Publications

When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory language, this means that the theory is twisted, which gives rise to an anomaly theory. In this paper, we give a detailed construction of this anomaly theory, as a functor that sends manifolds to infinite-dimensional Clifford algebras and bordisms to bimodules.

Related project(s):
5Index theory on Lorentzian manifolds

Let (Mi,gi)i∈N be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower dimensional Riemannian manifold (B,h) in the Gromov-Hausdorff topology. Lott showed that the spectrum converges to the spectrum of a certain first order elliptic differential operator D on B. In this article we give an explicit description of D. We conclude that D is self-adjoint and characterize the special case where D is the Dirac operator on B.

Related project(s):
5Index theory on Lorentzian manifolds

In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds in Rn+m. We first provide, following Cabré \cite{Cabre2008}, an ABP proof of the relative isoperimetric inequality proved in Choe-Ghomi-Ritoré \cite{CGR07}, by generalizing ideas of restricted normal cones given in \cite{CGR06}. Then we prove a relative isoperimetric inequalities for minimal submanifolds in Rn+m, which is optimal when the codimension m≤2. In other words we obtain a relative version of isoperimetric inequalities for minimal submanifolds proved recently by Brendle \cite{Brendle2019}. When the codimension m≤2, our result gives an affirmative answer to an open problem proposed by Choe in \cite{Choe2005}, Open Problem 12.6. As another application we prove an optimal logarithmic Sobolev inequality for free boundary submanifolds in the Euclidean space following a trick of Brendle in \cite{Brendle2019b}.

Related project(s):
22Willmore functional and Lagrangian surfaces

In this work we describe horofunction compactifications of metric spaces and finite dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, ultrapowers of the spaces at hand. The compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.

Related project(s):
20Compactifications and Local-to-Global Structure for Bruhat-Tits Buildings

In this paper we consider a Ricci de Turck flow of spaces with isolated conical singularities, which preserves the conical structure along the flow. We establish that a given initial regularity of Ricci curvature is preserved along the flow. Moreover under additional assumptions, positivity of scalar curvature is preserved under such a flow, mirroring the standard property of Ricci flow on compact manifolds. The analytic difficulty is the a priori low regularity of scalar curvature at the conical tip along the flow, so that the maximum principle does not apply. We view this work as a first step toward studying positivity of the curvature operator along the singular Ricci flow.

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry23Spectral geometry, index theory and geometric flows on singular spaces

Given a generic stable strongly parabolic $SL(2,\mathbb{C})$-Higgs bundle

$(\mathcal{E}, \varphi)$, we describe the family of harmonic metrics $h_t$ for

the ray of Higgs bundles $(\mathcal{E}, t \varphi)$ for $t\gg0$ by perturbing

from an explicitly constructed family of approximate solutions

$h_t^{\mathrm{app}}$. We then describe the natural hyperK\"ahler metric on

$\mathcal{M}$ by comparing it to a simpler "semi-flat" hyperK\"ahler metric. We

prove that $g_{L^2} - g_{\mathrm{sf}} = O(\mathrm{e}^{-\gamma t})$ along a

generic ray, proving a version of Gaiotto-Moore-Neitzke's conjecture.

  Our results extend to weakly parabolic $SL(2,\mathbb{C})$-Higgs bundles as

well.

  In the case of the four-puncture sphere, we describe the moduli space and

metric more explicitly. In this case, we prove that the hyperk\"ahler metric is

ALG and show that the rate of exponential decay is the conjectured optimal one,

$\gamma=4L$, where $L$ is the length of the shortest geodesic on the base curve

measured in the singular flat metric $|\mathrm{det}\, \varphi|$.

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

Let X be a compact Calabi-Yau 3-fold, and write \(\mathcal{M}, \overline{\mathcal{M}}\) for the moduli stacks of objects in coh(X) and the derived category D^b coh(X). There are natural line bundles \(K_{\mathcal{M}} \to \mathcal{M}, K_{\overline{\mathcal{M}}} \to \overline{\mathcal{M}}\) analogues of canonical bundles. Orientation data is an isomorphism class of square root line bundles \(K_{\mathcal{M}}^{1/2}, K_{\overline{\mathcal{M}}}^{1/2}\), satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman in their theory of motivic Donaldson-Thomas invariants, and is also important in categorifying Donaldson-Thomas theory using perverse sheaves. We show that natural orientation data can be constructed for all compact Calabi-Yau 3-folds X, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi-Yau 3-folds X that admit a spin smooth projective compactification.

Related project(s):
33Gerbes in renormalization and quantization of infinite-dimensional moduli spaces

We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

We show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

Pseudo H-type Lie groups \(G_{r,s}\) of signature (r,s) are defined via a module action of the Clifford algebra \(C\ell_{r,s}\) on a vector space V≅\(\mathbb{R}^{2n}\). They form a subclass of all 2-step nilpotent Lie groups and based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let \(\mathcal{N}_{r,s}\) denote the Lie algebra corresponding to \(G_{r,s}\). A choice of left-invariant vector fields [\(X_1, \ldots, X_{2n}\)] which generate a complement of the center of \(\mathcal{N}_{r,s}\) gives rise to a second order operator

\(\Delta_{r,s}:=\big{(}X_1^2+ \ldots + X_n^2\big{)}- \big{(}X_{n+1}^2+ \ldots +X_{2n}^2 \big{)}\)

which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutions of \(\Delta_{r,s}\) in the case r=0, s>0 and study their properties. In the case of r>0 we prove that \(\Delta_{r,s}\) admits no fundamental solution in the space of tempered distributions. Finally we discuss the local solvability of \(\Delta_{r,s}\) and the existence of a fundamental solution in the space of Schwartz distributions.

Related project(s):
6Spectral Analysis of Sub-Riemannian Structures

We show that the space introduced by Vasy in order to construct a pseudodifferential calculus adapted to the N-body problem can be obtained as the primitive ideal spectrum of one of the N-body algebras considered by Georgescu. In the process, we provide an alternative description of the iterated blow-up space of a manifold with corners with respect to a clean semilattice of adapted submanifolds (i.e. p-submanifolds). Since our constructions and proofs rely heavily on manifolds with corners and their submanifolds, we found it necessary to clarify the various notions of submanifolds of a manifold with corners.

Related project(s):
3Geometric operators on a class of manifolds with bounded geometry

We prove that the sign of the Euler characteristic of arithmetic groups with CSP is determined by the profinite completion.  In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F.  Our methods imply similar results for L2-torsion as well as a strong profiniteness statement for Novikov--Shubin invariants.

Related project(s):
18Analytic L2-invariants of non-positively curved spaces58Profinite perspectives on l2-cohomology

For a smooth manifold \(M\), possibly with boundary and corners, and a Lie group \(G\), we consider a suitable description of gauge fields in terms of parallel transport, as groupoid homomorphisms from a certain path groupoid in \(M\) to \(G\).  Using a cotriangulation \(\mathscr{C}\) of \(M\), and collections of finite-dimensional families of paths relative to \(\mathscr{C}\), we define a homotopical equivalence relation of parallel transport maps, leading to the concept of an extended lattice gauge (ELG) field. A lattice gauge field, as used in Lattice Gauge Theory, is part of the data contained in an ELG field, but the latter contains further local topological information sufficient to reconstruct a principal \(G\)-bundle on \(M\) up to equivalence. The space of ELG fields of a given pair \((M,\mathscr{C})\) is a covering for the space of fields in Lattice Gauge Theory, whose connected components parametrize equivalence classes of principal \(G\)-bundles on \(M\). We give a criterion to determine when ELG fields over different cotriangulations define equivalent bundles.

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

Photon surfaces are timelike, totally umbilic hypersurfaces of Lorentzian spacetimes. In the first part of this paper, we locally characterize all possible photon surfaces in a class of static, spherically symmetric spacetimes that includes Schwarzschild, Reissner--Nordström, Schwarzschild-anti de Sitter, etc., in n+1dimensions. In the second part, we prove that any static, vacuum, "asymptotically isotropic" n+1-dimensional spacetime that possesses what we call an "equipotential" and "outward directed" photon surface is isometric to the Schwarzschild spacetime of the same (necessarily positive) mass, using a uniqueness result by the first named author.

Related project(s):
5Index theory on Lorentzian manifolds

We show \(R\)-sectoriality for the fractional powers of possibly non-invertible \(R\)-sectorial operators. Applications concern existence, uniqueness and maximal \(L^{q}\)-regularity results for solutions of the fractional porous medium equation on manifolds with conical singularities. Space asymptotic behavior of the solutions close to the singularities is provided and its relation to the local geometry is established. Our method extends the freezing-of-coefficients method to the case of non-local operators that are expressed as linear combinations of terms in the form of a product of a function and a fractional power of a local operator.

Related project(s):
30Nonlinear evolution equations on singular manifolds

The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen [2016] developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to compute the Bartnik mass for minimal spheres satisfying a stability condition. In this work, we develop extensions and gluing tools, à la Mantoulidis and Schoen, for time-symmetric initial data sets for the Einstein-Maxwell equations that allow us to compute the value of an ad-hoc notion of charged Barnik mass for suitable charged minimal Bartnik data.

Related project(s):
5Index theory on Lorentzian manifolds

Relying on the theory of agrarian invariants introduced in previous work, we solve a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, when the groups are additionally torsion-free, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl-Tillmann and Friedl-Lück-Tillmann.

Related project(s):
8Parabolics and invariants

Parallel transport as dictated by a gauge field determines a collection of local reference systems. Comparing local reference systems in overlapping regions leads to an ensemble of algebras of relational kinematical observables for gauge theories including general relativity. Using an auxiliary cellular decomposition, we propose a discretization of the gauge field based on a decimation of the mentioned ensemble of kinematical observables. The outcome is a discrete ensemble of local subalgebras of 'macroscopic observables' characterizing a measuring scale. A set of evaluations of those macroscopic observables is called an extended lattice gauge field because it determines a G-bundle over M (and over submanifolds of M that inherit a cellular decomposition) together with a lattice gauge field over an embedded lattice. A physical observable in our algebra of macroscopic observables is constructed. An initial study of aspects of regularization and coarse graining, which are special to this description of gauge fields over a combinatorial base, is presented. The physical relevance of this extension of ordinary lattice gauge fields is discussed in the context of quantum gravity.

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

We provide a unified treatment of several results concerning full groups of ample groupoids and paradoxical decompositions attached to them. This includes a criterion for the full group of an ample groupoid being amenable as well as comparison of its orbit, Koopman and groupoid-left-regular representations. Besides that, we unify several recent results about paradoxicality in semigroups and groupoids, relating embeddings of Thompson's group V into full groups of ample étale groupoids.

Related project(s):
2Asymptotic geometry of sofic groups and manifolds

We show a rigidity result for subfactors that are normalized by a representation of a lattice Γ in a higher rank simple Lie group with trivial center into a finite factor. This implies that every subfactor of LΓ which is normalized by the natural copy of Γ is trivial or of finite index.

Related project(s):
2Asymptotic geometry of sofic groups and manifolds