BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//CERN//INDICO//EN
BEGIN:VEVENT
SUMMARY:Loops and Bubbles (4/4)
DTSTART;VALUE=DATE-TIME:20210421T150000Z
DTEND;VALUE=DATE-TIME:20210421T163000Z
DTSTAMP;VALUE=DATE-TIME:20230128T142048Z
UID:indico-contribution-278-133@indico.eimi.ru
DESCRIPTION:Speakers: Roberta Musina ()\nAn important class of problems in
Riemannian geometry can be stated as follows: given a smooth and orientab
le Riemanninan manifold M\, find an hypersphere U in M having prescribed m
ean curvature K at each point. We will be mainly focused on the case when
the target M is the Euclidean plane and the unknown U is a planar loop. Be
sides its geometrical interpretation\, this (apparently) simple problem na
turally arises in the study of the planar motion of an electrified particl
e that experiences a Lorentz force produced by a magnetostatic field. It c
an be regarded as a model for a more general question raised by V.I. Arnol
d in [Uspekhi Mat. Nauk 1986]. We will first discuss some Alexandrov-type
uniqueness results in case the prescribed curvature is a positive constant
or\, more generally\, a positive and radially non increasing function. To
obtain existence results we will choose a parametric point of view\, whic
h will lead us to study certain variational\, noncompact systems of second
order ODE's for functions on the circle. This will give us the opportunit
y to briefly introduce some variational (mountain pass lemma) and nonvaria
tional (Lyapunov-Schmidt dimension reduction)basic techniques. In the last
part of the course will overview some recent results and open problems in
case the target space M is the hyperbolic plane\, or the Euclidean/hyperb
olic 3-dimensional space.\n\nProgram:\n\n The curvature of planar curve
s. Planar loops and physical interpretation: a related ODE system and Arno
l'd problem. Planar loops of positive curvature. Homework: the curvature o
f circles and ellipses\, radially symmetric prescribed curvatures.\n Th
e four vertex theorem (Osserman's proof).\n Uniqueness results: Alexand
rov (1956)\, Aeppli (1960) and more (2011).\n The (pseudo)-length funct
ional and the weighted\, signed area functional.\n The variational appr
oach. A quick introduction to variational methods.\n Palais-Smale condi
tion\, the Mountain Pass Lemma\, saddle points.\n A non-variational app
roach: the Lyapunov-Schmidt dimension reduction.\n Loops of prescribed
curvature in the hyperbolic plane. Bubbles in the Euclidean and in hyperbo
lic spaces: recent results and open problems.\n\nhttps://indico.eimi.ru/ev
ent/278/contributions/133/
LOCATION:Zoom 443 726 1792
URL:https://indico.eimi.ru/event/278/contributions/133/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Loops and Bubbles (3/4)
DTSTART;VALUE=DATE-TIME:20210420T150000Z
DTEND;VALUE=DATE-TIME:20210420T163000Z
DTSTAMP;VALUE=DATE-TIME:20230128T142048Z
UID:indico-contribution-278-132@indico.eimi.ru
DESCRIPTION:Speakers: Roberta Musina ()\nAn important class of problems in
Riemannian geometry can be stated as follows: given a smooth and orientab
le Riemanninan manifold M\, find an hypersphere U in M having prescribed m
ean curvature K at each point. We will be mainly focused on the case when
the target M is the Euclidean plane and the unknown U is a planar loop. Be
sides its geometrical interpretation\, this (apparently) simple problem na
turally arises in the study of the planar motion of an electrified particl
e that experiences a Lorentz force produced by a magnetostatic field. It c
an be regarded as a model for a more general question raised by V.I. Arnol
d in [Uspekhi Mat. Nauk 1986]. We will first discuss some Alexandrov-type
uniqueness results in case the prescribed curvature is a positive constant
or\, more generally\, a positive and radially non increasing function. To
obtain existence results we will choose a parametric point of view\, whic
h will lead us to study certain variational\, noncompact systems of second
order ODE's for functions on the circle. This will give us the opportunit
y to briefly introduce some variational (mountain pass lemma) and nonvaria
tional (Lyapunov-Schmidt dimension reduction)basic techniques. In the last
part of the course will overview some recent results and open problems in
case the target space M is the hyperbolic plane\, or the Euclidean/hyperb
olic 3-dimensional space.\n\nProgram:\n\n The curvature of planar curve
s. Planar loops and physical interpretation: a related ODE system and Arno
l'd problem. Planar loops of positive curvature. Homework: the curvature o
f circles and ellipses\, radially symmetric prescribed curvatures.\n Th
e four vertex theorem (Osserman's proof).\n Uniqueness results: Alexand
rov (1956)\, Aeppli (1960) and more (2011).\n The (pseudo)-length funct
ional and the weighted\, signed area functional.\n The variational appr
oach. A quick introduction to variational methods.\n Palais-Smale condi
tion\, the Mountain Pass Lemma\, saddle points.\n A non-variational app
roach: the Lyapunov-Schmidt dimension reduction.\n Loops of prescribed
curvature in the hyperbolic plane. Bubbles in the Euclidean and in hyperbo
lic spaces: recent results and open problems.\n\nhttps://indico.eimi.ru/ev
ent/278/contributions/132/
LOCATION:Zoom 443 726 1792
URL:https://indico.eimi.ru/event/278/contributions/132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Loops and Bubbles (2/4)
DTSTART;VALUE=DATE-TIME:20210414T150000Z
DTEND;VALUE=DATE-TIME:20210414T163000Z
DTSTAMP;VALUE=DATE-TIME:20230128T142048Z
UID:indico-contribution-278-131@indico.eimi.ru
DESCRIPTION:Speakers: Roberta Musina ()\nAn important class of problems in
Riemannian geometry can be stated as follows: given a smooth and orientab
le Riemanninan manifold M\, find an hypersphere U in M having prescribed m
ean curvature K at each point. We will be mainly focused on the case when
the target M is the Euclidean plane and the unknown U is a planar loop. Be
sides its geometrical interpretation\, this (apparently) simple problem na
turally arises in the study of the planar motion of an electrified particl
e that experiences a Lorentz force produced by a magnetostatic field. It c
an be regarded as a model for a more general question raised by V.I. Arnol
d in [Uspekhi Mat. Nauk 1986]. We will first discuss some Alexandrov-type
uniqueness results in case the prescribed curvature is a positive constant
or\, more generally\, a positive and radially non increasing function. To
obtain existence results we will choose a parametric point of view\, whic
h will lead us to study certain variational\, noncompact systems of second
order ODE's for functions on the circle. This will give us the opportunit
y to briefly introduce some variational (mountain pass lemma) and nonvaria
tional (Lyapunov-Schmidt dimension reduction)basic techniques. In the last
part of the course will overview some recent results and open problems in
case the target space M is the hyperbolic plane\, or the Euclidean/hyperb
olic 3-dimensional space.\n\nProgram:\n\n The curvature of planar curve
s. Planar loops and physical interpretation: a related ODE system and Arno
l'd problem. Planar loops of positive curvature. Homework: the curvature o
f circles and ellipses\, radially symmetric prescribed curvatures.\n Th
e four vertex theorem (Osserman's proof).\n Uniqueness results: Alexand
rov (1956)\, Aeppli (1960) and more (2011).\n The (pseudo)-length funct
ional and the weighted\, signed area functional.\n The variational appr
oach. A quick introduction to variational methods.\n Palais-Smale condi
tion\, the Mountain Pass Lemma\, saddle points.\n A non-variational app
roach: the Lyapunov-Schmidt dimension reduction.\n Loops of prescribed
curvature in the hyperbolic plane. Bubbles in the Euclidean and in hyperbo
lic spaces: recent results and open problems.\n\nhttps://indico.eimi.ru/ev
ent/278/contributions/131/
LOCATION:Zoom 443 726 1792
URL:https://indico.eimi.ru/event/278/contributions/131/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Loops and Bubbles (1/4)
DTSTART;VALUE=DATE-TIME:20210413T150000Z
DTEND;VALUE=DATE-TIME:20210413T163000Z
DTSTAMP;VALUE=DATE-TIME:20230128T142048Z
UID:indico-contribution-278-130@indico.eimi.ru
DESCRIPTION:Speakers: Roberta Musina ()\nAn important class of problems in
Riemannian geometry can be stated as follows: given a smooth and orientab
le Riemanninan manifold M\, find an hypersphere U in M having prescribed m
ean curvature K at each point. We will be mainly focused on the case when
the target M is the Euclidean plane and the unknown U is a planar loop. Be
sides its geometrical interpretation\, this (apparently) simple problem na
turally arises in the study of the planar motion of an electrified particl
e that experiences a Lorentz force produced by a magnetostatic field. It c
an be regarded as a model for a more general question raised by V.I. Arnol
d in [Uspekhi Mat. Nauk 1986]. We will first discuss some Alexandrov-type
uniqueness results in case the prescribed curvature is a positive constant
or\, more generally\, a positive and radially non increasing function. To
obtain existence results we will choose a parametric point of view\, whic
h will lead us to study certain variational\, noncompact systems of second
order ODE's for functions on the circle. This will give us the opportunit
y to briefly introduce some variational (mountain pass lemma) and nonvaria
tional (Lyapunov-Schmidt dimension reduction)basic techniques. In the last
part of the course will overview some recent results and open problems in
case the target space M is the hyperbolic plane\, or the Euclidean/hyperb
olic 3-dimensional space.\n\nProgram:\n\n The curvature of planar curve
s. Planar loops and physical interpretation: a related ODE system and Arno
l'd problem. Planar loops of positive curvature. Homework: the curvature o
f circles and ellipses\, radially symmetric prescribed curvatures.\n Th
e four vertex theorem (Osserman's proof).\n Uniqueness results: Alexand
rov (1956)\, Aeppli (1960) and more (2011).\n The (pseudo)-length funct
ional and the weighted\, signed area functional.\n The variational appr
oach. A quick introduction to variational methods.\n Palais-Smale condi
tion\, the Mountain Pass Lemma\, saddle points.\n A non-variational app
roach: the Lyapunov-Schmidt dimension reduction.\n Loops of prescribed
curvature in the hyperbolic plane. Bubbles in the Euclidean and in hyperbo
lic spaces: recent results and open problems.\n\nhttps://indico.eimi.ru/ev
ent/278/contributions/130/
LOCATION:Zoom 443 726 1792
URL:https://indico.eimi.ru/event/278/contributions/130/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Geometric flows of networks (4/4)
DTSTART;VALUE=DATE-TIME:20210401T150000Z
DTEND;VALUE=DATE-TIME:20210401T163000Z
DTSTAMP;VALUE=DATE-TIME:20230128T142048Z
UID:indico-contribution-278-129@indico.eimi.ru
DESCRIPTION:Speakers: Alessandra Pluda (Università di Pisa)\, Matteo Nova
ga (Università di Pisa)\nhttps://indico.eimi.ru/event/278/contributions/1
29/
LOCATION:Zoom 443 726 1792
URL:https://indico.eimi.ru/event/278/contributions/129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Geometric flows of networks (3/4)
DTSTART;VALUE=DATE-TIME:20210330T150000Z
DTEND;VALUE=DATE-TIME:20210330T163000Z
DTSTAMP;VALUE=DATE-TIME:20230128T142048Z
UID:indico-contribution-278-128@indico.eimi.ru
DESCRIPTION:Speakers: Alessandra Pluda (Università di Pisa)\, Matteo Nova
ga (Università di Pisa)\nhttps://indico.eimi.ru/event/278/contributions/1
28/
LOCATION:Zoom 443 726 1792
URL:https://indico.eimi.ru/event/278/contributions/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Geometric flows of networks (2/4)
DTSTART;VALUE=DATE-TIME:20210325T150000Z
DTEND;VALUE=DATE-TIME:20210325T163000Z
DTSTAMP;VALUE=DATE-TIME:20230128T142048Z
UID:indico-contribution-278-127@indico.eimi.ru
DESCRIPTION:Speakers: Alessandra Pluda (Università di Pisa)\, Matteo Nova
ga (Università di Pisa)\nhttps://indico.eimi.ru/event/278/contributions/1
27/
LOCATION:Zoom 443 726 1792
URL:https://indico.eimi.ru/event/278/contributions/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Geometric flows of networks (1/4)
DTSTART;VALUE=DATE-TIME:20210323T150000Z
DTEND;VALUE=DATE-TIME:20210323T163000Z
DTSTAMP;VALUE=DATE-TIME:20230128T142048Z
UID:indico-contribution-278-126@indico.eimi.ru
DESCRIPTION:Speakers: Alessandra Pluda (Università di Pisa)\, Matteo Nova
ga (Università di Pisa)\nhttps://indico.eimi.ru/event/278/contributions/1
26/
LOCATION:Zoom 443 726 1792
URL:https://indico.eimi.ru/event/278/contributions/126/
END:VEVENT
END:VCALENDAR