the heat and the schroedinger equation in almost ...dme.ufro.cl/somachi2019/doc/sesion09.pdf · the...

LXXXVIII Encuentro Anual – Temuco 2019Sociedad de Matematica de Chile

The heat and the Schroedinger equation inalmost-Riemannian geometry

Ugo Boscain *

Resumen

Almost-Riemannian geometry is a generalization of Riemannian geometry that na-turally arises in the framework of control theory. Let X and Y be two smooth vectorfields on a two-dimensional manifold M . If X and Y are everywhere linearly indepen-dent, then they define a classical Riemannian metric on M (the metric for which theyare orthonormal) and they give to M the structure of metric space. If X and Y be-come linearly dependent somewhere on M , then the corresponding Riemannian metrichas singularities, but under generic conditions the metric structure is still well defined.Metric structures that can be defined locally in this way are called almost-Riemannianstructures. They are special cases of rank-varying sub-Riemannian structures. Almost-Riemannian structures show interesting phenomena, in particular those which concernthe relation between curvature, presence of conjugate points, and topology of the ma-nifold.

In this talk I will discuss how to define an intrinsic Laplacian in almost-Riemanniangeometry and I will discuss the relation between its self-adjointness and the propertiesof the geodesics. Surprising results for the heat and the Schroedinger equation appearalready in simple examples as the Grushin plane.

Referencias

[1] A. Agrachev, U. Boscain, M. Sigalotti, A Gauss-Bonnet-like Formula on Two-Dimensional Almost-Riemannian Manifolds, Discrete and Continuous DynamicalSystems-A, vol. 20, pp. 801-822, 2008.

[2] U. Boscain, C. Laurent, The Laplace-Beltrami operator in almost-Riemannian Geo-metry. Ann. Inst. Fourier (Grenoble) 63 (2013), no. 5, 1739-1770.

[3] U. Boscain, D. Prandi, Self-adjoint extensions and stochastic completeness of theLaplace-Beltrami operator on conic and anticonic surfaces anticonic-type surfaces. J.Di↵erential Equations 260 (2016) 3234-3269

*CNRS, Sorbonne Universite, Inria, Universite de Paris, Laboratoire Jacques-Louis Lions, Paris, Francee-mail: [email protected]

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Smooth fields of Hilbert space and applications

Harold Bustos*

Resumen

The study of infinite dimensional smooth bundles is not well understand. The adap-

tation of the theory from the finite dimensional case to Banach bundles takes several

complications and in the practice, only a few trivial examples can be treated [1]. Moreo-

ver, there are several problems and examples which could be developed in the framework

of Hilbert bundles [3], necessitating new characterizations or a more general treatment.

In this talk we are going to discuss a new approach for Hilbert bundles and fields over a

smooth manifold and some consequences, like connections and curvature. We also will

treat some examples motiving our approach.

This talk is based in a joint research collaboration with Fabian Belmote of Univer-

sidad Catolica del Norte

Referencias

[1] J.M.G. Fell and R.S. Doran, Representations of ⇤-algebras, locally compact groups, and

Banach ⇤-algebraic bundles. Vol. 1. Basic representation theory of groups and algebras.

Pure and Applied Mathematics, 125; Academic Press, Inc., Boston, MA, 1988.

[2] N.P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics,

Springer Monographs in Mathematics, Springer-Verlag, New York, 1998.

[3] Lempert L. and Szoke R. Direct Images, Fields of Hilbert Spaces, and Geometric Quan-

tization, Commun. Math. Phys. (2014) 327, 49–99

[4] D. Williams: Crossed Products of C⇤-Algebras, Mathematical Surveys and Mono-

graphs, 134, American Mathematical Society, 2007.

[5] P. Xu, Morita equivalence and Symplectic Realizations of Poisson Manifolds, Ann. Sc.

Ec. Norm. Sup. 25 (1992), 307–333.

*Departamento de Matematicas, Facultad de Ciencias, Universidad de Chile, e-mail:[email protected]

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Minimales y convexidad

Carolina Canales *

Resumen

Un minimal en dinamica es un conjunto cerrado, no vacıo y saturado de orbitasque no contiene conjuntos mas pequenos con las mismas propiedades. Veremos comorelacionar algunas caracterısticas dinamicas de estos con caracterısticas geometricascomo convexidad y curvatura.

*Departamento de Matematicas, Pontificia Universidad Catolica de Chile e-mail:[email protected].

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Mass in Riemannian and Kahler manifolds

Karen Corrales *

Resumen

The Positive Mass Theorem states that any asymptotically Euclidean (AE) 3-manifold

with positive scalar curvature has non-negative mass; and the mass is zero only for the

Euclidean space [3]. Afterwards, it was conjectured that the Positive Mass Theorem

holds for any asymptotically locally Euclidean (ALE) 3-manifold with positive scalar

curvature. Ten years after, it was found a counterexample.

Since, recently a mass formula for ALE scalar-flat Kahler manifolds was defined [2],

a natural question arise, what does happen for ALE Kahler manifolds?. It was proved

that the mass is zero for ALE Ricci-flat Kahler manifold [1], but the converse is not

true. Hence, there exist ALE scalar-flat Kahler manifolds with zero mass not Ricci-flat.

In this talk I will present the mass formula for ALE Kahler 3-manifolds and examples

of ALE scalar-flat Kahler manifolds where the Positive Mass Theorem does not hold.

The talk is based on joint work with Claudio Arezzo.

Referencias

[1] C. Arezzo, R. Lena, and L. Mazzieri, On the resolution of extremal and constant scalar

curvature Kahler orbifolds, nt. Math. Res. Not. IMRN, (2016).

[2] H. Hein and C. LeBrun, Mass in Kahler Geometry , Commun. Math. Phys. (2016)

347: 183–221.

[3] R. Schoen and S. Yau, On the proof of the positive mass conjecture in general relativity,

Commun. Math. Phys. (1979) 65: 45–76.

*Facultad de Matematicas, Pontificie Universidad Catolica de Chile, e-mail: [email protected]

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A universal gap for controllability of quantum systems

J.P. Gauthier *

Resumen

We discuss here a result, proved by Gorodski and Lytchak, a consequence of which

is that there exists a universal gap for the controllability of any finite dimensional

controlled quantum system. That is, the maximum of the minimum-times to connect

two points on the Hilbert sphere by trajectories of the controlled system, is bounded

from below by a certain universal constant.

We propose here (with a remaining very little gap) a very short proof of this result.

The tools are only from representation theory of compact groups.

This is a joint work with Francesco Rossi (University of Padova, [email protected])

and Remi Robin (Ecole polytechnique, [email protected])

Referencias

[1] U. Boscain, J-P. Gauthier, F. Rossi, M. Sigalotti, Approximate controllability, exactcontrollability, and conical eigenvalue intersections for quantum mechanical systems,Comm. Math. Phys. 333 (2015), no. 3, 1225–1239.

[2] Claudio Gorodski, Alexander Lytchak, Christian Lange, Ricardo Mendes, A diametergap for quotients of the unit sphere. Preprint.

*University of Toulon. e-mail: [email protected]

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Reduccion de Lazos en diagramas de Feynman y la teoria devariable compleja

Igor Kondrashuk ⇤

Abstract

I introduce a concept of Feynman diagrams and explain why this tool is very usefulfor study of quantum phenomena. Then, I consider a special family of ladder diagramsand show why this family is important for quantum physics, and review briefly oldresults about the calculation of the ladder diagrams. In particular, I pay attention tothe loop reduction technique for this family of diagrams. For this purpose, I use theBelokurov-Usyukina reduction method for four-dimensional scalar integrals in positionspace. I consider the Mellin-Barnes (MB) transform of the triangle ladder-like scalardiagram in d = 4 dimensions. It is shown how the multi-fold MB transform of themomentum integral corresponding to an arbitrary number of rungs is reduced to thetwo-fold MB transform. New formulas for the MB two-fold integration in the complexplanes of two complex variables are derived. I demonstrate that these formulas solvethe Bethe-Salpeter equation. I show that the recurrent property of the MB transformsobserved in the present work for that kind of diagrams has nothing to do with quantumfield theory, the theory of integral transforms, or the theory of polylogarithms in general,but has its origin in a simple recursive property of smooth functions. In this talk Iexplain the meaning of this recurrent property on the theory of complex variable side.These results are based on the joint work with my PhD student Pedro JulcaCordova.

References

[1] P. ALLENDES, B. KNIEHL, I. KONDRASHUK, E.A. NOTTE-CUELLO, M.ROJAS-MEDAR, Solution to Bethe-Salpeter equation via Mellin-Barnes transform,Nuclear Physics B, vol. 870, pp. 243-277, (2013).

[2] B. KNIEHL, I. KONDRASHUK, E.A. NOTTE-CUELLO, I. PARRA-FERRADA, M. ROJAS-MEDAR, Two-fold Mellin-Barnes transforms of Usyukina-Davy-dychev functions, Nuclear Physics B, vol. 876, pp. 322-333, (2013).

[3] I. GONZALEZ AND I. KONDRASHUK, Belokurov-Usyukina loop reduction innon-integer dimension, Physics of Particles and Nuclei, vol. 44, pp. 268-271, (2013)

⇤Grupo de Matematica Aplicada, Departamento de Ciencias Basicas, Universidad del Bio-Bio, Chilee-mail: [email protected]

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[4] I. GONZALEZ AND I. KONDRASHUK, Box ladders in a non-integer dimension,Theoretical and Mathematical Physics, vol. 177, pp. 1515-1540 , (2013).

[5] I. GONZALEZ, B.A. KNIEHL, I. KONDRASHUK, E.A. NOTTE-CUELLO, I.PARRA-FERRADA AND M.A. ROJAS-MEDAR, Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1, NuclearPhysics B, vol. 925, pp. 607-614, (2017).

[6] I. GONZALEZ, I. KONDRASHUK, E.A. NOTTE-CUELLO, I. PARRA-FERRADA, Multi-fold contour integrals of certain ratios of Euler gamma func-tions from Feynman diagrams: orthogonality of triangles, Analysis and MathematicalPhysics, vol. 8, pp 589-602 (2018)

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The Cauchy problem for the Kadomtsev-Petviashvili hierarchyof di↵erential equations

Enrique G. Reyes *

Resumen

I will summarize recent work on the solution to the Cauchy problem of the Kadomtsev-Petviashvili (KP) hierarchy. This hierarchy is constructed as follows:

Define P =PN

�1<n an(x, t1, t2, · · ·)@nx and set up the equation

Ptn = [(Pn)+, P ]

for n = 1, 2, 3, · · ·, in which (Pn)+ indicates the projection of Pn on the space ofdi↵erential operators. This equation translates into an infinite number of nonlinearequations for the coe�cients an.

One can solve all the equations of the KP hierarchy at once, by using factorizationsof infinite dimensional Lie groups of pseudo-di↵erential operators. I will present thisresult in two di↵erent contexts:

- Algebraic : formal pseudo-di↵erential operators are defined on algebras equippedwith derivations and valuations, and the groups are formal objects.

- Geometric : formal pseudo-di↵erential operators are defined on algebras equippedwith a Frolicher (or Frechet) structure, and the groups are Frolicher Lie groups.

Referencias

[1] Mulase, M., Solvability of the super KP equation and a generalization of the Birkho↵decomposition, Invent. Math. (1988) 92: 1–46.

[2] Magnot, J.P. and Reyes, E.G., Well-posedness of the Kadomtsev-Petviashvili hierarchy,Mulase factorization, and Frolicher Lie groups, arXiv 1608.03994.

[3] Eslami Rad, A.; Magnot, J.-P.; Reyes, E.G., The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coe�cient algebra, Journal of Nonlinear Mathe-matical Physics (2017) 24: sup 1, 103–120.

*Departamento de Matematica y Ciencia de la Computacion, Universidad de Santiago de Chile, e-mail:[email protected].

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Short-time existence for the network flow

Mariel Saez *

Resumen

The network flow is a system of parabolic di↵erential equations that describes themotion of a family of curves in which each of them evolves under curve-shortening flow.This problem arises naturally in physical phenomena and its solutions present a richvariety of behaviors.

The goal of this talk is to describe some properties of this geometric flow and todiscuss an alternative proof of short-time existence for non-regular initial conditions.The methods of our proof are based on techniques of geometric microlocal analysis thathave been used to understand parabolic problems on spaces with conic singularities.

This is a joint work with Jorge Lira, Rafe Mazzeo, and Alessandra Pluda.

*Departamento de Matematicas, Pontificia Universidad Catolica de Chile. e-mail: [email protected]

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