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pormedio

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dela

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caré

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Todaslas

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onferencias,situadaen

laplanta

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atemáticas,Facultad

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iencias.

JUE

VE

S9

17:00A

hmad

ElSoufi

(Université

François-Rabelais,

Tours)

On

thegeom

etryofm

anifoldsw

hoseLaplace-

Beltram

ioperatoradm

itslarge

eigenvalues

18:00Joaquim

Martín

(Universidad

Autónom

ade

Barcelona)

Pointwise

symm

etrizationinequalities

forSo-

bolevfunctions

onP

robabilityM

etricSpaces

VIE

RN

ES

10

10:00M

iguelSánchez(U

niversidadde

Granada)

Lorentzianm

anifoldsisom

etricallyem

beddi-ble

inLorentz-M

inkowski

11:00C

afé.

11:30M

arkH

askins(Im

perialCollege

London)

Exceptionalholonom

yand

calibratedsubm

a-nifolds.

14:00C

omida.

Resúm

enes

On

thegeom

etryof

manifolds

whose

Laplace-

Beltram

ioperatoradm

itslargeeigenvalues

Ahm

adE

lSoufiT

hesequence

ofeigenvaluesoftheD

irichletLaplacian

ona

boundedE

uclideandom

ainsatisfies

severalrestrictivecon-

ditionssuch

as:Faber-K

rahnisoperim

etricinequality,that

isthe

principaleigenvalueis

boundedabove

interm

softhe

volume

ofthe

domain,Payne-Pólya-W

einbergertype

uni-versalinequalities,thatis

thek-th

eigenvalueis

controlledin

terms

ofthek−

1previous

ones,etc.

The

situationchanges

completely

assoon

asE

uclideando-

mains

arereplaced

bycom

pactmanifolds.Forexam

ple,ac-cording

toresultsby

Colin

deV

erdièreand

Lohkam

p,givenany

compact

manifold

Mof

dimension

n≥

3,it

ispos-

sibleto

prescribearbitrarily

andsim

ultaneously,through

thechoice

ofa

suitableR

iemannian

metric

onM

,afinite

partofthe

spectrumof

theL

aplacian,thevolum

eand

theintegral

ofthe

scalarcurvature.

Hence,

Faber-Krahn

andPayne-Pólya-W

einbergerinequalities

haveno

analoguein

thiscontext.

Inthistalk,w

ew

illdiscusstheeffectofthe

geometry

onthe

eigenvalues.We

willtry

tounderstand

whatkind

ofgeome-

tricsituations

leadto

largeeigenvalues

fortheL

aplacianon

manifolds

offixed

volume,and

whatdoes

sucha

Riem

an-nian

manifold

looklike

oncerealized

asa

submanifold

ofaE

uclideanspace.

On

theother

hand,w

eshow

thatw

henthe

Laplacian

ispenalized

bythe

squarednorm

ofthe

mean

curvature,then

we

obtaina

Schrödingertype

operatorw

hosespectral

behavioris

similar

tothat

ofthe

Dirichlet

Laplacian

onE

uclideandom

ains.

Pointwise

symm

etrizationinequalities

forSobolev

functionsonProbability

Metric

SpacesJoaquim

Martín

Toform

ulateSobolev

inequalitiesone

needsto

answer

questionslike:w

hatisthe

roleof

dimension?

Whatnorm

sare

appropriateto

measure

theintegrability

gains?Just

tonam

ea

few...For

example,

incontrast

tothe

Euclidean

case,the

integrabilitygains

inG

aussianm

easureare

logarithmic

butdim

ensionfree

(logSobolev

inequalities).So

itis

easyto

understandthe

difficultiesto

derivea

generaltheory.I

will

discusssom

enew

methods

toprove

generalSobolev

inequalitiesthat

unifythe

Euclidean

andthe

Gaussian

cases,as

well

asseveral

important

model

manifolds.

Lorentzian

manifolds

isometrically

embeddible

inL

orentz-Minkow

ski.M

iguelSánchez

Our

aimis

togive

asim

plecharacterization

ofthe

classof

Lorentzian

manifolds

which

canbe

isometrically

embedded

inL

orentz-Minkow

skiLN

forsom

elarge

N(in

thespirit

ofclassical

Nash

theorem)

and,then,

toshow

thatthisclass

includesthe

mostrelevanttype

ofrelativisticspacetim

es,i.e.,

theglobally

hyperbolicones.

This

lastresultw

asclaim

edby

CJS

Clarke

(1970),buthisproofw

asaffected

bythe

so-calledfolk

problems

ofsmoothability

inL

orentzianG

eometry.

These

problems

will

bespecially

discussed.

The

talkis

basedin

ajoint

work

with

O.

Müller

(ar-X

iv:0812.4439v4).

Exceptionalholonom

yand

calibratedsubm

anifolds.M

arkH

askins

We

givean

introductionto

recentdevelopments

inthe

geo-m

etryof

compact

manifolds

with

exceptionalholonom

y,focusing

onrecent

work

with

Corti,N

ordstromand

Paci-ni;

we

provethe

existenceof

many

compact

7-manifolds

with

holonomy

G2

thatcontain

rigidassociative

subma-

nifolds.T

hem

ainingredients

inthe

proofare:

anappro-

priatenoncom

pactversionofthe

Calabiconjecture,gluing

methods

anda

certainclass

ofcom

plexprojective

3-folds(w

eakFano

3-folds).