bound on the Ricci curvature. This Bakry-Emery curvature have been extensively studied and have lead to the notion of synthetic lower bound on Ricci curvature for metric measure space by J.Lott-C.Villani et K-T. Sturm.
The Kato condition for the Ricci curvature is a weakening of the lower Ricci bound condition. With Ilaria Mondello and David Tewodrose, we have investigated limit of Riemannian manifold with such a Kato condition. Recently using a change times in Brownian motion, we have highlighted a relationship between the Kato condition and the Bakry-Emery condition. I will explain these conditions and this relationship and its potential usefulness.

We consider nonlinear elliptic equations with a gradient-dependent nonlinearity and a Hardy-type potential in a bounded domain punctured at zero. We present new results on the existence and classification of the positive solutions near zero. In the case of a ball, using a dynamical systems approach, we prove the existence of positive radial solutions, manifesting near zero each the profiles we ascertain from our classification results. This proves not only the optimality of our classification results in a variety of ranges for our parameters but also gives further insight into the refined asymptotic behaviour of the solutions near zero. By a modified Kelvin transform, our results also render the asymptotic behaviour of solutions at infinity. This is joint work with Dr Maria Farcaseanu (ISMMA Institute of the Romanian Academy).

In this talk, following an approach pioneered by Melrose, we prove that Nakajima quiver varieties are quasi-asymptotically conical (QAC) manifolds when the moment map parameters are generic. The iterated structure at infinity together with recent results of Kottke-Rochon on quasi-fibered boundary (QFB) metrics allows us to further prove a Sen type theorem for the L^2 cohomology of any Nakajima quiver variety under the above genericity condition. This is an ongoing project, joint with Frédéric Rochon.

For a quasilinear equation involving the n−Laplacian and an exponential nonlinearity, I will discuss quantization issues for blow-up masses in the non-compact situation, where the exponential nonlinearity concentrates as a sum of Dirac measures. A fundamental tool is provided here by some Harnack inequality of sup + inf type, a question of independent interest that we prove in the quasilinear context through a new and simple blow-up approach.

The Caffarelli-Silvestre extension theorem allows one to obtain the fractional Laplacian of fractional order in (0,1) via a Dirichlet-to-Neumann map which maps solutions to a weighted Dirichlet problem on Euclidean halfspace to functions on its boundary. Dirichlet’s principle gives a resulting sharp Sobolev trace inequality. Given that the fractional Laplacian may be defined for higher fractional orders and analogous fractional operators are naturally defined on conformal of certain geometries, it is natural to extend Caffarelli-Silvestre’s results to the higher order and for general geometries. In this talk, we discuss relevant recent results obtained in collaboration with G. Lu and Q. Yang.

We construct the Feynman propagator for Klein-Gordon (KG) equation on Minkowski space perturbed by a decaying spatial potential. This extends works of Vasy and Gerard-Wrochna, in which potential perturbations of KG are assumed to decay in spacetime, in particular to vanish as time goes to infinity. Thus, for a large class of potentials which decay in space but not necessarily in time, we construct global in time solutions to the inhomogeneous KG equation, each of whose wavefront set is the flowout of the wavefront set of the source, in the direction of the Hamilton flow. (Such solutions were shown to exist locally by Duistermaat-Hormander.) To accomplish this, we prove a global Fredholm estimate. The persistence of the potential in time means that estimates for KG can be obtained using positive commutator estimates with operators in the three-body calculus of Vasy. This is joint work with Dean Baskin and Moritz Doll.

This talk is closely related to David Sher’s talk at this meeting on Tuesday.
We consider families $P_h$ of differential operators on an interval that depend on a parameter $h\geq 0$ and degenerate as $h\to 0$. We consider the problem of constructing quasimodes, i.e. (families of) solutions $u_h$, $h>0$, of $P_h u_h = O(h^\infty)$ as $h\to 0$. A classical example is the semi-classical Schroedinger Operator $P_h = h^2 \partial^2 + V$ where $\partial = d/dx$ and $V$ is a smooth function. If $V$ is positive then quasimodes can be found using the standard WKB method. At zeroes of $V$ additional difficulties arise (solved by Olver long ago) due to different scaling behavior near and away from the zeroes. Another classical example is Bessel’s equation with parameter $\nu=1/h$, where the behavior of solutions uniformly for large parameter and large argument is of interest (and well-known). One application of quasimode constructions is to find approximations of the spectrum of $P_h$ for small positive $h$.
We construct, and give a precise description of, full sets of quasimodes for very general families $P_h=P(x,\partial_x,h)$ of any order, including the examples above as well as operators where the coefficients depend analytically on $x$ and $h$, under a mild genericity hypothesis. The generality of the setup leads to a high degree of combinatorial and analytic complexity, which can be handled by an efficient representation of the data by Newton polygons and of the result in terms of iterated blow-ups and a suitable class of oscillatory-polyhomogeneous functions.
This is joint work with Dennis Sobotta.

Dimension 4 is the next horizon for applications of Ricci flow to topology, where the main goal is to understand the topological operations that Ricci flow performs both at singular times, and in its long-term behavior.With Alix Deruelle, we explain how Ricci flow develops or resolves orbifold singularities by a notion of stability for orbifold Ricci solitons that we introduce depending only on the curvature at the singular points. We construct ancient and immortal Ricci flows spontaneously forming or desingularizing arbitrarily complicated orbifold singularities by bubbling-off Ricci-flat ALE metrics.

This unexpectedly predicts that singular spherical and cylindrical orbifolds should not appear as finite-time singularity models. On the other hand, (complex) hyperbolic orbifolds appear as limits of immortal « thick » 4-dimensional Ricci flows as t\to+\infty.

I will talk about sharp upper bounds for the first eigenvalue $\lambda_{1,p}$ of the p-Laplacian on asymptotically hyperbolic manifolds and (some of) their submanifolds. As a corollary, I’ll show that for any minimal conformally compact submanifold Y^{k+1}$ within \mathbb{H}^{n+1}(-1), \lambda_{1,p}(Y)=\left(\frac{k}{p}\right)^{p} I will then talk about lower bounds on \lambda_{1,2}(Y) in the case where minimality is replaced with a bounded mean curvature assumption and where the ambient space is a general Poincar\’e-Einstein space whose conformal infinity is of non-negative Yamabe type. This is joint work with Aaron Tyrrell.

I will present some old and new results concerning existence of sign-changing solutions to the Yamabe problem on the round sphere and existence of positive solutions to a class of systems of PDE’s with critical growth in the whole euclidean space in presence of a competitive regime.

We consider in this talk the second conformal eigenvalue of the conformal laplacian in dimensions larger than 3. It is defined as the infimum, over all metrics in a given conformal class of positive Yamabe type, of the second eigenvalue of the conformal laplacian for the metric. We prove that in dimensions 3 to 7 the second conformal eigenvalue is never attained for metric close to the round metric on the sphere. This is ongoing work with J. Vétois (Mc Gill).

Solutions to the Einstein vacuum equations are Ricci flat Lorentzian manifolds and among them the static and the stationary solutions are particularly relevant. Using techniques in comparison geometry, we will show that there are vacuum static 3+1 black hole solutions, metrically complete but with a non-standard spatial topology and asymptotics, that cannot be put into stationary rotation, that is, there are no non-static stationary metrics close to them. To our knowledge, this is the first result of this kind in the literature. This is joint work with Javier Peraza.

We will explore the Casimir energy for 2-dimensional hyperbolic orbifold surfaces that may have finitely many conical singularities. In computing the contribution to the energy from a conical singularity, we derive an expression of an elliptic orbital integral as an infinite sum of special functions that converges exponentially quickly. Under a natural assumption (known to hold asymptotically) on the growth of the lengths of primitive closed geodesics of the (2,3,7)-triangle group orbifold we will see that its Casimir energy is positive (repulsive). Apparently this has implications for warp travel in the presence of gravitational singularities like black holes. This talk is based on joint work with K. Fedosova and G. Zhang.

In mathematical general relativity, the notion of mass has been defined for certain classes of manifolds that are asymptotic to a fixed model background. Typically, the mass is an invariant computed in a chart at infinity, which is related to the scalar curvature and has certain positivity properties. When the model is hyperbolic space, under certain assumptions on the geometry at infinity one can compute the mass using the so-called mass aspect function, a function on the unit sphere extracted from the term describing the leading order deviation of the metric from the hyperbolic background. This definition of mass, due to Xiaodong Wang, is a particular case of the definition by Chruściel and Herzlich which proceeds by taking the limit of certain surface integrals and applies to asymptotically hyperbolic manifolds with less stringent asymptotics. In this talk I will present our joint work with Romain Gicquaud defining the mass aspect function and the mass for asymptotically hyperbolic manifolds of low regularity. We show that in this setting one can use cut-off functions to define suitable replacements to the potentially ill-defined surface integrals of Chruściel and Herzlich. Moreover, we are able to define the mass aspect function as a distribution on the unit sphere for metrics having slower fall-off towards hyperbolic metric than those originally considered by Xiaodong Wang. The related notion of mass is well-behaved under changes of coordinates and we expect that the positivity can be proven.

We apply the techniques of geometric microlocal analysis to a classical and well-studied analysis problem: the asymptotic expansions of Bessel functions as both their order and argument go to infinity. By focusing on the modulus and phase of these functions, we are able to give a complete description of their asymptotic behavior on an appropriate blown-up space. The result is a unified picture encompassing all previously known expansions and covering all intermediate regimes.

Yi Wang (Johns Hopkins University)

Existence of fully nonlinear Yamabe metrics on noncompact manifolds