Pédagogie > summer school
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Fri, 05 Jun 2020 10:36:07 GMTAlexander Gorodnik - Diophantine approximation and flows on homogeneous spaces (Part 2)The
fundamental problem in the theory of Diophantine approximation is to
understand how well points in the Euclidean space can be approximated by
rational vectors with given bounds on denominators. It turns out that
Diophantine properties of points can be encoded using flows on
homogeneous spaces, and in this course we explain how to use techniques
from the theory of dynamical systems to address some of questions in
Diophantine approximation. In particular, we give a dynamical proof of
Khinchin’s theorem and discuss Sprindzuk’s question regarding
Diophantine approximation with dependent quantities, which was solved
using non-divergence properties of unipotent flows. In conclusion we
explore the problem of Diophantine approximation on more general
algebraic varieties.
http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22922
oai:canal-u.fr:22922URL: http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22922 ]]>Alexander Gorodnik - Diophantine approximation and flows on homogeneous spaces (Part 3)The
fundamental problem in the theory of Diophantine approximation is to
understand how well points in the Euclidean space can be approximated by
rational vectors with given bounds on denominators. It turns out that
Diophantine properties of points can be encoded using flows on
homogeneous spaces, and in this course we explain how to use techniques
from the theory of dynamical systems to address some of questions in
Diophantine approximation. In particular, we give a dynamical proof of
Khinchin’s theorem and discuss Sprindzuk’s question regarding
Diophantine approximation with dependent quantities, which was solved
using non-divergence properties of unipotent flows. In conclusion we
explore the problem of Diophantine approximation on more general
algebraic varieties.
http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22926
oai:canal-u.fr:22926URL: http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22926 ]]>Andras Vasy - Microlocal analysis and wave propagation (Part 2)In these lectures I will explain the basics of microlocal analysis, emphasizing non elliptic problems, such as wave propagation, both on manifolds without boundary, and on manifolds with boundary. In the latter case there is no `standard' algebra of differential, or pseudodifferential, operators; I will discuss two important frameworks: Melrose's totally characteristic, or b, operators and scattering operators. Apart from the algebraic and mapping properties, I will discuss microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as normally hyperbolic trapping. The applications discussed will include Fredholm frameworks (which are thus global even for non elliptic problems!) for the Laplacian on asymptotically hyperbolic spaces and the wave operator on asymptotically de Sitter spaces, scattering theory for `scattering metrics' (such as the `large ends' of cones), wave propagation on asymptotically Minkowski spaces and generalizations (`Lorentzian scattering metrics') and on Kerr de Sitter type spaces. The lectures concentrate on linear PDE, but time permitting I will briefly discuss nonlinear versions. The lecture by the speaker in the final workshop will use these results to solve quasilinear wave equations globally, including describing the asymptotic behavior of solutions, on Kerr de Sitter spaces.
http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22537
oai:canal-u.fr:22537URL: http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22537 ]]>Andras Vasy - Microlocal analysis and wave propagation (Part 3)In these lectures I will explain the basics of microlocal analysis, emphasizing non elliptic problems, such as wave propagation, both on manifolds without boundary, and on manifolds with boundary. In the latter case there is no `standard' algebra of differential, or pseudodifferential, operators; I will discuss two important frameworks: Melrose's totally characteristic, or b, operators and scattering operators. Apart from the algebraic and mapping properties, I will discuss microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as normally hyperbolic trapping. The applications discussed will include Fredholm frameworks (which are thus global even for non elliptic problems!) for the Laplacian on asymptotically hyperbolic spaces and the wave operator on asymptotically de Sitter spaces, scattering theory for `scattering metrics' (such as the `large ends' of cones), wave propagation on asymptotically Minkowski spaces and generalizations (`Lorentzian scattering metrics') and on Kerr de Sitter type spaces. The lectures concentrate on linear PDE, but time permitting I will briefly discuss nonlinear versions. The lecture by the speaker in the final workshop will use these results to solve quasilinear wave equations globally, including describing the asymptotic behavior of solutions, on Kerr de Sitter spaces.
http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22543
oai:canal-u.fr:22543URL: http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22543 ]]>Andras Vasy - Microlocal analysis and wave propagation (Part 4)In these lectures I will explain the basics of microlocal analysis, emphasizing non elliptic problems, such as wave propagation, both on manifolds without boundary, and on manifolds with boundary. In the latter case there is no `standard' algebra of differential, or pseudodifferential, operators; I will discuss two important frameworks: Melrose's totally characteristic, or b, operators and scattering operators. Apart from the algebraic and mapping properties, I will discuss microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as normally hyperbolic trapping. The applications discussed will include Fredholm frameworks (which are thus global even for non elliptic problems!) for the Laplacian on asymptotically hyperbolic spaces and the wave operator on asymptotically de Sitter spaces, scattering theory for `scattering metrics' (such as the `large ends' of cones), wave propagation on asymptotically Minkowski spaces and generalizations (`Lorentzian scattering metrics') and on Kerr de Sitter type spaces. The lectures concentrate on linear PDE, but time permitting I will briefly discuss nonlinear versions. The lecture by the speaker in the final workshop will use these results to solve quasilinear wave equations globally, including describing the asymptotic behavior of solutions, on Kerr de Sitter spaces.
http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22547
oai:canal-u.fr:22547URL: http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22547 ]]>Andrew Lorent - The Aviles-Giga functional: past and presentindisponible
http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22291
oai:canal-u.fr:22291URL: http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22291 ]]>Bruno Lévy - A numerical algorithm for L2 semi-discrete optimal transport in 3Dindisponible
http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22281
oai:canal-u.fr:22281URL: http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22281 ]]>Camillo De Lellis - Center manifolds and regularity of area-minimizing currents (Part 1)A celebrated
theorem of Almgren shows that every integer rectifiable current which
minimizes (locally) the area is a smooth submanifold except for a
singular set of codimension at most 2. Almgren’s theorem is sharp in
codimension higher than 1, because holomorphic subvarieties of Cn are
area-minimizing. In fact the typical singularity of a 2-dimensional
area-minimizing current is modelled by branch points of holomorphic
curves. These singularities are rather difficult to analyze because they
might be very high order phenomena.
http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22249
oai:canal-u.fr:22249URL: http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22249 ]]>Camillo De Lellis - Center manifolds and regularity of area-minimizing currents (Part 3)A celebrated
theorem of Almgren shows that every integer rectifiable current which
minimizes (locally) the area is a smooth submanifold except for a
singular set of codimension at most 2. Almgren’s theorem is sharp in
codimension higher than 1, because holomorphic subvarieties of Cn are
area-minimizing. In fact the typical singularity of a 2-dimensional
area-minimizing current is modelled by branch points of holomorphic
curves. These singularities are rather difficult to analyze because they
might be very high order phenomena.
http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22145
oai:canal-u.fr:22145URL: http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22145 ]]>Camillo De Lellis - Center manifolds and regularity of area-minimizing currents (Part 4)A celebrated
theorem of Almgren shows that every integer rectifiable current which
minimizes (locally) the area is a smooth submanifold except for a
singular set of codimension at most 2. Almgren’s theorem is sharp in
codimension higher than 1, because holomorphic subvarieties of Cn are
area-minimizing. In fact the typical singularity of a 2-dimensional
area-minimizing current is modelled by branch points of holomorphic
curves. These singularities are rather difficult to analyze because they
might be very high order phenomena.
http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22153
oai:canal-u.fr:22153URL: http://indexation.univ-fcomte.fr/ori-oai-search/notice/view/oai%253Acanal-u.fr%253A22153 ]]>