# Geometric and Spectral Methods in Partial Differential Equations (16w5066)

Arriving in Oaxaca, Mexico Sunday, December 11 and departing Friday December 16, 2016

## Organizers

Richard Melrose (Massachusetts Institute of Technology)

Pablo Suárez Serrato (Universidad Nacional Autónoma de México)

Pierre Albin (University of Illinois, Urbana-Champaign)

Jesse Gell-Redman (Johns Hopkins University)

## Objectives

This workshop builds on recent progress on the geometric analysis of non-compact and singular spaces arising in geometry, especially that related to the study of linear partial differential equations. For example, take the work on moduli space of Riemann surfaces by Ji, Mazzeo, M\"uller, and Vasy [JMMV]; this addresses a central and in some ways primative question, the basic nature of the spectrum, on an incomplete manifold with a certain structure at the singular locus. Question more related to index theory and invariant theory can be found in the work on signature theory for iterated cone edge spaces in [ALMPI, ALMPII], on the anti-self-dual deformation complex [LV2013], on signature formulas and Euler characteristics in [AL2013], and on Chern-Simons theory on Teichm\"uller space [GM2014]. Further work related to important recently resolved questions in differential geometry used tools similar to those above (as we elaborate below) in the work on K\"ahler-Einstein metrics [JMR, D2012, T2013], and also the applications to physics, in particular general relativity, in [VD2013, HVsemi], as well as some infinite dimensional contexts [Mathai-Melrose, KM].

It will differ from other meetings in geometric analysis chiefly due to its focus on microlocal methods. It will compliment nicely the November 2014 BIRS workshop ``Geometric Scattering Theory an Applications,'' where microlocal methods will also be emphasized, but the topics will be related primarily to scattering theory and general relativity. Not only the topics then, but also the microlocal methods themselves will be quite different. The main areas of focus will be:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The workshop program will place an emphasis on the work of the many talented junior researchers in the field. A typical day will have two 50 minute talks by junior researchers, and two 30 minute talks by senior researchers. Each day's talks will be on the same or closely related topics. On two days, one of the 50 minute talks will be supplanted by a survey talk on a broad topic given by a senior researcher. We will have two morning talks, one long and one short, followed by a long break to give participants space to begin and extend collaborations, followed by two afternoon talks. We will also include an open problem discussion session.

Our list of participants includes experts in microlocal analysis, dynamics, topology, spectral theory and scattering theory. Our aim is to help junior researchers lay the groundwork for collaborations that will enlarge on connections between different fields.

We now outline the research programs involved together with their connections and applications

A substantial portion of all recent work in differential geometry has taken place in the context of singular spaces. Such singular spaces are in fact of central importance, as they arise in natural and crucial contexts; as moduli spaces of geometric structures on smooth, compact manifolds without boundary, as quotients of smooth manifolds by the action of a smooth compact lie group, or as zero sets of smooth polynomials. Thus, they arise even when one considers only smooth compact spaces, and one is forced to confront them. This has happened notably in the recent work on singular K\"ahler-Einstein manifolds, which saw a flurry of research recently by Brendle, Donaldson, Mazzeo-Jeffers-Rubinstein, and Tian. Another major example is the work of Li, Sun, and Yao, and also Ji, Mazzeo, M\"uller, and Vasy on the moduli space with the Weil-Peterrson metric

The central role that invariant theory, particularly index theory of linear elliptic operators plays in the study of compact manifolds without boundary has become, thanks to very recent developments, increasingly applicable in the study of such singular spaces. One has for example the recent work of Block and Viaclovsky on the index of the anti-self-dual deformation complex, and the index and Euler characteristic formulas of Atiyah and Lebrun on for dimensional cone-edge spaces. In fact, this recent work is related to a longstanding set of results in the theory of linear elliptic operators on singular spaces which goes back to Bismut and Cheeger, and whose modern imprint lies at the core of the modern theory pseudodifferential operators.

Indeed, pseudodifferential calculi (filtered algebras of pseudodifferential operators which under appropriate assumptions are closed under composition) on non-compact manifolds and singular spaces are the topic of much recent work, notably the aforementioned results of Mazzeo, Jeffers, and Rubinstein. In fact, there is a close connection between the non-compact and singular settings. In many cases of interest (including most of those mentioned) the construction of, for example, fundamental solutions for elliptic operators on singular spaces can be accomplished by leveraging preexisting constructions of fundamental solutions of related problems on complete, non-compact spaces. More importantly, in geometric microlocal analysis, the structures of such Green's functions (in both settings) can now be described in detail as conormal distributions on certain manifolds with corners obtained via radial blow up. In this context, one has both the work of Mazzeo-Melrose [Mazzeo-Melrose:Zero] on asymptotically hyperbolic spaces, and the more recent work of Mazzeo-Vertman and Krainer-Mendoza which uses the structure of parametrices for elliptic operators in the edge setting to tackle boundary value problems there. This is in turn related to the work for example of Albin, Leichtnam, Mazzeo, and Piazza on iterated edge spaces, where they use a good structure theorem for the Green's function for the Hodge Laplacian; its structure is related to the edge Green's function just discussed.

The determinant of the Laplacian and the associated torsion invariants have applications in topology, as they give analytic formul\ae{} for Reidemeister torsion, knot theory, where they generalize the Alexander polynomial, and physics, where they are related to Chern-Simons theory. Recently there has been a lot of interest in using analytic torsion of locally symmetric spaces as a way of studying the torsion subgroup of group cohomology [Bergeron-Venkatesh, Muller-Pfaff}. While these approaches rely on the Selberg trace formula and its extensions, and are thus restricted to a rigid class of spaces, they have inspired direct analytic approaches valid in much greater generality. Finding topological expressions for analytic torsion of noncompact and singular spaces is an open and important problem. The strong analytic surgery techniques developed in \cite{Mazzeo-Melrose:Surgery, Hassell-Mazzeo-Melrose:Surgery} and applied to spaces with cylindrical ends by Hassell in \cite{Hassell:AT] are currently being extended to spaces asymptotically modeled by locally symmetric spaces of rank one. [ARS1,ARS2].

Essentially, the possibility for interaction between spectral theory, invariant theory, and topology with the modern theory of linear PDE is a central and exciting topic in geometric analysis, and our purpose at this meeting is to foster connections between experts in all these topics.

It is worth pointing out here that we have not attempted to make a complete survey of the topics in geometry which may arise and indeed which play important roles in the research of many of our proposed invitation list. Indeed, this would not be possible in the space available here, but it is worth emphasizing further two of the topics mentioned above. First, dynamical systems, apart from being a topic of independent interest, plays an important role in spectral theory and PDE, indeed in semiclassical analysis, where it is connected via the classical-quantum correspondence to the high energy behaviour of eigenfunctions of the Laplacian, in the simplest instance on smooth planar domains with boundary. Second, we have the cutting edge work of Vasy, including work with Hintz, which has simplified the study of scattering theory and wave equations on space-times arising in general relativity.

\bibliographystyle{abbrv}

It will differ from other meetings in geometric analysis chiefly due to its focus on microlocal methods. It will compliment nicely the November 2014 BIRS workshop ``Geometric Scattering Theory an Applications,'' where microlocal methods will also be emphasized, but the topics will be related primarily to scattering theory and general relativity. Not only the topics then, but also the microlocal methods themselves will be quite different. The main areas of focus will be:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

- Index theory of elliptic linear differential operators, in particular index formulas, signature theory, K-theory, especially on noncompact and singular spaces. Relationships between index theory and the study of moduli spaces.
- Spectral asymptotics and spectral invariants: heat kernels and their asymptotic expansions, determinants, analytic torsion, again in singular and noncompact contexts.
- Special geometries, e.g. conformal geometry and hyperbolic geometry: spectral theory of hyperbolic quotients, conformally invariant differential operatos.
- Applications of linear PDE and geometric microlocal analysis to general relativity other topics in physics: scattering various space-times, linear and non-linear wave equations, Einstein's equations, and magnetic monopoles.

The workshop program will place an emphasis on the work of the many talented junior researchers in the field. A typical day will have two 50 minute talks by junior researchers, and two 30 minute talks by senior researchers. Each day's talks will be on the same or closely related topics. On two days, one of the 50 minute talks will be supplanted by a survey talk on a broad topic given by a senior researcher. We will have two morning talks, one long and one short, followed by a long break to give participants space to begin and extend collaborations, followed by two afternoon talks. We will also include an open problem discussion session.

Our list of participants includes experts in microlocal analysis, dynamics, topology, spectral theory and scattering theory. Our aim is to help junior researchers lay the groundwork for collaborations that will enlarge on connections between different fields.

We now outline the research programs involved together with their connections and applications

A substantial portion of all recent work in differential geometry has taken place in the context of singular spaces. Such singular spaces are in fact of central importance, as they arise in natural and crucial contexts; as moduli spaces of geometric structures on smooth, compact manifolds without boundary, as quotients of smooth manifolds by the action of a smooth compact lie group, or as zero sets of smooth polynomials. Thus, they arise even when one considers only smooth compact spaces, and one is forced to confront them. This has happened notably in the recent work on singular K\"ahler-Einstein manifolds, which saw a flurry of research recently by Brendle, Donaldson, Mazzeo-Jeffers-Rubinstein, and Tian. Another major example is the work of Li, Sun, and Yao, and also Ji, Mazzeo, M\"uller, and Vasy on the moduli space with the Weil-Peterrson metric

The central role that invariant theory, particularly index theory of linear elliptic operators plays in the study of compact manifolds without boundary has become, thanks to very recent developments, increasingly applicable in the study of such singular spaces. One has for example the recent work of Block and Viaclovsky on the index of the anti-self-dual deformation complex, and the index and Euler characteristic formulas of Atiyah and Lebrun on for dimensional cone-edge spaces. In fact, this recent work is related to a longstanding set of results in the theory of linear elliptic operators on singular spaces which goes back to Bismut and Cheeger, and whose modern imprint lies at the core of the modern theory pseudodifferential operators.

Indeed, pseudodifferential calculi (filtered algebras of pseudodifferential operators which under appropriate assumptions are closed under composition) on non-compact manifolds and singular spaces are the topic of much recent work, notably the aforementioned results of Mazzeo, Jeffers, and Rubinstein. In fact, there is a close connection between the non-compact and singular settings. In many cases of interest (including most of those mentioned) the construction of, for example, fundamental solutions for elliptic operators on singular spaces can be accomplished by leveraging preexisting constructions of fundamental solutions of related problems on complete, non-compact spaces. More importantly, in geometric microlocal analysis, the structures of such Green's functions (in both settings) can now be described in detail as conormal distributions on certain manifolds with corners obtained via radial blow up. In this context, one has both the work of Mazzeo-Melrose [Mazzeo-Melrose:Zero] on asymptotically hyperbolic spaces, and the more recent work of Mazzeo-Vertman and Krainer-Mendoza which uses the structure of parametrices for elliptic operators in the edge setting to tackle boundary value problems there. This is in turn related to the work for example of Albin, Leichtnam, Mazzeo, and Piazza on iterated edge spaces, where they use a good structure theorem for the Green's function for the Hodge Laplacian; its structure is related to the edge Green's function just discussed.

The determinant of the Laplacian and the associated torsion invariants have applications in topology, as they give analytic formul\ae{} for Reidemeister torsion, knot theory, where they generalize the Alexander polynomial, and physics, where they are related to Chern-Simons theory. Recently there has been a lot of interest in using analytic torsion of locally symmetric spaces as a way of studying the torsion subgroup of group cohomology [Bergeron-Venkatesh, Muller-Pfaff}. While these approaches rely on the Selberg trace formula and its extensions, and are thus restricted to a rigid class of spaces, they have inspired direct analytic approaches valid in much greater generality. Finding topological expressions for analytic torsion of noncompact and singular spaces is an open and important problem. The strong analytic surgery techniques developed in \cite{Mazzeo-Melrose:Surgery, Hassell-Mazzeo-Melrose:Surgery} and applied to spaces with cylindrical ends by Hassell in \cite{Hassell:AT] are currently being extended to spaces asymptotically modeled by locally symmetric spaces of rank one. [ARS1,ARS2].

Essentially, the possibility for interaction between spectral theory, invariant theory, and topology with the modern theory of linear PDE is a central and exciting topic in geometric analysis, and our purpose at this meeting is to foster connections between experts in all these topics.

It is worth pointing out here that we have not attempted to make a complete survey of the topics in geometry which may arise and indeed which play important roles in the research of many of our proposed invitation list. Indeed, this would not be possible in the space available here, but it is worth emphasizing further two of the topics mentioned above. First, dynamical systems, apart from being a topic of independent interest, plays an important role in spectral theory and PDE, indeed in semiclassical analysis, where it is connected via the classical-quantum correspondence to the high energy behaviour of eigenfunctions of the Laplacian, in the simplest instance on smooth planar domains with boundary. Second, we have the cutting edge work of Vasy, including work with Hintz, which has simplified the study of scattering theory and wave equations on space-times arising in general relativity.

\bibliographystyle{abbrv}

#### Bibliography

- [ALMPI] P.~Albin, {\'E}.~Leichtnam, R.~Mazzeo, and P.~Piazza.

The signature package on {W}itt spaces.

*Ann. Sci. \'Ec. Norm. Sup\'er. (4)*, 45(2):241--310, 2012. - [ALMPII] P.~Albin, {\'E}.~Leichtnam, R.~Mazzeo, and P.~Piazza.

Hodge theory on {C}heeger spaces.

*arxiv:1307.5473.v2*, 2013. - [AL2013] M.~Atiyah and C.~Lebrun.

Curvature, cones and characteristic numbers.

*Math. Proc. Cambridge Philos. Soc.*, 155(1):13--37, 2013. - [Bergeron-Venkatesh] N.~Bergeron and A.~Venkatesh.

The asymptotic growth of torsion homology for arithmetic groups.

*J. Inst. Math. Jussieu*, 12(2):391--447, 2013. - [D2012] S.~K. Donaldson.

K\"ahler metrics with cone singularities along a divisor.

In*Essays in mathematics and its applications*, pages 49--79. Springer, Heidelberg, 2012. - [GM2014] C.~Guillarmou and S.~Moroianu.

Chern-{S}imons line bundle on {T}eichm\"uller space.

*Geom. Topol.*, 18(1):327--377, 2014. - [Hassell:AT] A.~Hassell.

Analytic surgery and analytic torsion.

*Comm. Anal. Geom.*, 6(2):255--289, 1998. - [Hassell-Mazzeo-Melrose:Surgery] A.~Hassell, R.~Mazzeo, and R.~B. Melrose.

Analytic surgery and the accumulation of eigenvalues.

*Comm. Anal. Geom.*, 3(1-2):115--222, 1995. - [HVsemi] P.~Hintz and A.~Vasy.

Semilinear wave equations on asymptotically de {S}itter, {K}err-de {S}itter, and {M}inkowski spacetimes.

*arXiv:1306.4705*, 2013. - [JMR] T.~Jeffers, R.~Mazzeo, and Y.~Rubinstein.

K\"ahler-einstein metrics with edge singularities.

*arXiv:1105.5216*, 2011. - [JMMV] L.~Ji, R.~Mazzeo, W.~M\"uller, and A.~Vasy.

Spectral theory for the weil-petersson laplacian on the riemann moduli space.

*arXiv:1206.4010*, 2012. - [KM] C.~Kottke and R.~Melrose.

Equivalence of string and fusion loop-spin structures.

*arXiv:1309.0210*, 2013. - [LV2013] M.~T. Lock and J.~A. Viaclovsky.

An index theorem for anti-self-dual orbifold-cone metrics.

*Adv. Math.*, 248:698--716, 2013. - [Mathai-Melrose] V.~Mathai and R.~B. Melrose.

Geometry of {P}seudodifferential algebra bundles and {F}ourier {I}ntegral {O}perators.

Available online at arXiv:1210.0990. - [Mazzeo-Melrose:Zero] R.~Mazzeo and R.~B. Melrose.

Meromorphic extension of the resolvent on complete spaces with with asymptotically negative curvature.

*J. Funct. Anal.*, pages 260--310, 1987. - [Mazzeo-Melrose:Surgery] R.~Mazzeo and R.~B. Melrose.

Analytic surgery and the eta invariant.

*Geom. Funct. Anal.*, 5(1):14--75, 1995. - [Muller-Pfaff] W.~M{\"u}ller and J.~Pfaff.

Analytic torsion of complete hyperbolic manifolds of finite volume.

*J. Funct. Anal.*, 263(9):2615--2675, 2012. - [T2013] G.~Tian.

Partial {$C^0$}-estimate for {K}\"ahler-{E}instein metrics.

*Commun. Math. Stat.*, 1(2):105--113, 2013. - [VD2013] A.~Vasy.

Microlocal analysis of asymptotically hyperbolic and {K}err-de {S}itter spaces (with an appendix by {S}emyon {D}yatlov).

*Invent. Math.*, 194(2):381--513, 2013.