MICROLOCAL AND GLOBAL ANALYSIS, INTERACTIONS WITH...

MICROLOCAL AND GLOBAL ANALYSIS, INTERACTIONS WITH GEOMETRY

University of Potsdam, March 4-8, 2019

Presenters

Herbert Amann 4

Fabrice Baudoin 4

Wolfram Bauer 4

Francesco Bei 5

Karsten Bohlen 6

Maxim Braverman 6

Jochen Bruning 6

Marco Cappiello 7

Paulo Carrillo-Rouse 7

Hua Chen 7

Li Chen 8

Sandro Coriasco 8

Claire Debord 9

Nils Dencker 9

Michael Dreher 10

Karsten Fritzsch 10

Anahit Galstyan 10

Daniel Grieser 11

Gerd Grubb 11

Georges Habib 12

Bernard Helffer 12

Magda Khalile 13

Klaus Kirsten 13

Yuri Kordyukov 13

Matthias Lesch 141

2 MICROLOCAL AND GLOBAL ANALYSIS, INTERACTIONS WITH GEOMETRY

Xiaochun Liu 14

Ursula Ludwig 15

Irina Markina 15

Calin Martin 15

Gerardo Mendoza 16

Werner Muller 16

Vladimir Nazaikinskii 16

Victor Nistor 17

Paolo Piazza 18

Vladimir Rabinovich 18

Luigi Rodino 19

Julie Rowlett 19

Zhuoping Ruan 19

Anton Savin 20

Simon Scott 20

Alexander Strohmaier 21

Joachim Toft 21

Andras Vasy 22

Boris Vertman 22

Yawei Wei 22

Jens Wirth 23

Karen Yagdjian 23

Poster Session

Fernando de Avila Silva 25

Lashi Bandara 25

Valerii Galkin 26

Anton Kutsenko 26

Gianmarco Molino 26

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https://www.uni-potsdam.de/db/zeik-portal/gm/lageplan-up.php?komplex=1

http://www.krongut-bornstedt.de/level9_cms/?LC=EN


Herbert Amann

Institut fur Mathematik

Universitat Zurich

Ramistrasse 71

CH-8006 Zurich

Switzerland

[email protected]

Title: Parabolic equations on Riemannian manifolds.

Abstract: We describe local existence and regularity results for linear and nonlinear parabolic boundary valueproblems on Riemannian manifolds which can be noncompact and may have singularities.

Our work is based on maximal regularity in Sobolev and in Holder spaces. It covers quasilinear problems inSobolev space settings, as well as fully nonlinear equations in a Holder space framework.

〈 Back to Schedule 〉

Fabrice Baudoin

Department of Mathematics

University of Connecticut

341 Mansfield Road U1009

Storrs, CT 06269-1009

USA

[email protected]

Title: H-type foliations.

Abstract: With a view toward sub-Riemannian geometry, we introduce and study H-type foliations. Thesestructures are natural generalizations of K-contact geometries which encompass as special cases K-contact manifolds,twistor spaces, 3K-contact manifolds and H-type groups. Under an horizontal Ricci curvature lower bound, we proveon those structures sub-Riemannian diameter upper bounds and first eigenvalue estimates for the sub-Laplacian.Then, using a result by Moroianu-Semmelmann, we classify the H-type foliations that carry a parallel horizontalClifford structure. Finally, we prove an horizontal Einstein property and compute the horizontal Ricci curvature ofthose spaces in codimension more than 2. This is joint work with Erlend Grong, Gianmarco Molino and Luca Rizzi.


Wolfram Bauer

Institut fur Analysis

Fakultat fur Mathematik und Physik

Leibniz Universitat Hannover

Welfengarten 1

DE-30167 Hannover

Germany

[email protected]

Title: Ultra-hyperbolic operators on pseudo-H-type groups.

Abstract: Pseudo H-type Lie groups Gr,s of signature (r, s) are defined via a module action of the Clifford algebraC`r,s on a vector space V ∼= R2n. They form a subclass of all 2-step nilpotent Lie groups and based on their algebraicstructure they can be equipped with a left-invariant pseudo-Riemannian metric. Let Nr,s denote the Lie algebra

MICROLOCAL AND GLOBAL ANALYSIS, INTERACTIONS WITH GEOMETRY 5

corresponding to Gr,s. A choice of left-invariant vector fields [X1, · · · , X2n] which generate a complement of thecenter of Nr,s gives rise to a second order operator

∆r,s :=(X2

1 + · · ·+X2n

)−(X2n+1 + · · ·+X2

2n

),

which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutionsof ∆r,s in the case r = 0, s > 0 and study their properties. It is shown that for r > 0 the operator ∆r,s admitsno fundamental solution in the space of tempered distributions. We discuss the local solvability of ∆r,s and theexistence of a fundamental solution in the larger class of Schwartz distributions. This is a joint work with I. Markina(University of Bergen) and A. Froehly (formerly Leibniz Universitat Hannover).

References:

[1] W. Bauer, A. Froehly and I. Markina, The fundamental solution of a class of ultra-hyperbolic operators onPseudo H-type groups, arXiv:1901.08318.

[2] P. Ciatti, Scalar products on Clifford modules and pseudo-H-type Lie algebras, Ann. Mat. Pura Appl. 178(4) (2000), 1-31.

[3] K. Furutani, I. Markina, Complete classification of pseudo-H-type Lie algebras: I, Geom. Dedicata 190(2017) 23-51.

[4] D. Muller, and F. Ricci, Analysis of second order differential operators on Heisenberg groups I, Invent. Math.101 (1990), 545-582.


Francesco Bei

Dipartimento di Matematica

Universita degli Studi di Padova

Via 8 Febbraio, 2

IT-35122 Padova

Italy

[email protected]

Title: Degenerating Hermitian metrics and spectral convergence.

Abstract: The goal of this seminar is to report about some recent results in the setting of degenerating Hermitianmetrics and spectral convergence. More precisely let (X,h) be a compact and irreducible Hermitian complex spaceof complex dimension m. Let π : M → X be a resolution of X. Let p : M × [0, 1]→M be the natural projection onM . Let g ∈ C∞(M × [0, 1], p∗T ∗M ⊗ p∗T ∗M) be such that:

(1) gs is a Hermitian product on M for each s ∈ [0, 1],(2) gs is a positive definite Hermitian product on M for each s ∈ (0, 1](3) g0 = p∗h.

Roughly speaking gs is a smooth family of Hermitian metrics on M that degenerates at s = 0. For each s ∈ (0, 1]let ∆∂,m,0,s : L2Ωm,0(M, gs) → L2Ωm,0(M, gs) be the unique closed extension of the Hodge-Kodaira Laplacian

∆∂,m,0,s : Ωm,0(M)→ Ωm,0(M) acting on the canonical bundle of M . For s = 0 let us consider

∆∂,m,0,abs := ∂t

m,0,min ∂m,0,max : L2Ωm,0(A, g0|A)→ L2Ωm,0(A, g0|A)

where A := M \D and D ⊂M is the normal crossings divisor such that D = π−1(sing(X)). The first aim is to showthat ∆∂,m,0,abs has entirely discrete spectrum. The the second and main goal is to show that

lims→0

λk(s) = λk(0)

where λk(s) and λk(0) are the eigenvalues of ∆∂,m,0,s and ∆∂,m,0,abs, respectively.



Karsten Bohlen

Fakultat fur Mathematik

Universitat Regensburg

DE-93040 Regensburg

Germany

[email protected]

Title: K-homology and index theory on Lie manifolds.

Abstract: I consider so-called Lie manifolds, which can be viewed as an axiomatization of numerous differenttypes of compactifications of complete non-compact manifolds with bounded geometry and prescribed behavior “atinfinity”. On such manifolds there is a pseudodifferential calculus and one can consider fully elliptic operators whichgive rise to Fredholm operators on appropriate Sobolev spaces. A problem, proposed by Victor Nistor, asks for ageneral index formula of Atiyah-Singer type, valid for Fredholm pseudodifferential operators contained in the Liecalculus. In this talk, which is based on joint work with Jean-Marie Lescure, I present a solution to the problem.


Maxim Braverman


Northeastern University

360 Huntington Avenue

Boston, MA 02115

USA

[email protected]

Title: The index of a local boundary value problem for strongly Callias-type operators.

Abstract: We consider a complete Riemannian manifold M whose boundary is a disjoint union of finitely manycomplete connected Riemannian manifolds. We compute the index of a local boundary value problem for a stronglyCallias-type operator on M . Our result extends an index theorem of D. Freed to non-compact manifolds, thusproviding a new insight on the Horava-Witten anomaly.


Jochen Bruning


Humboldt-Universitat

Rudower Chaussee 25

DE-12489 Berlin

Germany

[email protected]

Title: Some remarks on equivariant spectral theory.

Abstract: We consider a compact G-manifold, M , a G-vector bundle, E, over M , and a G-equivariant self-adjointelliptic operator, A, of order one acting on the smooth sections of E. We restrict the operator to the ρ-isotypicalsubspace of sections and show that its η-function is meromorphic in the whole complex plane, but possibly with polesof higher order; 0 is not a pole, though. This result is derived from the equivariant heat expansion, that in this casecontains nonlocal coefficient and logarithmic terms. This approach leads to a proof of the equivariant APS-Theorem.This is joint work with Ken Richardson.



Marco Cappiello


Universita degli Studi di Torino

Via Carlo Alberto, 10

IT-10123 Torino

Italy

[email protected]

Title: Pseudodifferential operators of infinite order and applications to Schrodinger-type equations.

Abstract: We introduce a class of pseudodifferential operators of infinite order, that is with symbols admittingexponential growth at infinity. This type of operators represent a powerful tool when dealing with initial valueproblems for evolution equations with data in Gevrey-type spaces. In particular, in this talk we describe someapplications to the Cauchy problem associated to a Schrodinger operator with lower order terms with complexvalued coefficients and initial data having an exponential behavior at infinity. These results are obtained jointly withAlessia Ascanelli (University of Ferrara).


Paulo Carrillo-Rouse

Institut de Mathematiques de Toulouse

Universite Paul Sabatier, Toulouse III

118, Route de Narbonne

FR-31062 Toulouse

France

[email protected]

Title: Topological formulas for obstructions for Fredholm boundary conditions for manifolds withcorners.

Abstract: For every manifold with corners (mwc) there is a homology theory called conormal homology, definedin terms of faces and orientations of their conormal bundle, and whose cycles correspond geometrically to corner’scycles. Its Euler characteristic is given by the alternating sum of the number of (open) faces of a given codimension.With Jean-Marie Lescure, we proved some years ago that for a mwc of low codimension (2 or 3) X if any b-ellipticpseudodifferential on X can be perturbed by a b-regularizing operator so it becomes Fredholm then the even Eulercorner characteristic of X vanishes; and we gave an almost converse depending on torsion and/or up to stablehomotopy. In the present talk I will explain the above results for any codimension and moreover I will describea Boundary analytic morphism (depending only on K-theory principal symbol classes) with values in conormalhomology that measures the obstruction to be Fredholm up to perturbation of a given b-elliptic operator. Next, Iwill give a topological description of this morphism that allows in particular to give explicit topological formulas forthese obstructions. If time allows it I will discuss the relation of these morphisms with the Fredholm index morphismsand their topological versions, as well as the relation with corner cobordism invariance. This expose is based on jointwork with Jean-Marie Lescure (Clermont-Ferrand) and Mario Velasquez (Bogota).


Hua Chen

School of Mathematics and Statistics

Wuhan University

Wuhan 430072

Hubei Province

China


[email protected]

Title: Estimates of Dirichlet Eigenvalues for a Class of Sub-elliptic Operators.

Abstract: Let Ω be a bounded connected open subset in Rn with a smooth boundary ∂Ω. Given the systems ofreal smooth vector fields X = (X1, X2, · · · , Xm) defined on a neighborhood of Ω, which satisfying the Hormander’scondition, and ∂Ω is non-characteristic for X. For a self-adjoint sub-elliptic operator ∆X = −

∑mi=1X

∗i Xi on Ω,

we denote its kth Dirichlet eigenvalue by λk. We obtain an uniform upper bound for the sub-elliptic Dirichlet heatkernel, and then we give an explicit sharp lower bound estimate of λk which is polynomial increasing in k with theorder relating to the generalized Metivier index. Furthermore, we establish an explicit asymptotic formula of λkwhich generalize the Metivier’s results in 1976. This asymptotic formula implies that under a certain condition ourlower bound estimate for λk is optimal in sense of the order of k. On the other hand, the upper bound estimates ofDirichlet eigenvalues for general sub-elliptic operators are also given, which in some sense will be precise from theresult of this talk.


Li Chen


Universitat Mannheim

DE-68131 Mannheim

Germany

[email protected]

Title: Analysis on Keller-Segel Models in Chemotaxis.

Abstract: I this talk I will summerize some of our contributions in the analysis of parabolic elliptic Keller-Segelsystem, a typical model in chemotaxis. For the case of linear diffusion, after introducing the critical mass in twodimension, I will show our result for blow-up conditions for higher dimension. The second part of the talk isconcentrated in the critical exponent for Keller-Segel system with porus media type diffusion. In the end, motivatedfrom the result on nonlocal Fisher-KPP equation, we show that the nonlocal reaction will also help in preventingthe blow-up of the solutions.


Sandro Coriasco




IT-10123 Torino

Italy

[email protected]

Title: Lifting properties for ultra-modulation spaces and one-parameter groups of Gevrey typepseudo-differential operators.

Abstract: We deduce one-parameter group properties for pseudo-differential operators op(a), where a belongs to

symbol classes of Gevrey type, associated with suitable, non-vanishing, weight functions ω0, and denoted by Γ(ω0)∗ .

This allows to show that there are pseudo-differential operators op(a) and op(b) which are inverses to each others,

and satisfy a ∈ Γ(ω0)∗ and b ∈ Γ

(1/ω0)∗ .

We apply these results to deduce lifting properties for modulation spaces and construct isomorphisms betweenthem. In particular, for any couple of admissible weight functions ω, ω0, we prove that the Toeplitz operator (or


localization operator) Tp(ω0) is an isomorphism from the weighted modulation spaceMp,q(ω) to the weighted modulation

space Mp,q(ω/ω0)

, for every p, q ∈ (0,∞].

This is joint work with A. Abdeljawad and J. Toft.


Claire Debord

Laboratoire de Mathematiques Blaise Pascal

Universite Clermont-Auvergne

3, place Vasarely

TSA 60026, CS 60026

FR-63178 Aubiere Cedex

France

[email protected]

Title: Index theory through Lie groupoids.

Abstract: The aim of this lecture is to explain why Lie groupoids are very naturally linked to Atiyah-Singer indextheory. After introducing Lie groupoids, and giving various examples, I will explain how these geometrical objectscan be used:

• to construct the pseudodifferential calculus,• to construct the index of pseudodifferential operators without using the pseudodifferential calculus,• to prove and generalize index theorems.

This talk, inspired by ideas of A. Connes, will be based on joint works with JM. Lescure and with G. Skandalis.


Nils Dencker

Center for Mathematical Sciences

Lund University

Box 118

SE-221 00 Lund

Sweden

[email protected]

Title: The Solvability of Differential Equations.

Abstract: Since Hans Lewy sixty years ago presented his famous counterexample, it has been known that non-symmetric linear partial differential equations are generically not solvable. For differential operators with simplecharacteristics, solvability is equivalent to the Nirenberg-Treves condition (Ψ). This condition involves the signchanges of the symbol of the nonsymmetric part of the highest order terms. Condition (Ψ) also has consequences forthe stability of the spectrum and for nonlinear differential equations.

In this talk, we shall consider differential operators that have double characteristics. Examples are weakly hyper-bolic operators and parabolic operators. Then one can define conditions corresponding to (Ψ) on the lower orderterms at the double characteristics. We shall show that these conditions are necessary for solvability in several cases.A particularly interesting case are the weakly hyperbolic operators.



Michael Dreher


Universitat Rostock

DE-18051 Rostock

Germany

[email protected]

Title: General symmetrisable systems.

Abstract: We consider various systems of fluid dynamics and see how they can lead to mixed order systems. Wepresent a general pseudodifferential framework for symmetrising those systems and use this framework to present aunified theory of singular limits such as the low Mach number limit or the quasi-neutral limit.


Karsten Fritzsch




Welfengarten 1

DE-30167 Hannover

Germany

[email protected]

Title: The Calderon Projector on φ-Manifolds with Boundary.

Abstract: In recent years, there has been great interest in the plasmonic eigenvalue problem on singular spaces.This is a two-sided boundary value problem that describes the coupling of electromagnetic fields to the electron gasof a conducting body, given by the geometry. The interest comes from the fact that the geometry can be used tospecifically tailor properties of the resulting surface waves and hence electromagnetic properties of the body. As theplasmonic eigenvalue problem is directly linked to the Dirichlet-to-Neumann maps, it is useful to construct and studyCalderon projectors for interesting pairs of operators and geometries.

One such geometry can be described by φ-manifolds with boundary. For instance, given two touching spheres,a quasi-homogeneous blow-up of the point of tangency will give rise to a further, singular boundary hypersurfaceof the exterior domain. This singular face comes equipped with a fibration onto a closed manifold, resembling thesituation of Mazzeo-Melrose’s φ-calculus, but with additional, regular boundary hypersurfaces.

Seeing the regular faces as carriers of boundary conditions, I will present the construction of the Calderon projectorfor the Laplacian (of a φ-metric) in the setting of a general φ-manifold with boundary and derive some of its keyproperties.

This is part of ongoing joint work with Daniel Grieser and Elmar Schrohe.


Anahit Galstyan

School of Mathematical and Statistical Sciences

University of Texas Rio Grande Valley

1201 W. University Dr.

Edinburg, TX 78539

USA

[email protected]

Title: Semilinear Klein-Gordon Equation in the Friedmann-Lamaitre-Robertson-Walker spacetime.


Abstract: We present some results on the semilinear massless waves propagating in the Einstein-de Sitter spacetimeand semilinear Klein-Gordon Equation in the de Sitter spacetime.

We examine the solutions of the semilinear wave equation, and, in particular, of the ϕp model of quantum fieldtheory in the curved space-time. More precisely, for 1 < p < 4 we prove that solution of the massless self-interactingscalar field equation in the Einstein-de Sitter universe has finite lifespan.

Furthermore, we present a condition on the self-interaction term that guaranties the existence of the global intime solution of the Cauchy problem for the semilinear Klein-Gordon equation in the FLRW (Friedmann-Lamaitre-Robertson-Walker) model of the contracting universe. For the equation with the Higgs potential we give an estimatefor the lifespan of solution.


Daniel Grieser


Carl von Ossietzky Universitat Oldenburg

DE-26111 Oldenburg

Germany

[email protected]

Title: The Fubini anomaly for regularized integrals.

Abstract: Regularized integrals appear in many contexts of analysis. It is natural to ask whether an analogue ofFubinis theorem is true for the regularized integral. This is indeed the case if the function under consideration has asuitable joint regularity in the two variables (polyhomogeneity). Without this assumption Fubinis theorem may failto hold. In some interesting cases joint polyhomogeneity is not satisfied, but its failure can be resolved by blow-up.We analyze how the failure of Fubinis theorem can be quantified in terms of the resolution data.


Gerd Grubb

Department of Mathematical Sciences

University of Copenhagen

Universitetsparken 5

DK-2100 Copenhagen

Denmark

[email protected]

Title: Fractional-order operators and transmission spaces.

Abstract: For a strongly elliptic pseudodifferential operator P of order 2a (generally noninteger) — for examplethe fractional Laplacian (−∆)a, 0 < a < 1 — there is a useful theory for boundary value problems on smooth subsetsΩ of Rn, when P satisfies Hormander’s a-transmission condition at the boundary. For the homogeneous Dirichletproblem (u supported in Ω), the solution space for data in Hs(Ω), s ≥ 0, is then the a-transmission space

Ha(2a+s)(Ω) = Λ(−a)+ e+Ha+s(Ω), where Λ

(t)+ denotes an order-reducing pseudodifferential operator of plus-type. A

similar result holds in Holder-Zygmund spaces. We shall give a more down-to-earth description of the transmissionspaces, involving a power da of the distance d(x) = dist(x, ∂Ω) and a Poisson solution operator for the Laplacian.

There are nice solution properties for the equation r+Pu = f itself (we can let s → ∞ above), but it turns outthat for Schrodinger equations (r+P + V (x))u = f and heat equations (∂t + r+P )u(x, t) = F (x, t), as well as instudies of eigenvalues, there are nontraditional limitations on the regularity of solutions at the boundary. They arelinked with the special structure of the transmission spaces.

References:

[1] G. Grubb, Fractional Laplacians on domains, a development of Hormander’s theory of µ-transmission pseu-dodifferential operators. Adv. Math. 268 (2015), 478–528.


[2] G. Grubb, Limited regularity of solutions to fractional heat and Schrodinger equations. To appear in Dis.Cont. Dyn. Syst. 39 (June 2019), arXiv:1806.10021.


Georges Habib


Faculty of Sciences II

Lebanese University

P.O. Box 90656

Fanar-Matn

Lebanon

[email protected]

Title: The Bochner formula for Riemannian flows.

Abstract: We consider a Riemannian manifold (M, g) endowed with a Riemannian flow and we study the curvatureterm in the Bochner-Weitzenbock formula of the basic Laplacian on M. We prove that this term splits into two parts.The first part depends mainly on the curvature operator of the underlying manifold M and the second part isexpressed in terms of the O’Neill tensor of the flow. After getting a lower bound for this term, depending on thesetwo parts, we establish an eigenvalue estimate of the basic Laplacian on basic forms. We then discuss the limitingcase of the estimate and prove that when equality occurs the manifold M is a local product. This work is joint withFida El Chami.


Bernard Helffer

Laboratoire de Mathematiques Jean Leray

Universite de Nantes

2 Rue de la Houssiniere

FR-44 322 Nantes Cedex 3

France

[email protected]

Title: On Courant’s nodal domain property for linear combinations of eigenfunctions (after P. Berardand B. Helffer).

Abstract: We revisit Courant’s nodal domain property for linear combinations of eigenfunctions. This propertywas proven by Sturm (1836) in the case of dimension 1. Although stated as true for the Dirichlet Laplacian indimension > 1 in a footnote of the celebrated book of Courant-Hilbert (and wrongly attributed to H. Herrmann, aPhD student of R. Courant), it appears to be wrong in dimension > 1. This was first observed by V. Arnold in theseventies.

In this talk, we present simple and explicit counterexamples to this so-called ”Herrmann’s statement” for domainsin Rd, S2, or T2. We also discuss the existence of a counterexample in a C∞, convex domain Ω in R2 in relationwith the analysis of the number of domains delimited by the level sets of a second eigenfunction for the Neumannproblem. We finally discuss the question to have positive statements. This work has been done in collaboration withP. Berard.



Magda Khalile




Welfengarten 1

DE-30167 Hannover

Germany

[email protected]

Title: Asymptotics of Robin eigenvalues in domains with corners.

Abstract: We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvi-linear polygons. There exists a number N ∈ N depending on the domain such that the asymptotics of the N firsteigenvalues is essentially determined by the corner openings. While only a rough estimate is available for the nexteigenvalues, we prove under some geometric assumptions the existence of an effective self-adjoint operator, acting onthe boundary of the domain with boundary conditions at the corners, which leads the asymptotic behavior of anyeigenvalue beyond the critical number N .


Klaus Kirsten


Baylor University

Waco, TX 76796

USA

Klaus [email protected]

Title: Conformal transformations and gluing formulas.

Abstract: Let M1 and M2 be two Riemannian manifolds each of which have the boundary N . Consider theLaplacian on M1 and M2 augmented with Dirichlet boundary conditions on N . A natural question to ask is ifthere is any relation between spectral properties of the Laplacian on M1, M2, and the Laplacian on the manifoldM (without boundary) obtained gluing together M1 and M2, namely M = M1 ∪N M2. A partial answer is givenby the Burghelea-Friedlander-Kappeler-gluing formula for zeta-determinants. This formula contains an (in general)unknown polynomial which is completely determined by some data on a collar neighborhood of the hypersurfaceN . In this talk I will use conformal transformations to understand the geometric content of this polynomial. Theunderstanding obtained will pave the way for a fairly straightforward computation of the polynomial (at least for lowdimensions of M). Furthermore it leads to a partial understanding of the heat invariant for the Dirichlet-to-Neumannmap. This is joint work with Yoonweon Lee (Inha University, Korea).


Yuri Kordyukov

Institute of Mathematics

Ufa Federal Research Centre RAS

K. Marx str., 16/2

Ufa RU-450077

Russia

[email protected]

Title: Riemannian structures and Laplacians for generalized smooth distributions.


Abstract: A (generalized) smooth distribution on a smooth manifold M is a locally finitely generated C∞(M)-submodule D of the C∞(M)-module of smooth compactly supported vector fields on M . First, we introduce a notionof Riemannian structure on D. Then, for a smooth distribution D on M , a Riemannian metric on D and a positivedensity µ on M , we construct the associated Laplacian ∆D. If M is compact, this operator is essentially self-adjointas an unbounded operator on the Hilbert space L2(M,µ) with domain C∞(M).

A distribution is called involutive, if it is closed under the Lie bracket operation on vector fields. An involutivedistribution is called a singular foliation. I. Androulidakis and G. Skandalis constructed a longitudinal pseudodif-ferential calculus and the corresponding scale of longitudinal Sobolev spaces for an arbitrary singular foliation ona compact manifold. Given a smooth distribution D on a compact manifold M , let F be the smallest involutiveC∞(M)-submodule, which contains D. Assume that it is finitely generated, then F is a singular foliation. We showthat the Laplacian ∆D is longitudinally hypoelliptic in the scale of longitudinal Sobolev spaces associated with F .When the foliation F is regular, we prove that the Laplacian determines an unbounded multiplier on the foliationC∗-algebra. This allows us to get some information on its spectrum in terms of the spectra of the correspondingoperators along the leaves of the foliation. This is joint work with I. Androulidakis.


Matthias Lesch

Mathematisches Institut

Universitat Bonn

Endenicher Allee 60

DE-53115 Bonn

Germany

[email protected]

Title: Operators of Fuchs type, conical singularities, and asymptotic methods.

Abstract: As we are celebrating Professor Schulze’s 75th birthday I would like to take the liberty to use my talkfor some personal mathematical reminiscences. Professor Schulze’s work on singular analysis was very influencial formy Habilitation Schrift. In the early 90s I learned Schulze’s point of view through various visits to the WeierstrassInstitute, Potsdam, and through private lessons.

In my talk I will explain the heat expansion for elliptic operators of Fuchs type, mainly focusing on my abovementioned Schrift, but also hinting at subsequent work.


Xiaochun Liu

School of Mathematics and Statistics

Wuhan University

Wuhan 430072

Hubei Province

China

[email protected]

Title: A quasilinear elliptic equation with critical growth on compact Riemannian manifolds.

Abstract: In this talk, we will discuss a class of quasilinear elliptic equations involving critical nonlinearity oncompact Riemannian manifolds. With the help of critical point theory and some analysis techniques, we obtain theexistence theorem under certain assumptions.



Ursula Ludwig

Fakultat fur Mathematik

Universitat Duisburg-Essen

DE-45117 Essen

Germany

[email protected]

Title: A Morse-Bott type complex for intersection homology and the Bismut-Zhang torsion.

Abstract: An important comparison theorem in global analysis is the comparison theorem of analytic and topo-logical torsion, aka Cheeger-Muller theorem, for smooth compact manifolds equipped with a unitary flat vectorbundle. In recent years the study of extensions of the Cheeger-Muller theorem to singular spaces for different typesof singularities has become a fruitful area of research.

For spaces with conical and wedge singularities it is well-known that the analytic torsion is not a topologicalinvariant in general. In this talk we define a torsion, which conjecturally could serve as a “topological counterpart”in a Cheeger-Muller theorem for wedge singularities.


Irina Markina


University of Bergen

Johannes Brunsgate 12

Bergen NO-5008

Norway

[email protected]

Title: On Cauchy-Szego kernel for quaternionic Siegel upper half space.

Abstract: In the talk we introduce a quaternionic analogue of the Heisenberg group and explain the relationbetween this group and the quaternionic analogous of the Siegel upper half space. We discuss regular functions,that are counterpart of complex holomorphic functions for quaternionic setting. The Hardy space is the space ofregular functions in the Siegel upper half space with L2 boundary values. We construct the Cauchy-Szego kernelfor the Cauchy-Szego projection integral operator from the space of L2-integrable functions defined on the boundaryof the quaternionic Siegel upper half space to the space of boundary values of the quaternionic regular functionsof the Hardy space over the quaternionic Siegel upper half space. We also present the fundamental solution for ahypoelliptic operator related to the boundary of the Siegel upper half space. The talk is based on the joint work withDer Chen Chang (Georgetown University, Washington DC, USA) and Wei Wang (Zhejiang University, Zhejiang, PRChina).


Calin Martin


Universitat Wien

Oskar-Morgenstern-Platz 1

AT-1090 Wien

Austria

[email protected]

Title: Hamiltonian Formulation for Wave-Current Interactions in Stratified Rotational Flows.

Abstract: We show that the Hamiltonian framework permits an elegant formulation of the nonlinear governing


equations for the coupling between internal and surface waves in stratified water flows with piecewise constantvorticity. This is joint work with Adrian Constantin (University of Vienna) and Rossen Ivanov (Dublin Institute ofTechnology).


Gerardo Mendoza


Temple University

Philadelphia, PA 19122

USA

[email protected]

Title: Singular foliations by tori.

Abstract: Let N be a compact manifold and T a nowhere vanishing real vector field on N whose one-parametergroup of diffeomorphisms, at, consists of isometries of some Riemannian metric; write F for the family of such pairs(N , T ). All objects are of class C∞. Letting B be the space of closures of orbits of a, one can prove a number ofresults on B, in particular an analogue of the Gysin sequence and a classification theorem via elements in H2(B,Z)of elements of F modulo a certain natural notion of isomorphism. In general, the space B is not a manifold, but isa Hausdorff space in which there is an open dense subset Br which is a smooth manifold such that its preimage inN by the quotient map N → B is a principal torus bundle, the elements of the torus (which is just the closure ofthe group at in the group of isometries) acting as isometries on all of N . Thus N admits a singular foliation by tori,and B is the space of leaves. In special cases B is an orbifold, but in general it need not be. I plan to describe theabove quoted results and some others, loosely described as doing analysis on B, in some detail.


Werner Muller

Mathematisches Institut

Universitat Bonn

Endenicher Allee 60

DE-53115 Bonn

Germany

[email protected]

Title: Approximation of L2-invariants of locally symmetric space.

Abstract: L2-invariants are defined in terms of the universal covering of a compact manifold as counterparts ofclassical invariants such as Betti numbers, the index of elliptic operators, or the analytic torsion. They have importantapplications in topology and geometry. In particular, they are related to the study of the asymptotic behavior ofBetti numbers, analytic torsion and other spectral invariants for sequences of finite coverings converging in theBenjamini-Schramm sense to the universal covering. In this talk I will consider this problem for locally symmetricspaces. This is closely related to the limit multiplicity problems of DeGeorge-Wallach, Delorme and others. I willreview some recent results and discuss some open problems.


Vladimir Nazaikinskii

Ishlinsky Institute for Problems in Mechanics RAS

Pr. Vernadskogo, 101-1

Moscow RU-119526

Russia


[email protected]

Title: Partial Spectral Flow and the Aharonov–Bohm Effect in Graphene.

Abstract: Consider a graphene tube in the shape of a right circular open-ended cylinder whose height and radiusare both much greater than the distance between neighboring carbon atoms. A magnetic field everywhere vanishingon the tube surface (or at least tangent to it) is adiabatically switched on, causing the eigenvalues of the tight-binding Hamiltonian describing the electron π-states in graphene in the nearest neighbor approximation to move inthe process. If the magnetic flux through the tube in the final configuration has an integer number of flux quanta,then the final Hamiltonian is unitarily equivalent to the initial one, and hence the electron energy spectrum returnsto its original position. Moreover, the lattice function space on which the tight-binding Hamiltonian acts beingfinite-dimensional, the spectral flow of the Hamiltonian, defined as the net number of eigenvalues (counted withmultiplicities) that pass through 0 in the positive direction, is necessarily zero. However, a more detailed analysisreveals that the eigenfunctions corresponding to these eigenvalues prove to be localized near the Dirac points Kand K ′ in the momentum space, and if one separately counts the spectral flow for the eigenfunctions localized nearK and near K ′, then one obtains two “partial spectral flows,” which have opposite signs and the same modulus(equal to the number of magnetic flux quanta through the tube). The physical interpretation is that switching onthe magnetic field creates electron–hole pairs (or, more precisely, pairs of “electron” and “hole” energy levels) ingraphene, the number of pairs being determined by the magnetic flux. Whenever an electron level is created near K,the corresponding hole level is created near K ′, and vice versa. Further, the number of electron/hole levels creatednear K equals the spectral flow of the family of Dirac operators approximating the tight-binding Hamiltonian nearK (and the same is true with K replaced by K ′). We assign a precise mathematical meaning to the notion of partialspectral flow in such a way that all the preceding assertions make rigorous sense.

This is joint work with M. I. Katsnelson and J. Bruning.


Victor Nistor

UFR Mathematiques, Informatique, Mecanique

Universite de Lorraine

3 Rue Augustin Fresnel

FR-57073 Metz Cedex 03

France

[email protected]

Title: Spaces with ‘oscillating singularities’ after Schulze and their relation to bounded geometry.

Abstract: In several joint works, Schulze and his coauthors have introduced a new class of singular domainsand spaces that go beyond the “classical” polyhedral-type ones, namely the class of domains with ”oscillatingsingularities”. They considered in detail the conical and cuspidal cases, which can be thought of as being obtainedby replacing the asymptotically straight cylindrical ends appearing in the work of Kondratiev (among many others)with periodically oscillating cylindrical ends. Schulze and his coauthors have then studied pseudodifferential operatorson these spaces, including the symbolic structure of these operators, their resolvents, and their Fredholm property.In particular, new features arise in the characterization of the Fredholm property, and these new features go beyondthe ones considered by Kondratiev, Krainer, Melrose, Mendoza, Schrohe, Schulze and many others in the case of(asymptotically straight) cones. A similar construction to that of Schulze and his coauthors has recently appeared inthe work of Melo on comparison algebras. In my talk, I will first review some results of these authors and then I willdiscuss their relation to manifolds with boundary and bounded geometry, in general, and with some recent resultsof H. Amann and of myself joint with Ammann and Grosse, in particular.



Paolo Piazza


Universita di Roma ‘La Sapienza’

Piazzale Aldo Moro 2

IT-00185 Roma

Italy

[email protected]

Title: K-theory classes of Dirac-type operators on stratified pseudo manifolds.

Abstract: Let G be a discrete group and let X be a G-Galois covering with quotient X/G, a smooth compactmanifold. Given a G-equivariant operator of Dirac-type one can define a K-homology class in the equivariant K-homology of X and an index class in the K-theory of the reduced C∗-algebra of G. If the operator is L2-invertible,then we also have a rho class. In this talk, extending previous joint results with Pierre Albin and with Vito Zenobi,I will report on recent work with Pierre Albin and Jesse Gell-Redman addressing the existence of these classes for ageneral Dirac-type operator on (the regular part of) a stratified pseudomanifold.


Vladimir Rabinovich

Instituto Politecnico Nacional

ESIME Zacatenco

Ciudad de Mexico 07738

Mexico

[email protected]

Title: Differential operators on infinite graphs with general conditions on vertices.

Abstract: Let Γ be an infinite metric oriented graph embedded in Rn, E , V be infinite countable sets of edgese ∈ E , and vertices v ∈ V of Γ. The graph Γ is equipped with a differential operator

Au(x) =

r∑j=0

aj(x)u(j)(x), x ∈ Γ \ V

with piece-wise smooth coefficients aj , j = 0, 1, ..., r, and general connection operators at the vertices v ∈ V

Bku(v) =

mk∑j=0

bj,k(v)u(j)Ev (v) ∈ Cd(v), v ∈ V, k = 1, ...,m,

where d(v) is the number of edges incident to v, bj,k(v) are d(v)× d(v) complex matrices,

u(j)Ev (v) = (u

(j)1 (v), ..., u

(j)d(v)(v)) ∈ Cd(v),

u(j)i (v), i = 1, ..., d(v), are limit values at the vertex v ∈ V of the derivatives u

(j)i (x) taken along the edges ei ∈ Eν

according their orientation. We associate with the operator A and the operators Bk, k = 1, ...,m, an operator A =(A,B1, B2, ..., Bm) acting from the Sobolev space Hs(Γ) to the space Hs−r(Γ)⊕L2(V)

mwhere L2(V) = ⊕v∈VCd(v).

We study the smoothness, and exponential behavior at infinity solutions of the equation Au = (f, ϕ1, ..., ϕm), andfor periodic graphs we obtain the necessary and sufficient conditions of the Fredholmness of A, and a descriptionof the essential spectrum of the realization of A in L2(Γ). We give applications of these results to the Schrodingeroperators on periodic graphs with general conditions at the vertices.

References:

[1] V. Rabinovich, Fredholm theory of differential operators on periodic graphs. Mathematical Problems inQuantum Physics, October 8-11, 2016, Atlanta, Georgia. Contemp. Math. 717, AMS, 2018.

[2] V. Rabinovich, On the essential spectrum of quantum graphs. Integr. Equ. Oper. Theory 88 (2017), 339–362.



Luigi Rodino




IT-10123 Torino

Italy

[email protected]

Title: Non-commutative residue for metaplectic operators.

Abstract: The non-commutative residue was introduced by Manin in connection with the algebraic study of theKdV solutions. The result of Manin was extended by Wodzicki to pseudo-differential operators and by Guillemin toFourier integral operators. We compute the non-commutative residue for metaplectic operators, obtaining for theman explicit expression of the result of Guillemin.


Julie Rowlett

Department of Mathematical Sciences

Chalmers University of Technology

SE-412 96 Gothenburg

Sweden

[email protected]

Title: Spectral asymptotics for rough Riemannian manifolds.

Abstract: This is joint work with L. Bandara and M. Nursultanov. Our topological setting is a smooth compactmanifold of dimension two or higher with smooth boundary. Although this underlying topological structure issmooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a roughRiemannian manifold (rrm). These arise naturally in the context of harmonic analysis, as L. Bandara showed thatthey are geometric invariances of the Kato square root problem. The “roughness” of this geometric context can beseen for example by the fact that there is no canonical distance between points on a rrm. I will discuss a certainclass of weighted Laplace equations, with a range of boundary conditions including Dirichlet, Neumann, and mixed,for which we obtain spectral asymptotics.


Zhuoping Ruan


Nanjing University

22 Hankou Road

Nanjing 210093

Jiangsu Province

China

[email protected]

Title: Hyperbolic problems with totally characteristic boundary.


Abstract: We discuss first-order N ×N hyperbolic systems

(1)

∂tu+ xA(t, x, y)∂xu+

d∑j=1

Bj(t, x, y)∂ju = f(t, x, y),

u∣∣t=0

= g(x, y),

where (t, x, y) ∈ (0, T )×R+×Rd, ∂j = ∂/∂yj , and A,Bj ∈ C∞([0, T ]×R+×Rd;MN×N (C)). Note that the boundaryat x = 0 is totally characteristic due to the factor x in front of the coefficient A.

Assuming that the differential operator ∂t + xA(t, x, y)∂x +∑dj=1Bj(t, x, y)∂j is hyperbolic in the sense that the

matrix

A(t, x, y)ξ +

d∑j=1

Bj(t, x, y)ηj ,

for (t, x, y, ξ, η) ∈ [0, T ] × R+ × Rd × (R1+d \ 0) has real semisimple eigenvalues of constant multiplicities, we provethe well-posedness of Eq. (1) in Sobolev spaces and in weighted Sobolev spaces. More specifically, we show that thesolution u admits an asymptotic expansion of the form

u(t, x, y) ∼∑(p,k)

xp logkxupk(t, y) as x→ +0

provided that the right-hand side f(t, x, y) and the initial condition g(x, y) have such asymptotic expansions. More-over, the coefficients upk(t, y) are uniquely determined by the data (f, g) as solutions to certain first-order hyperbolicsystems in (0, T )× Rd.

We also briefly indicate physically interesting applications to the compressible Euler equations.This is joint work with Ingo Witt (Gottingen).


Anton Savin


Peoples’ Friendship University of Russia (RUDN University)

Ul. Miklukho-Maklaya, 6

Moscow RU-117198

Russia

[email protected]

Title: On Fredholm property of boundary value problems for the wave equation with conditions onthe entire boundary.

Abstract: Boundary value problems for hyperbolic equations with conditions on the entire boundary were consid-ered by many authors (e.g., see Bourgin and Duffin, Sobolev, Arnold, Antonevich, Ptashnik, see also recent work byBar and Strohmaier on index theory of Lorentzian Dirac operator).

We consider the wave equation with conditions on the entire boundary of the cylinder [0, T ] ×M , where M is acompact closed Riemannian manifold. Conditions for Fredholm solvability of the problem are obtained.

The results were obtained in a joint work with Andrei Boltachev (RUDN University). The research was supportedby RFBR (projects 16-01-00373, 19-01-00574) and RUDN University program 5-100.


Simon Scott


King’s College London

Strand

London WC2R 2LS


United Kingdom

[email protected]

Title: Fibre-oriented bordism homology and genera.

Abstract: The set of fibrations over manifold X which are fibrewise bordant has a canonical ring structure withrespect to which a vertical genus is determined by certain combinations of ‘vertical characteristic classes’ taking valuesin the singular cohomology of X. Family index densities provide natural examples. It turns out that the structuresinvolved are governed by a vertical Pontryagin-Thom spectrum and its associated generalised (co)homology. Thesethemes and, in particular, how the orientation along the fibres is a crucial ingredient, will be the focus of this talk.


Alexander Strohmaier

School of Mathematics

University of Leeds

Leeds LS2 9JT

United Kingdom

[email protected]

Title: A trace formula for stationary spacetimes.

Abstract: Stationary spacetimes are spacetimes that admit a global timelike Killing vector-field. I will explainaspects of spectral theory on stationary spacetimes, thus generalising spectral theory of the Laplace operator onRiemannian manifolds. I will then present a trace formula analogous to the Gutzwiller-Duistermaat-Guillemin traceformula. The classical geometric object underlying the singularity trace expansion is the contact manifold of lightlike geodesics. (Joint work with S. Zelditch.)


Joachim Toft


Linnæus University

SE-351 95 Vaxjo

Sweden

[email protected]

Title: Analytic pseudo-differential calculus via the Bargmann transform.

Abstract: The Bargmann transform is a transform which maps Fourier-invariant function spaces and their dualsto certain spaces of formal power series expansions, which sometimes are convenient classes of analytic functions.

In the 70th, Berezin used the Bargmann transform to translate problems in operator theory into a pseudo-differential calculi, where the involved symbols are analytic functions, and the corresponding operators map suitableclasses of entire functions into other classes of entire functions.

Recently, some investigations on certain Fourier invariant subspaces of the Schwartz space and their dual (dis-tribution) spaces have been performed by the author. These spaces are called Pilipovic spaces, and are defined byimposing suitable boundaries on the Hermite coefficients of the involved functions or distributions. The family ofPilipovic spaces contain all Fourier invariant Gelfand-Shilov spaces as well as other spaces which are strictly smallerthan any Fourier invariant non-trivial Gelfand-Shilov space.

In the talk we show that the Bargmann images of Pilipovic spaces and their distribution spaces are conve-nient classes of analytic functions or power series expansions which are suitable when investigating analytic pseudo-differential operators.


We also deduce continuity properties for such pseudo-differential operators when the symbols and target functionspossess certain (weighted) Lebesgue estimates. We also show that the counter image with respect to the Bargmanntransform of these results generalise some continuity results for (real) pseudo-differential operators with symbols inmodulation spaces, when acting on other modulation space.

The talk is based on collaborations with Nenad Teofanov from the University of Novi Sad.


Andras Vasy


Stanford University

450 Serra Mall

Stanford, CA 94305

[email protected]

Title: Boundary rigidity and the local inverse problem for the geodesic X-ray transform on tensors.

Abstract: In this talk, based on joint work with Plamen Stefanov and Gunther Uhlmann, I discuss the boundaryrigidity problem on manifolds with boundary (for instance, a domain in Euclidean space with a perturbed metric), i.e.determining a Riemannian metric from the restriction of its distance function to the boundary. This corresponds totravel time tomography, i.e. finding the Riemannian metric from the time it takes for solutions of the correspondingwave equation to travel between boundary points. A version of this relates to finding the speed of seismic wavesinside the Earth from travel time data, which in turn permits a study of the structure of the inside of the Earth.

This non-linear problem in turn builds on the geodesic X-ray transform on such a Riemannian manifold withboundary. The geodesic X-ray transform on functions associates to a function its integral along geodesic curves, sofor instance in domains in Euclidean space along straight lines. The X-ray transform on symmetric tensors is similar,but one integrates the tensor contracted with the tangent vector of the geodesics. I will explain how, under suitableconvexity assumptions, one can invert the geodesic X-ray transform on functions, i.e. determine the function from itsX-ray transform, in a stable manner, as well as the analogous tensor result, and the connection to the full boundaryrigidity problem.


Boris Vertman


Carl von Ossietzky Universitat Oldenburg

DE-26111 Oldenburg

Germany

[email protected]

Title: Resolvent trace expansion on stratified spaces.

Abstract: We present recent results establishing resolvent trace asymptotics for the Hodge Laplace operator onstratified spaces of arbitrary depth satisfying a spectral Witt condition. This is joint work with Luiz Hartmann andMatthias Lesch.


Yawei Wei

School of Mathematical Sciences

Nankai University

94 Weijin Road

Tianjin 300071

China

[email protected]


Title: Existence of solutions for nonlinear degenerate equations.

Abstract: This talk concerns Dirichlet problems for semi-linear degenerate equations. Degenerate elliptic operatorsare motivated by the structure of the corresponding singular manifolds. Here we will first discuss the definitions ofthe degenerate elliptic operators, and then the weighted Sobolev spaces. Some inequalities will also be introduced,which are useful for the proof of the existence of solutions for the corresponding nonlinear degenerate equations.


Jens Wirth

Institut fur Analysis, Dynamik und Modellierung


Universitat Stuttgart

Pfaffenwaldring 57

DE-70569 Stuttgart

Germany

[email protected]

Title: On singular hyperbolic problems.

Abstract: In the talk I will present some aspects of the phase space analysis for hyperbolic problems with (strongly)singular time-dependent coefficients having point-singularities. We will consider weak and very weak solutions andcharacterise properties of solution operators to the Cauchy problems.


Karen Yagdjian

School of Mathematical and Statistical Sciences

University of Texas Rio Grande Valley

1201 W. University Dr.

Edinburg, TX 78539

USA

[email protected]

Title: A novel integral transform approach to solving partial differential equations in the curvedspace-times.

Abstract: In the talk we will present the integral transform that allows us to construct solutions of the hyperbolicpartial differential equations with variable coefficients via solutions of a simpler equation. The transform was sug-gested by the author in the case when the last equation was a wave equation. Then it was used to investigate severalwell-known equations such as generalized Tricomi equation, the Klein–Gordon equation in the de Sitter, Einstein-deSitter, and FLRW space-times. More precisely, this integral transform is aimed to generate representation formulasfor the solutions to the equation

(1) utt − a2(t)A(x, ∂x)u−M2u = f, t ∈ (0, T ), x ∈ Ω ,

where M ∈ C and A(x, ∂x) is elliptic operator A(x, ∂x) =∑|α|≤m aα(x)∂αx , with smooth coefficients aα(x) ∈ C∞(Ω)

in the domain in Ω ⊆ Rn. For the given smooth function f = f(x, t) defined in Ω × [0, T ] let w = w(x, t; b) be asolution of the problem

(2)

wtt −A(x, ∂x)w = 0, t ∈ (0, T1), x ∈ Ω,

w(x, 0; b) = f(x, b), wt(x, 0; b) = 0, x ∈ Ω,


with the parameter b ∈ (0, T ) and 0 < T1 ≤ ∞. We define the operator EEA by w = EEA[f ] and introduce theintegral operator K : w 7−→ u, by

u(x, t) = K[w](x, t) :=

∫ t

0

db

∫ |φ(t)−φ(b)|0

K(t; r, b;M)w(x, r; b) dr, x ∈ Ω, t ∈ (0, T ),

which maps function w = w(x, t; b) into solution u of the equation (1). Here φ(t) =∫ t0a(τ) dτ is a distance function

produced by a = a(t) while K(t; r, b;M) is written via Gauss’ hypergeometric function. Thus, if EEA is a resolvingoperator of the problem (2), then we obtain the solution operator u = GA[f ] = K EEA[f ]. A class of operators forwhich we have obtained representation formulas for the solutions includes a(t) = t`, ` ∈ R, a(t) = e±t.

References:

[1] K. Yagdjian, A note on the fundamental solution for the Tricomi–type equation in the hyperbolic domain. J.Differential Equations 206 (2004), no. (1), 227–252.

[2] K. Yagdjian and A. Galstian, Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime.Comm. Math. Phys. 285 (2009), 293–344.

[3] K. Yagdjian, Global existence of the scalar field in de Sitter spacetime. J. Math. Anal. Appl. 396 (2012),no. (1), 323–344.

[4] K. Yagdjian, Huygens’ Principle for the Klein-Gordon equation in the de Sitter spacetime. J. Math. Phys.54 (2013), no. (9), 091503.

[5] K. Yagdjian, Integral transform approach to solving Klein-Gordon equation with variable coefficients. Math-ematische Nachrichten 288 (2015), no. (17-18), 2129-2152.

[6] K. Yagdjian, Integral transform approach to generalized Tricomi equations. J. Differential Equations 259(2015), 5927–5981.



Poster Session

Fernando de Avila Silva

Departamento de Matematica

Universidade Federal do Parana

CP 19081 - Jd das Americas

CEP 81531-990 Curitiba-Parana

Brazil

[email protected]

Title: A class of globally hypoelliptic Cauchy operators on the torus and generalized Siegel conditions.

Abstract: We present an investigation of global hypoellipticity problem for Cauchy operators defined on the torusTn+1 belonging to the class L =

∑mj=0Qj(t,Dx)Dm−j

t . Here, Qj(·, Dx) is a pesudo-differential operator on Tn,smoothly depending on t ∈ T and satisfying suitable hypothesis. We propose a study by combining Hormander andSiegel conditions on the symbols of Qj(·, Dx).

References:

[1] F. de Avila Silva, R.B. Gonzalez, A. Kirilov, and C. Medeira, Global Hypoellipticity for a Class of Pseudo-differential Operators on the Torus. To appear, Journal of Fourier Analysis and Applications (2018).

[2] D. Dickinson, T.V. Gramchev, and M. Yoshino, First order pseudodifferential operators on the torus: normalforms, diophantine phenomena and global hypoellipticity. Ann. Univ. Ferrara Sez. VII Sci. Mat. 41 (1997),51–64.

[3] D. Dickinson, T.V. Gramchev, and M. Yoshino, Perturbations of vector fields on tori: resonant normal formsand Diophantine phenomena. Proc. Edinb. Math. Soc. 45 (2002), 731–759.

[4] L. Hormander, The Analysis of Linear Partial Differential Operators III. Springer-Verlag, Berlin, 2007.[5] T. Gramchev, P. Popivanov, and M. Yoshino, Global solvability and hypoellipticity on the torus for a class of

differential operators with variable coefficients. Proc. Japan Acad. 68 (1992), 53–57.[6] T. Gramchev, P. Popivanov, and M. Yoshino, Global properties in spaces of generalized functions on the torus

for second order differential operators with variable coefficients. Rend. Sem. Mat. Pol. Torino 51 (1993),no.2, 145–172.

[7] S.J. Greenfield and N.R. Wallach, Global hypoellipticity and Liouville numbers. Proc. Amer. Math. Soc. 31(1972), no. 1, 112–114.

[8] M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries: Background Analysis andAdvanced Topics. Series: Pseudo-Differential Operators, vol. 2, Birkhauser Basel, 2010.


Lashi Bandara


Universitat Potsdam

Karl-Liebknecht-Str. 24-25

DE-14476 Potsdam

Germany

[email protected]

Title: Boundary value problems for general elliptic operators.

Abstract: We consider general elliptic operators on smooth manifolds with smooth boundary and develop aframework to study boundary value problems. This generalises previous work of Bar-Ballmann where it is assumedthat the induced operator on the boundary can be chosen to be self-adjoint. We proceed by understanding the


spectral theory of a general induced operator on the boundary as a bi-sectorial operator and use methods arisingfrom the H∞ functional calculus, semigroup theory, as well as the Kato square root problem.


Valerii Galkin

Department of Applied Mathematics

Surgut State University

Surgut RU-628412

Russia

[email protected]

Title: Microlocal Geometry Makes Global Structures in Porous Medium.

Abstract: We consider the problem of connectedness in the metric space which model is “porous medium”. Themain question is the description of measure for quantity of global connections between two points in the space providedthe local distribution of connections in microstructure is given. A porous medium is a network of intergrain channelsformed by internally connected intermediate spaces between particles. The main description of above is based on socalled functional solutions theory in the algebraically adjoint space to the L1

loc equipped by the Tikhonov topology.Natural examples of above problems are investigated in description of global structures produced by connected poresin matrix of oil-containing sands and similar problems arise in research of materials of nuclear reactors under influenceof neutron flow.

The Russian Foundation for Basic Research supported this work, project nos. 16- 29-15105, 18-01-00343.


Anton Kutsenko

Mathematical Sciences

Jacobs University

Campus Ring 1

DE-28759 Bremen

Germany

[email protected]

Title: Is there a difference between linear ODEs, non-linear stochastic ODEs, and linear PDEs.

Abstract: Assuming finite derivatives with arbitrary small rational steps of discretization, we show that nonlinearstochastic ODEs, linear ODEs, and linear PDEs (all of them with bounded continuous and discontinuous coefficients)generate the same algebra, namely the universal UHF-algebra. Thus, topologically and algebraically, they areequivalent. It is interesting to note that while the differential algebras are insensitive to any dimensions, the integro-differential algebras are sensitive to the number of variables, but still insensitive to the number of functions. Somephysical applications to the geometry of waves will be discussed.


Gianmarco Molino


University of Connecticut

341 Mansfield Road U1009

Storrs, CT 06269-1009

USA

[email protected]


Title: The Horizontal Einstein property for H-type foliations.

Abstract: We generalize the notion of H-type sub-Riemannian manifolds introduced by Baudoin and Kim, andthen introduce a notion of parallel Clifford structure related to a recent work of Moroianu and Semmelmann. Onthose structures, we prove an Einstein property for the horizontal distribution using ideas from Ishihara’s work onhyper-Kahler and quaternionic Kahler manifolds.This is a joint work with F. Baudoin, E. Grong, and L. Rizzi.


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