EMS-RSME JOINT MATHEMATICAL WEEKEND · 2011-10-26 · PRESENTATION WelcometoBilbao. As part of the commemorations of its centennial, the Royal Spanish SocietyofMathematics(RSME)organizesjointlywiththeEuropeanMath

EMS-RSME JOINT

MATHEMATICAL WEEKEND

Bilbao, October 7-9, 2011

Organized by

CONTENTS

Presentation 2

Conference Committees 3

Schedule 4

EMS Distinguished Speaker: Gabriel Navarro 7

Session 1: Groups and Representations 9Main Speaker: Dan Segal 9Invited Speakers 10

Session 2: Symplectic Geometry 13Main Speaker: Miguel Abreu 13Invited Speakers 14

Session 3: PDEs in Mechanics and Physics 17Main Speaker: María J. Esteban 17Invited Speakers 18

Sessione 4: Functional Analysis Methods in Quantum Information 21Main Speaker: David Pérez 21Invited Speakers 22

Poster Presentations 25Session 1: Groups and Representations 25Session 2: Symplectic Geometry 27Session 3: PDEs in Mechanics and Physics 28Session 4: Functional Analysis Methods in Quantum Information 29

Participants 30

EMS-RSME Joint Mathematical Weekend 1

PRESENTATION

Welcome to Bilbao.As part of the commemorations of its centennial, the Royal Spanish

Society of Mathematics (RSME) organizes jointly with the European Math-ematical Society (EMS) this Mathematical Weekend in Bilbao from October7 to 9, 2011. As is usual in the EMS Mathematical Weekends, the meet-ing begins on a Friday afternoon and ends on Sunday at lunch time. Allmathematicians, from Europe and elsewhere, are welcome to the conference.

As in previous editions of the EMS Mathematical Weekends, there areseveral special sessions where some of the most prominent researchers in theirfields have agreed to give a talk. In this edition, these are the four topics thathave been selected for the special sessions: Groups and Representations,Symplectic Geometry, PDEs in Mechanics and Physics, and Func-tional Analysis Methods in Quantum Information. Also, a plenarytalk is going to be delivered by Gabriel Navarro, from the University ofValencia (Spain), who has been appointed EMS Distinguished Speakerby the European Mathematical Society. In addition, there is a poster ses-sion open to all participants who also have the possibility of giving a shorttalk on their contribution.

We hope this conference will be fruitful for all of you.

The Organizing Committee

2 EMS-RSME Joint Mathematical Weekend

CONFERENCE COMMITTEES

The Organizing Committee:

• Gustavo A. Fernández Alcober

• Luis Martínez Fernández

• Josu Sangroniz Gómez(University of the Basque Country)

The Program Committee:

• Rui Loja Fernandes (Technical University of Lisbon)

• Alexander Moretó (University of Valencia)

• Joaquim Ortega Cerdá (University of Barcelona)

• Silvie Paycha (Blaise Pascal University)

• Luis Vega González (University of the Basque Country)


SCHEDULE

Friday, October 7th.

Saturday, October 8th.


Sunday, October 9th.

Short talks by poster presenters. Saturday, October 8th.

Rooms.


EMS DISTINGUISHED SPEAKER

Gabriel NavarroUniversity of Valencia

Main problems in the representation theory of finite groupsThe Representation Theory of Finite Groups is a thriving subject, with

many fascinating and deep open problems, and some recent successes. Weshall give an overview of some of these.


Session 1: GROUPS AND REPRESENTATIONS

Main Speaker

Dan SegalUniversity of Oxford

Some algebraic properties of compact topological groupsWriters on topological groups often adopt the tacit convention that ‘sub-

groups’ are always closed and that homomorphisms are always continuous.We simple-minded algebraists sometimes forget the topology, and so areled to ask: what about non-closed subgroups, or non-continuous homomor-phisms?

In general there is probably not much to be said. But in the case ofcompact groups it turns out that there are some interesting restrictions. Iwill report on some recent results obtained by Nikolay Nikolov and myself.These include

Theorem 1. Let G be a compact group and f : G→ Q a group epimor-phism onto an abstract group Q.(i) If Q is finitely generated then Q is finite;(ii) if also G is (topologically) finitely generated, then ker f is open in G.

A special case of (ii) solves an old problem posed by Serre:

Corollary 1. If G is a (topologically) finitely generated profinite groupthen every subgroup of finite index in G is open, and every group homomor-phism from G to another profinite group is continuous.

This shows that finitely generated profinite groups have a ‘rigidity’ prop-erty: their topology is uniquely determined by the underlying abstract groupstructure.

More general results characterize those finitely generated compact groupsthat possess a countably infinite abstract quotient, or have a virtually-densenormal subgroup of infinite index, or a proper dense normal subgroup (infact it suffices to assume that our compact groups are (topologically) finitelygenerated modulo their identity component).

The methods involve both finite group theory and the theory of compactconnected Lie groups; in fact the parallelism between the classifications ofsimple groups in these two categories plays a certain role in some of theinvestigations.


Invited Speakers

Laurent BartholdiUniversity of Göttingen

Growth and Poisson boundaries of groups(joint work with Anna Erschler)

Let G be a finitely generated group. A rich interplay between algebraand geometry arises by viewing G as a metric space, or as a metric measuredspace. I will describe two invariants of finitely generated groups, namelygrowth and Poisson boundary, and explain by new examples that their rela-tionship is deep, but still mysterious.

Its growth function γ(n) counts the number of group elements that can bewritten as a product of at most n generators. This function depends on thechoice of generators, but only mildly: say γ ∼ δ if γ(n) ≤ δ(Cn) ≤ γ(C2n)for some C > 0; then the ∼-equivalence class of γ is independent of thechoice of generators.

For example, the growth of Zd is asymptotic to nd, while the growth ofa free group is asymptotic to 2n. There are groups whose growth function isknown to lie strictly between polynomials and exponentials; I will describethe first examples for which the asymptotic growth is known. I will also de-scribe an example of a group of exponential growth, whose Poisson boundaryis trivial for all finitely-supported random walks. Perhaps surprisingly, bothexamples come from the same general construction, permutational wreathproducts.

Carles BrotoAutonomous University of Barcelona

Equivalences between fusion systems of finite groups of Lietype

I will review the concept of fusion system at a prime p of a finite groupand the connections with the homotopy type of the classifying space. As anapplication, I will show some equivalences between fusion systems of finitegroups of Lie type.

Pierre-Emmanuel CapraceCatholic University of Louvain

Trees, contractions and SL2

We shall present necessary and sufficient conditions for a locally compactgroup acting properly on a locally finite tree to be isomorphic to SL2 over alocal field k of arbitrary characteristic. As an application, a characterizationof SL2(k) among locally compact groups can be deduced, in the spirit ofHilbert’s 5th problem.


Andrei JaikinAutonomous University of Madrid

Finite simple quotients of groups satisfying Kazhdan’sproperty (T )

(joint work with Mikhail Ershov and Martin Kassabov)

Let X be a graph and let ε > 0 be a real number. We say that X is anε-expander if, for all sets A consisting of at most half of the vertices of X,we have |∂A| ≥ ε|A|, where ∂A denotes the boundary of A, i.e. the verticesof distance 1 from A. An infinite family {Xi}i∈N of k-regular finite graphsis called a family of expanders if there exists ε > 0 such that all the graphsXi are ε-expanders and the number of vertices of Xi tends to infinity. Thefirst proof of the existence of families of expanders by Pinsker is based oncounting arguments. For applications one wants explicit constructions. Mar-gulis realized that using infinite residually finite groups satisfying Kazhdan’sproperty (T ) one can construct explicit examples of families of expanders.

The construction is done in the following way. Let Γ be an infiniteresidually finite group satisfying Kazhdan’s property (T ) (for example, Γ =SLn(Z)). Let S be a symmetric finite generating set of Γ and {Γi}i∈N a familyof normal subgroups of Γ of finite index such that |Γ/Γi| tends to infinity.Then the Cayley graphs Xi = Cay(Γ/Γi;S) form a family of expanders.

We say that a family {Gi} of finite groups is a family of expanders ifthere are k ∈ N and ε > 0 such that every group Gi has a symmetric subsetSi of k generators for which Cay(Gi;Si) is an ε-expander. It was conjecturedby Babai, Kantor and Lubotzky that the family of all the finite (nonabelian)simple groups is a family of expanders. The conjecture first was proved byKassabov, Nikolov and Lubotzky for all the simple groups with the exceptionof the Suzuki groups and recently for the Suzuki groups by Breuillard, Greenand Tao.

In my talk I want to consider the following question.

Question. Let F be a family of finite simple groups. Is there a groupΓ satisfying Kazhdan’s property such that the groups from F are quotientsof Γ?

Our main result is the following:

Theorem. There exists a group Γ satisfying property (T ) such thatevery finite simple group of Lie type of rank at least 2 is a quotient of Γ.

Geoffrey RobinsonUniversity of Aberdeen

Some links between fusion and representation theoryIt has been known for a long time, starting with the work of R. Brauer,

that there are strong links between the p-local structure of a finite group Gand its representation theory in characteristic p. Similarly, the conjugacy ofp-subgroups (p-fusion) of G is determined completely by p-local information,as demonstrated by J.L. Alperin. In this talk, we will discuss how to obtain


representations of a finite group from knowledge of its p-fusion, includingexplicit examples of group representations so constructed.

Raphaël RouquierUniversity of Oxford

Perverse equivalences and representationsAn important theme in representation theory in the past twenty years

has been the study of correspondences between representations of differentgroups or algebras. One such form of correspondence is that of equivalencesof derived categories. We will present a combinatorial framework for thisand explain how it applies to representations of finite groups. This is jointwork with Joe Chuang and David Craven.


Session 2: SYMPLECTIC GEOMETRY

Main Speaker

Miguel AbreuIST-Technical University of Lisbon

Lagrangian intersections in toric manifoldsRigidity of Lagrangian intersections plays a fundamental role in sym-

plectic geometry and topology. In this talk, after a general introduction,I will address two natural Lagrangian intersection problems in the contextof symplectic toric manifolds: displaceability of torus orbits and of a torusorbit with the real part of the toric manifold. The emphasis will be on somerecent joint work with Leonardo Macarini, showing how one can use simplecartesian product and symplectic reduction considerations to go from basicexamples to much more sophisticated ones.


Invited Speakers

Anna FinoUniversity of Turin

Calabi-Yau equation on symplectic T 2-bundles over T 2

I shall discuss some Calabi-Yau type geometries and equations in thecontext of non-integrable almost complex structures. The focus will be oncertain symplectic non-Kähler 4-manifolds admitting a 2-torus fibration overa 2-torus base.

William KirwinUniversity of Cologne

Adapted complex structuresI will discuss a new perspective on adapted complex structures which

is related to “imaginary-time” geodesic flow. I will also discuss some gener-alizations of adapted complex structures to the “magnetic” case, as well asmention some applications. (This is mostly joint work with B. Hall)

Gabriele La NaveUniversity of Illinois, Urbana-Champaign

The V-soliton equation, symplectic reduction and theKähler-Ricci flow

I will describe how the Ricci flow on a Kähler manifold X with initialKähler metric g is equivalent to a static equation of soliton type, called theV-soliton equation on a manifoldM with a holomorphic symplectic action ofC∗, such that one of the symplectic reductions is (X,ω), the initial datum ofthe flow. I will then describe regularity questions of the equation and someimplications of the equivalence for finite type singularities of the Kähler-Ricciflow.

Andrea LoiUniversity of Cagliari

Global symplectic coordinates on Kähler manifolds.We show a global version of Darboux theorem and we find global explicit

symplectic coordinates on bounded symmetric domains of noncompact type,on certain complex domains of Cn equipped with rotation invariant Kählerforms and on Calabi’s tubular domains of Cn equipped with complete andKähler-Einstein forms.


Eva MirandaPolytechnic University of Catalonia

The symplectic geometry of b-manifoldsA b-manifold is a pair (M,H) where H is a hypersurface of M . Calculus

on b-manifolds (b-calculus) started with the work of Melrose in connectionto the Atiyah-Patody-Singer formula and the work of Nest and Tsygan onformal deformations of symplectic manifolds with boundary.

It turns out that a symplectic structure on M that “blows up linearly”along a hypersurface H can be formulated either as a “generic” Poisson struc-ture Π on M or as a b-symplectic two form (in the language of b-calculus).

In this talk we will investigate the symplectic geometry of b-manifolds.These manifolds provide two geometrical structures which are relevant froma symplectic point of view:

(1) The b-symplectic form itself on M .(2) A (regular) codimension-one symplectic foliation, F, induced on H

by the Poisson structure Π.We will present these two geometrical structures from a symplectic per-

spective and provide theorems that connect both aspects. In particular, wewill state some Moser-like theorems in the context of b-calculus (using b-cohomology) which allow to prove local and semilocal normal form result.We will also present some results concerning these b-cohomology groups (DeRham analogues in b-calculus). These Moser tricks for b-symplectic mani-folds, in their turn, give the right tools to construct a b-symplectic structurein a neighbourhood of a given manifold H endowed with codimension-onesymplectic foliations (with two vanishing foliated cohomology invariants).We will also give a complete description of H and its Poisson structure Πwhen one of the leaves of the symplectic foliation F is compact. Finally, wewill explore some interesting issues from the Poisson point of view like themodular class or the integrability of M and H as Poisson manifolds.

References:[1] V. Guillemin, E. Miranda, A.R. Pires, Codimension one symplectic

foliations and regular Poisson structures, to appear at the Bulletin of theBrazilian Mathematical Society (special issue for the Poisson 2010 confer-ence), arXiv:1009.1175v2 [math.SG], 2010.

[2] V. Guillemin, E. Miranda, A.R Pires, Symplectic and Poisson geom-etry on b-manifolds, preprint 2010.

Dmitri PanovKing’s College London

Hyperbolic geometry and symplectic Calabi-Yau manifoldsA question of Gromov asks: Is it true that every compact manifold is

a quotient of a hyperbolic space by a discrete group of isometries? I willexplain how one can use this question to prove that the fundamental groupsof compact symplectic Calabi-Yau six-manifolds can be arbitrary. The talkis based on the joint work with Anton Petrunin and Joel Fine.


Session 3: PDEs IN MECHANICS AND PHYSICS

Main Speaker

María J. EstebanCNRS and University of Paris-Dauphine

Eigenvalues of operators in spectral gaps and applications inrelativistic quantum mechanics

In this talk I will present a variational approach for the characterizationand computation of the eigenvalues of operators in their (possibly existing)spectral gaps. The variational methods presented here are not classical be-cause they correspond to the study of critical points of totally indefinitefunctionals.

As a particular application, I will present some results on the dependenceon the strength of the magnetic field for the first eigenvalue of the magneticCoulomb-Dirac operator. This mathematical model describes the physics ofa relativistic electron subject to the action of a strong electro-magnetic field.


Invited Speakers

Virginie Bonnaillie-NoëlUniversity of Rennes 1, IRMAR CNRS

Eigenpairs of a model Schrödinger operator with Neumannconditions on the half-plane

We study the eigenpairs of a model Schrödinger operator with a quadraticpotential and Neumann boundary conditions on a half-plane. The potentialis degenerate in the sense that it reaches its minimum all along a line whichmakes the angle θ with the boundary of the half-plane. We investigate thebehavior of the eigenpairs in the limit θ → 0 and we prove a full asymptoticexpansion for these eigenvalues and their associated eigenvectors. We alsopresent numerical experiments obtained by a finite element method withthe library Mélina [4]. The numerical results confirm and enlighten thetheoretical approach.This problem is motivated by the analysis of the third critical fieldHC3 in thetheory of superconductivity (see for instance [2], [3]). Results mentioned inthis talk have been obtained in collaboration with Monique Dauge, NicolasPopoff and Nicolas Raymond, see [1].

References:[1] V. Bonnaillie-Noël, M. Dauge, N. Popoff and N. Raymond. Discrete

spectrum of a model Schrödinger operator on the half-plane with Neumannconditions. Documenta Mathematica 9 (2004) 283–299.

[2] S. Fournais, B. Helffer. Spectral methods in surface superconductiv-ity. Progress in Nonlinear Differential Equations and their Applications, 77.Birkhäuser Boston Inc., Boston, MA 2010.

[3] K. Lu, X.-B. Pan. Surface nucleation of superconductivity in 3-dimensions. J. Differential Equations 168(2) (2000) 386–452. Special issuein celebration of Jack K. Hale’s 70th birthday, Part 2 (Atlanta, GA/Lisbon,1998).

[4] D. Martin. The finite element library Mélina.http://perso.univ-rennes1.fr/daniel.martin/melina (2006).

Diego CórdobaSpanish National Research Council

Turning waves and breakdown for incompressible flowsWe consider the evolution of an interface generated between two im-

miscible, incompressible and irrotational fluids. Specifically we study theMuskat and water wave problems. We show that starting with a family ofinitial data given by (x, f(x)), the interface reaches a regime in finite timein which is no longer a graph. In particular, for the Muskat problem, thisresult allows us to reach an unstable regime, for which the Rayleigh-Taylorcondition changes sign and the solution breaks down. In the case of waterwaves we have shown the existence of splash (interface that touches itself inone point) singularities in finite time.


Miguel EscobedoUniversity of the Basque Country

Some new remarks on a kinetic equation for Comptonscattering

We consider an approximation, for small values of the energy variable,of the kinetic equation that describes the Compton scattering. After prov-ing the existence of local solutions, we discuss their global existence. Wedescribe the asymptotic behavior of some global solutions and compare withthe results already known for the complete equation.

David LannesEcole Normale Supérieure

Shallow water and rough bottoms for the water wavesequations

We consider the Euler equations for free surface water waves in the caseof varying bathymetry, considering the problem in the shallow water scalingregime. In the case of rapidly varying periodic bottom boundaries this is aproblem of homogenization theory. The principal issue is that the shallowwater limit and the homogenization process do not commute and must beperformed simultaneously. We derive new model equations and exhibit theeffect of a “Bragg resonance” with the periodic bottom, which leads to seculargrowth and can influence the time interval of validity of the theory. This isa joint work with W. Craig and C. Sulem.

Mathieu LewinCNRS and University of Cergy-Pontoise Saint-Martin

Derivation of Pekar’s polaron from a microscopic model ofquantum crystal

A polaron is an electron interacting with a polar crystal, which is ableto form a bound state by using the distortions of the crystal induced by itsown density of charge.

The simplest model used by physicists to describe polarons was intro-duced by Pekar in 1954. In this theory, the electron is simply submittedto an effective attractive nonlinear Coulomb force, modeling the crystallinedeformations.

In this talk I will explain how one can derive Pekar’s polaron from aquantum theory of crystals, in a macroscopic limit where the polaron liveson a scale much larger than the crystal size.

This is joint work with Nicolas Rougerie (CNRS/Cergy).


Nicola ViscigliaUniversity of Pisa

On the minimization of semirelativistic S-P-S energyWe present a joint work with J. Bellazzini and T. Ozawa. We discuss

the compactness (up to translation) of minimizing sequences to the semirel-ativistic Schrödinger-Poisson-Slater (SPS) energy:

I(ρ) = inf∫|u|2dx=ρ

1

2‖u‖2

H12 (R3)

+1

4

∫ ∫R3×R3

|u(x)|2|u(y)|2

|x− y|dxdy− 3

8

∫R3

|u|83dx

with ρ > 0 small enough.

The main obstructions to apply the classical concentration-compactness tech-nique in the above problem are related to the following facts:

• the energy involves three terms with different degree of homogeneities;• the quadratic term of the energy is not rescaling invariant and non-local;• the natural limit minimization problem

inf‖u‖L2=ρ

1

2‖u‖2

H12− 3

8‖u‖

83

L83

does not satisfy a strict subadditivity property w.r.t. ρ.Those difficulties make the problem interesting from a mathematical point ofview. In fact, at the best of our knowledge, the techniques developed in theliterature to study SPS energy in a non-relativistic setting fail in our context.

We also discuss questions related to the qualitative properties of the mini-mizers to SPS in the non relativistic setting.


Session 4: FUNCTIONAL ANALYSIS METHODS IN QI

Main Speaker

David PérezThe Complutense University of Madrid

Playing quantum games with operator spacesMany protocols in computer science and cryptography, as well as many

complexity problems, can be analyzed by means of two-prover-one-round(2P1R) games. In this type of games a verifier sends challenges to twocooperative provers which have some communication limitations –which isusually no communication at all. Then they have to answer according to apreviously agreed strategy in order to optimize their probability of winningthe game. The game is defined by specifying the way the verifier chooses thequestions together with the way he verifies, based on questions and answers,whether the provers win the game. In these games one can give certainquantum abilities to some or all of the participants

The range of applicability of this type of games is quite wide. Theyare a common tool to test fundamental properties of Nature, such as thenecessity of Quantum Mechanics to explain experimental data. They lieat the heart of most inapproximability results in complexity theory basedon the PCP theorem. They are used to provide certifiable randomness anddevice independent cryptography via quantum mechanical effects. They alsoallow to analyze the recently introduced position-based cryptography, as wellas other quantum protocols such as coherent state exchange. Finally, theyallow to analyze the potential saves in communication to implement tasks inparallel in the regime where the different cores share quantum states.

Recently, in a series of works in collaboration with Thomas Cooney,Marius Junge, Miguel Navascues, Carlos Palazuelos, Ignacio Villanueva andMichael Wolf, we have analyzed these games connecting them with the math-ematical theory of Operator Spaces. In this talk I will show this connectionand sketch some of the applications derived from it.


Invited Speakers

David AlonsoUniversity of Alberta

On the volume of hyperplane sections and projections ofconvex bodies

The slicing problem asks the following: “Is it true that there exists anabsolute constant c such that for any convex body of volume 1 there exists ahyperplane section of volume greater than c?”. The best general result knownis that every n-dimensional convex body of volume 1 has a hyperplane sectionof volume greater than c

n14for some absolute constant c. We will give better

estimates for some particular classes of convex bodies.Similarly, one can ask a similar question for projections instead of sec-

tions and it happens that the Haar measure of the set of vectors θ ∈ Sn−1

such that |Pθ⊥K| ≥ c|K|n−1n is greater than or equal to 1−cn |K|

n−1|Π∗(K)||Bn

2 |n−1|Π∗(Bn2 )| ,

where Π∗(K) denotes the polar projection body of K. This affine invariant

quantity(|K|n−1|Π∗(K)||Bn

2 |n−1|Π∗(Bn2 )|

) 1n is bounded from above by 1 and from below by

c√n. We investigate the value of this quantity and show that for any K and

any c < 1, there exists a projection of K of dimension k ≥ cnα(c) such that(|PEK|k−1|Π∗(PEK)||Bk

2 |k−1|Π∗(Bk2 )|

) 1k ≥ c.

Guillaume AubrunUniversity of Lyon

Entanglement threshold for random states(joint work with Stanislaw Szarek and Deping Ye)

We consider random states on Cd ⊗ Cd obtained by partial tracing aHaar-distributed random pure state on Cd⊗Cd⊗Cs, and study the problemwhether they are typically separable or entangled (for large d, s).

We show the existence of a threshold s0 ≈ d3 (up to logarithmic factors).If s ≤ (1 − ε)s0, random states are typically entangled; if s ≥ (1 + ε)s0,random states are typically separable. We define the threshold s0 usinggeometric parameters associated to the convex body of all separable states.To estimate s0 we use tools from Asymptotic Geometric Analysis (notablythe MM∗-estimate).


Thomas CooneyThe Complutense University of Madrid

Operator algebras and locally quantum preparations of jointsystems

(joint work with David Pérez García and Ignacio Villanueva Díez.)

In a recent paper, Barnum, Beigi, Boixo, Elliott and Wehner proved thatif two physical systems A and B are the bounded operators on infinite dimen-sional Hilbert spaces HA, HB, then every locally quantum, non-signallingpreparation on the joint system can be simulated by a state in the Hilbertspace HA ⊗HB.

We show how this result can be adapted to the infinite-dimensional case.Operator algebra tensor products, von neumann algebras, and noncommu-tative Lp-spaces naturally appear and provide techniques ideally suited toobtaining the result.

Tobias FritzThe Institute of Photonic Sciences

Witnessing infinite-dimensional state spacesStandard quantum field theory predicts the state space of a quantum

field to be infinite-dimensional. For most purposes, this infinite-dimensionalHilbert space can be approximated by finite-dimensional ones, so that onecannot tell from experiment whether the actual dimension is infinite or onlyfinite but very large. Here we identify a context in which this approxima-bility fails and the infinite-dimensionality of Hilbert space can in principlebe witnessed. This is based on a result from combinatorial group theory, forwhich we give a new proof that adapts to the consideration of experimentalimperfections. In the presence of experimental imperfections, the witnessturns into a finite lower bound on the dimension. Since this lower boundgrows very slowly with experimental accuracy, we regard the present witnessas a proof of concept not worthy of experimental implementation.

Olivier GuedonUniversity Paris-Est Marne-la-Vallée

Sparsity and non-Euclidean embeddings(joint work with Omer Friedland)

We present a relation between sparsity and non-Euclidean isomorphicembeddings. We introduce a general restricted isomorphism property andshow how it enables to construct embeddings of `np , p > 0, into various typeof Banach or quasi-Banach spaces. In particular, for 0 < r < p < 2 withr ≤ 1, we construct a family of operators that embed `np into `(1+η)n

r , withoptimal polynomial bounds in η > 0.


Magdalena MusatUniversity of Copenhagen

Factorizable completely positive maps and quantuminformation theory

In recent work with Uffe Haagerup, we study factorization and dilationproperties of Markov maps between von Neumann algebras, motivated by thequestion of existence of non-factorizable Markov maps, as formulated by C.Anantharaman-Delaroche. We provide simple examples of non-factorizableMarkov maps on Mn(C) for all n ≥ 3 , as well as an example of a one-parameter semigroup (T (t))t≥0 of Markov maps on M4(C) such that T (t)fails to be factorizable for all small values of t > 0 .

The existence of non-factorizable Markov maps turned out to have aninteresting application to an open problem in quantum information theory,known as the asymptotic quantum Birkhoff conjecture (AQBP). We solve theconjecture in the negative by showing that every non-factorizable Markovmap on Mn(C) , n ≥ 3, provides a counterexample. We also discuss veryrecent developments concerning the question whether every factorizable mapdoes satisfy the AQBP.

In connection with this, we put into evidence a new asymptotic propertyof factorizable maps and establish connections to the Connes embeddingproblem.

Ion NechitaCNRS and Paul Sabatier University

Block-modified Wishart matrices and applications toentanglement theory

We study linear maps acting on the blocks of a Wishart matrix with atensor product structure. We obtain a free probabilistic description of thelimiting eigenvalue distribution of the resulting matrix and we apply theseresults to entanglement theory. We study separability criteria for randomquantum states of large dimension. This is joint work with Teodor Banica,Benoit Collins and Deping Ye.


POSTER PRESENTATIONS

Session 1: Groups and Representations

I Cristina Acciarri (University of L’Aquila).(jointly with Gustavo A. Fernández Alcober, Pavel Shumyatsky)Focal subgroups and outer commutator words. Let G be a finitegroup, and let P be a Sylow p-subgroup of G for a given prime p. It isknown that P ∩G′ is generated by the commutators lying in P . As a naturalgeneralisation, we pose the following question: if w is a group word, is ittrue that w(G) ∩ P can be generated by the values of w contained in P? Inthe poster we will present some positive results in this direction.

I Paz Arroyo Jordá (Polytechnic University of Valencia).(jointly with M. Arroyo-Jordá, A. Martínez-Pastor, M.D. Pérez-Ramos)On products of groups and conditional permutability. Products ofgroups whose factors are linked by certain permutability conditions have beenwidely investigated over the last years. The research on this topic was moti-vated by the interest in providing criteria for products of supersoluble groupsto be supersoluble. Initially Asaad and Shaalan (1989) obtained a supersolu-bility criterion via totally permutability. More recently Guo, Shum and Skiba(2005) have extended previous results by considering a weaker conditionof subgroups permutability, namely conditional permutability. We presentnew developments involving conditional permutability and other close per-mutability properties, also in the framework of formation theory.

I M. José Felipe (Polytechnic University of Valencia).(jointly with Elena Alemany, Antonio Beltrán)Structure of normal subgroups and G-class sizes. There exists a strongrelation between the structure of a finite group and the arithmetical proper-ties underlying the set of its conjugacy class sizes. In this poster we presentsome recent results showing how the structure of a normal subgroup N iscontrolled by the set csG(N), which is defined as the set of sizes of the G-conjugacy classes contained in N . For instance, we show that if csG(N) ={1,m} for some positive integer m, then N is nilpotent. More precisely,N either is abelian or is the product of a nonabelian p-group by a centralsubgroup of G ([1]). This result extends the celebrated N. Itô’s Theoremon groups having exactly two class sizes. On the other hand, this has beenextended for class sizes of elements of prime power order of N in [4].

We also prove the solvability and determine the structure of a normalsubgroup N when csG(N) = {1,m, n}, where m and n are positive integerssuch that m < n and m does not divide n ([2]). This partially extends therecent classification due to Dolfi and Jabara of groups having three class sizes.


Our approach is made by a suitable generalization for normal subgroups ofthe concept of F -group (which were classified by Rebmann), and also byusing Baer and Suzuki’s theorems on the classification of groups possessingnontrivial partitions.

References:[1] Alemany, E., Beltrán, A., Felipe, M.J. Nilpotency of normal subgroups having two

G-class sizes. Proc. Amer. Math. Soc. 139 (8) 2663-2669, (2011).[2] Alemany, E., Beltrán, A., Felipe, M.J. Itô’s theorem on groups with two class sizes

revisited. Accepted in Bull. Aus. Math. Soc. (2011)[3] Akhlaghi, Z., Beltrán, A., Felipe, M.J., Khatami, M. Structure of normal sub-

groups with three G-class sizes. Monatsh. Math. (2011) doi: 0.1007/s00605-011-0290-8[4] Beltrán, A., Felipe, M.J. Normal subgroups and class sizes of elements of prime

power order. Accepted in Proc. Amer. Math. Soc. (2011).

I Ainhoa Iñiguez (University of Oxford).Word equations on nilpotent groups on class 2. Let G be a finitep-group of class at most 2, and let w = w(x1, . . . , xn) be a group word inn variables. We prove that the number of solutions in G × n· · · × G to theequation w = 1 is at least |G|n−1. This result, also independently obtainedby Matthew Levy, solves a special case of a conjecture of Alon Amit.

I José M. Muñoz Escolano (University of Zaragoza).(jointly with Leonid A. Kurdachenko, Nadezhda Turbay)Infinite conjugate-permutable subgroups. A subgroup H of a group Gis called conjugate-permutable ifHHg = HgH for every g ∈ G. Clearly, everynormal or permutable subgroup is conjugate-permutable. In this poster, weclaim that for different classes of infinite groups such as Chernikov or poly-cyclic groups, every conjugate-permutable subgroup is subnormal; while forothers such as soluble–by–finite minimax groups, every conjugate-permutablesubgroup is ascendant. Furthermore, we also study the structure of infinitegroups whose cyclic subgroups are all conjugate-permutable.

I Lucía Sanus (University of Valencia).(jointly with Carlo Casolo, Silvio Dolfi, Emanuele Pacifici)Groups whose prime graph on conjugacy class sizes has few com-plete vertices. Let G be a finite group, and let Γ(G) denote the primegraph built on the set of conjugacy class sizes of G. We consider the situ-ation when Γ(G) has “few complete vertices”, and our aim is to investigatethe influence of this property on the group structure of G.

I Amaia Zugadi Reizabal (University of the Basque Country).On the Hausdorff dimension and the abelian group structure of theGupta-Sidki group. The Gupta-Sidki group and the Second Grigorchukgroup are particular instances of the so-called GGS-groups. In this poster,we give the Hausdorff dimension of the closures of these groups. And we alsodescribe an abelian group structure found recently in some of these groups(where the Gupta-Sidki group is included).


Session 2: Symplectic Geometry

I Ibai Basabe (University of Florida).New homotopy invariants, and their applications to robot mo-tion planning on topological groups and closed simply connectedsymplectic manifolds. Topological complexity, TC, is a homotopy invari-ant which was defined by Michael Farber at the beginning of this century.TC emerged during the search to find the minimum number of instructionsneeded by a robot to move inside its configuration space. Other homotopyinvariants appear when imposing certain characteristics to the movementof the robot, for example, symmetry. These definitions will be rigorouslydescribed in this poster, in which we will find the minimum number of in-structions for spaces of configurations such as topological groups and closedsimply connected symplectic manifolds. These results are part of a jointwork of Basabe, González, Rudyak and Tamaki.

I Giovanni Bazzoni (UCM-ICMAT).Minimal algebras and 2−step nilpotent Lie algebras in dimension 7The goal of this work is to obtain a classification of length 2 minimal algebrasof dimension 7 generated in degree 1 over a field k of characteristic char(k) 6=2. We give a precise answer in the case in the case k is algebraically closed;in other cases we only have partial results (in particular, we can deal withfinite or real closed fields). The classification of such algebras is equivalentto the classification of 2−step nilpotent Lie algebras in dimension 7. Whenk is a number field, we also obtain a classification of 2−step nilmanifolds ofdimension 7 up to k−homotopy type.

I Leonardo Colombo (ICMAT-CSIC).On variational problems on Lie groups. The purpose of this poster isto describe the so-called Tulczyjew’ s triple associated with the prolongationE = LτAA of the Lie algebroid τA : A → M. As an application, we deducethe second order Euler-Lagrange equations on Lie algebroids following a geo-metric point of view, generating a dynamic from a Lagrangian submanifold.

References:[1] J. Cariñena, J. Nunes da Costa, P. Santos. Quasi-coordinates from the point of

view of Lie algebroid J. Phys. A: Math. Theor. 40 10031 (2007).[2] D. Iglesias, J.C. Marrero, D.Martín de Diego and D. Sosa. Singular lagrangian

systems and variational constrained mechanics on Lie algebroids. Dynamical Systems: anInternational Journal 23 (2008) Issue 3, 351-397

[3] M. de León, E. Lacomba. Lagrangian submanifolds and higher-order mechanicalsystems, J. Phys. A.:Math. Gen. 22 (1989), 3809-3820.

[4] M. de León, J.C. Marrero, E. Martínez: Lagrangian submanifolds and dynamicson Lie algebroids, J. Phys. A: Math. Gen. 38 (2005)241-308.

[5] K. Mackenzie, General theory of Lie groupoids and Lie algebroids, volume 213 ofLondon Mathematical Society Lec. Notes Ser.. Cambridge university Press, Cambridge(2005).

[6] E. Martínez. Lagrangian Mechanics on Lie Algebroids. Acta Appl. Math.,67(2001), 295-320

I Rui Loja Fernandes (IST-Technical University of Lisbon).(jointly with Pedro Frejlich)A h-principle for symplectic foliations. We show that a classical re-sult of Gromov in symplectic geometry extends to the context of symplectic


foliations, which we regard as a h-principle for (regular) Poisson geometry.Namely, we formulate a sufficient cohomological criterion for a regular bivec-tor to be homotopic to a regular Poisson structure, in the spirit of Haefliger’scriterion for homotoping a distribution to a foliation. We give an exampleto show that this criterion is not too unsharp.

I Víctor Manero (University of the Basque Country).(jointly with Marco Freibert)Symplectic half-flat structures on solvable Lie algebras. We classifydecomposable 6-dimensional Lie algebras with a symplectic half-flat struc-ture. We use several obstructions to the existence of a calibrated G2 form.

I Romero Solha (Polytechnic University of Catalonia).Geometric quantisation of integrable systems. This poster shows anattempt to extend some results by Sniatycki, Guillemin and Sternberg ingeometric quantisation considering regular fibrations as real polarisationsto the singular setting. The generic real polarisations concerned here aregiven by integrable systems with nondegenerate (in the Morse-Bott sense)singularities. And the definition of geometric quantisation used is the onesuggested by Kostant; via higher cohomology groups. The case of nondegen-erate singularities was obtained in dimension 2 by Hamilton and Mirandaand the completely elliptic case was considered by Hamilton in any dimen-sion. The idea is to combine previous results of Miranda and Presas on aKünneth formula to reduce to the 2-dimensional case. One of the key toolsused here is an extension of the results of Rawnsley on the Kostant complex.This is based on a joint work in progress with Eva Miranda.

I Raquel Villacampa (CUD).(jointly with Marisa Fernández, Anna Fino, Luis Ugarte)Hermitian geometry with torsion. We present conditions that allowus to construct some new SKT metrics in dimension 2n + 2 (resp. HKT indimension 4n+4) starting from some almost contact structures in dimension2n+1 (resp. in dimension 4n+3). Explicit examples for each case are given.

I Marco Zambon (Autonomous University of Madrid).Moment maps up to homotopy. The notion of moment map is centralin symplectic geometry. We propose an extension of this notion to higherdifferential forms. We focus on the case of closed 3-forms invariant undera Lie group action. We show that moment maps up to homotopy existfor actions satisfying very reasonable assumptions, give an interpretation interms of equivariant cohomology, discuss an analog of Marsden-Weinsteinreduction and the relation to Courant algebroids.

Session 3: PDEs in Mechanics and Physics

I Naiara Arrizabalaga (University of the Basque Country).Distinguished self-adjoint extensions of Dirac operators with diag-onal potentials. We prove some Hardy-Dirac inequalities with two differentweights including measure valued and Coulombic ones. Those inequalities


are used to construct distinguished self-adjoint extensions of Dirac opera-tors for a class of diagonal potentials related to the weights in the abovementioned inequalities.

I Rafael Granero (ICMAT-CSIC).On the Muskat problem on a strip. We study the fluid problem ofthe evolution of the interface given by two incompressible fluids with dif-ferent densities in the porous strip R × [−l, l]. This problem is known asthe Muskat problem and is analogous to the two phase Hele-Shaw cell. Weobtain explicitly the contour equation and we show that the full problemis equivalent to a classical solution of the contour equation. We also givesome useful properties as the conservation of the mean and the equivalencebetween the same problem posed in the whole plane when l→∞. Our mainresult is the local well posedness in Sobolev spaces for the interfacial wavewhen the Rayleigh-Taylor condition is satisfied and the Maximum Principlefor ||f ||L∞(t). We also prove a decay estimate for ||f ||L∞(t) and a MaximumPrinciple for ||∂xf ||L∞(t) for a special class of initial data. Also we comparethe whole plane case and the problem in the strip.

Session 4: Functional Analysis Methods in QuantumInformation

I Zoltan Zimboras (ISI-Institute for Scientific Interchange).Quantum fluctuation algebra: a unifying tool for quantum centrallimit theorems and quantum memories. The concept of the Quan-tum Fluctuation Algebra (QFA) was developed in the context of QuantumStatistical Mechanics in order to describe the fluctuations of macroscopicobservables. In Classical Statistical Mechanics, one could prove for a largevariety of (non-critical) lattice models that the joint probability distributionof the so-called “fluctuation observables” of different macroscopic quantitiesbecomes a (multidimensional) Gaussian in the thermodynamic limit. Thesepropositions are often referred to as the Central Limit Theorems of StatisticalMechanics. However, in the quantum setting the “fluctuation operators” ofmacroscopic observables do not commute even in the thermodynamic limit,so one cannot talk about their joint probability distribution. Hence it wasalso an open question how one could formulate a corresponding QuantumCentral Limit Theorem. This issue was partially solved in the late 1980’sby the introduction of the QFA. In the present poster we demonstrate howthe mathematical concept of the QFA is also very useful in Quantum In-formation Theory, in particular in the description of Quantum Memories(atomic ensembles that can store the state of a light field). We also showhow the Tensor Algebra Formalism (borrowed from Constructive QuantumField Theory) provides a natural description of the QFA. With this flexibleformalism we could not only attack new problems about Quantum Memories,but could also prove new types of Quantum Central Limit Theorems.


PARTICIPANTS

Abdelhamid Mohammed Djaouti (Université d’ Oran), [email protected] Miguel (Instituto Superior Técnico Lisboa), [email protected] Cristina (University of L’Aquila), [email protected]ía Pedro (UPV/EHU), [email protected] Martínez Elena (Universidad Politécnica de Valencia), [email protected] David (University of Alberta), [email protected]é Carlos (University of Lisbon), [email protected] Jone (UPV/EHU), [email protected] Felipe (BCAM), [email protected] Naiara (UPV/EHU), [email protected]á Paz (Universitat Politècnica de València), [email protected]á Milagros (Universitat Politècnica de València), [email protected] Guillaume (Université de Lyon), [email protected] M. Vittoria (Technische Universität München - BCAM), [email protected]ó Juan Antonio (Universidad Politécnica de Madrid), [email protected] Laurent (University of Göttingen), [email protected] Ibai (University of Florida), [email protected] Jesús (Universidad de Zaragoza), [email protected] Giovanni (UCM-ICMAT), [email protected]án Antonio (Universitat Jaume I), [email protected]és Julio (Universidad de Zaragoza), [email protected] Mariagrazia (Università di Milano), [email protected]ël Virginie (IRMAR CNRS), [email protected] Carles (Universitat Autònoma de Barcelona), [email protected] Andrea (Universidad Complutense de Madrid), [email protected] Antonio (Universidad de Valladolid), [email protected] Pierre-Emmanuel (Université Catholique de Louvain), [email protected] Cristian (BCAM), [email protected] Leonardo (ICMAT-CSIC), [email protected] José Manuel (Universidad Autónoma de Madrid) [email protected] Thomas (Universidad Complutense de Madrid), [email protected]órdoba Diego (ICMat-CSIC), [email protected] Toby (Universidad Complutense de Madrid), [email protected] Carlota (ICMAT), [email protected] la Hoz Méndez Francisco (UPV/EHU), [email protected] Prada María Ángeles (UPV/EHU), [email protected] Sophie (Imperial College London), [email protected] Silvio (Università di Firenze), [email protected] Javier (UPV/EHU), [email protected] Luis (UPV/EHU), [email protected] Miguel (UPV/EHU), [email protected] María J. (CNRS University Paris-Dauphine), [email protected] Luca (UPV/EHU), [email protected] María José (Universidad Politécnica de Valencia), [email protected] Rui Loja (Instituto Superior Técnico), [email protected]ández Marisa (UPV/EHU), [email protected]ández Carlos (UNED - UCM), [email protected]


Fernández Alcober Gustavo A. (UPV/EHU), [email protected] Anna (Università di Torino), [email protected] Tobias (Institut de Ciéncies Fotóniques), [email protected]ía Andoni (University of Helsinki), [email protected]ía-López Jesús (Universidad Politécnica de Madrid), [email protected]ómez Ayala Eugenio Jesús (UPV/EHU), [email protected]ómez Cubillo Fernando (University of Valladolid), [email protected] Rafael (CSIC), [email protected] Olivier (Université Paris-Est Marne La Vallée), [email protected] García Javier(UPV/EHU), [email protected]ñiguez Ainhoa (University of Oxford), [email protected] Zapirain Andrei (Universidad Autónoma de Madrid), [email protected] Rehan Ahmad (Universidad Autónoma de Madrid), [email protected]énez Gómez Carlos Hugo (Universidad de Sevilla), [email protected]énez Vargas Antonio (Universidad de Almería), [email protected] Elamin (Universidad de Almería), [email protected] Nikolaos (BCAM), [email protected] William (University of Cologne), [email protected] Teppei (University of Oxford), [email protected] Pankaj (Perm State University), [email protected] Nave Gabriele (University of Illinois Urbana-Champaign), [email protected] David (ENS Paris), [email protected] Leire (UPV/EHU), [email protected] Mathieu (CNRS University of Cergy-Pontoise), [email protected] Mikel (UPV/EHU), [email protected] Andrea (Università di Cagliari), [email protected]ópez Sánchez Luis Daniel (CSIC - ICMAT), [email protected] Qi (BCAM), [email protected] Angelo (Università di Pisa), [email protected] Víctor (UPV/EHU), [email protected] Iraide (UPV/EHU), [email protected]ínez Luis (UPV/EHU), [email protected]ínez-Pastor Ana (Universidad Politécnica Valencia), [email protected]ía José (UPV/EHU), [email protected] Eva (Universitat Politècnica de Catalunya), [email protected] Guillermo (Universidad Complutense de Madrid), [email protected]ó Quintana Alexander (University of Valencia), [email protected]ñoz Escolano José M. (Universidad de Zaragoza), [email protected] Magdalena (University of Copenhagen), [email protected] Yukihiko (BCAM), [email protected] Gabriel (University of Valencia ), [email protected] Ion (CNRS Université de Toulouse), [email protected] Cerdà Joaquim (University of Barcelona), [email protected] Emanuele (Università di Milano), [email protected] Sudip Kumar (Jadavpur University), [email protected] Dmitri (King’s College London), [email protected] Silvie (Blaise Pascal University), [email protected]érez Javier (UNED), [email protected]érez-García David (Universidad Complutense de Madrid), [email protected] Federica (UPV-EHU), [email protected]írez Maribel (Universidad de Almería), [email protected] Geoffrey (University of Aberdeen), [email protected] Rey Teresa (Universidad Complutense de Madrid), [email protected] Mikael (University of Copenhagen), [email protected] Raphaël (University of Oxford), [email protected]ánchez Sanz Julia (BCAM), [email protected] Gómez Josu (UPV/EHU), [email protected] Joseba (UPV/EHU), [email protected] Lucía (Universitat de València), [email protected]é Marta (European Mathematical Society), [email protected]


Schneider Csaba (University of Lisbon), [email protected] Dan (University of Oxford), [email protected] Pavel (University of Brasilia), [email protected] Romero (Universitat Politécnica de Catalunya), [email protected] Gediminas (Vilnius University), [email protected] Luis (Universidad de Zaragoza), [email protected] González Luis (UPV/EHU), [email protected] Rafael (Universidad de Sevilla), [email protected] Raquel (Centro Universitario de la Defensa de Zaragoza), [email protected] Ignacio (Universidad Complutense de Madrid), [email protected] Nicola (Università di Pisa), [email protected] (Universidad Autónoma de Madrid - ICMAT), [email protected] Zoltan (ISI), [email protected] Reizabal Amaia (UPV/EHU), [email protected]


Documents

EMS-RSME JOINT MATHEMATICAL WEEKEND · 2011-10-26 · PRESENTATION WelcometoBilbao. As part of the commemorations of its centennial, the Royal Spanish SocietyofMathematics(RSME)organizesjointlywiththeEuropeanMath