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19.9.2018 14:55

MATS4400 Density Functional Theory for strong correlated systems and Optimal Transport, 5 op [kotisivu]

Matematiikan ja tilastotieteen laitos

Tiedotus:
Kurssin ilmoittautumisaika päättyi 15.1.18 klo 23:59.

Yleisiä tietoja

Kotisivu: http://users.jyu.fi/~joaugero/teaching.html
Alkaa - päättyy: 8.1.18 - 14.3.18
Ilmoittautumisaika: 1.10.17 klo 0:00 - 15.1.18 klo 23:59
Opettaja(t): Tutkijatohtori Augusto Gerolin Gavea (augusto.gerolin@jyu.fi)
Laajuus: 5 op
Kielet: opetuskielet: English; suorituskielet: English
Ilmoittautuneita: 7
Maksimi osallistujamäärä: 25
Sopii vielä: 18
Organisaatiot:Matematiikan ja tilastotieteen laitos (MATHS), Matematiikka (MAT)
Ajankohtaista:

The Strictly-Correlated-Electrons (SCE) density functional theory (SCE DFT) approach, originally proposed by Michael Seidl [Phys. Rev. A 60, 4387 (1999)], could pave the way to a formulation of density functional theory, alternatively to widely used Kohn-Sham DFT, especially aimed at the study of strongly-correlated systems. The course aim to present that theory focusing on some mathematical aspects of SCE DFT relying to optimal transportation theory with Coulomb costs. We will present both the physical intuition of the problem and rigorous mathematical results (guided by physical intuition) on the interface between Density Functional Theory and Optimal Transport.

Optimal Transport Theory is a very active field of mathematics with many applications in geometry, PDEs, meterology, probability, statistics, immaging (see the books written by the Fields medalists Cédric Villani). In this course, we will revisited that theory by considering Coulomb cost functions motivated by SCE DFT.

Sisältö:

1st part (M. Seidl, ∼ 3 weeks): Ground state problem for Many-body Schödinger Equation; A brief introduction of Density Functional Theory: Hohenberg-Kohn functional, Kohm Sham Equations; Density Functional Theory for strongly correlated systems (adiabatic limit). Co-motion functions for spherically symmetric systems.

2nd part (A. Gerolin, ∼ 5 weeks): Duality between the space of finite measures and continuous bounded functions; Monge and Monge-Kantorovich problems, existence of optimal plans, Kantorovich duality and existence of Kantorovich potentials. Monge problem (two marginal case) and Wasserstein distances. Multi-marginal Optimal Transport for the attractive harmonic case (existence of Monge minimizers). Study the two electrons (marginals) case for Coulomb costs and the N electrons (multi-marginal) case for radially symmetric densities.

Depending of the interests of the students the following topics (not limited of) can be covered as well: Regularity of Kantorovich potentials for Coulomb costs. Entropic Transport. Optimal Transport for Repulsive harmonic costs. Semi-classical limit of the Hohenberg-Kohn functional.

Osaamistavoitteet:

From the mathematical point of view, we expect at the end of the courses the students will be familiarized with and combine techniques on Calculus of Variations, Functional Analysis, Convex Analysis and Measure Theory. The course has a good environment to introduce master students to applied analysis. In particular, in putting in mathematical grounds a problem that comes from physics. Finally, we hope to present some numerical aspects and point out open problems in the field.

Esitiedot:

This is a master level course in mathematics. No background in physics will be necessary to follow this course.

Suoritustavat:

There will be around three homework assignments and a final exam. Grades will be based on the homeworks (1/3) and the final exam (2/3). Collaboration between student are encouraged during homework (but not during the exams). If you collaborate with other students during a homework you must specify it. Moreover, each student must submit a homework solution individually.

Aikataulu:

8 weeks, starting in January 2018.

Kirjallisuus:

Lecture notes will be posted online or send by e-mail every week. Other references will be provided
during the lectures.

Yhteystiedot:

Teachers:

Michael Seidl (Physics, VU-Amsterdam) and Augusto Gerolin (Mathematics, University of Jyväskylä).
E-mail: michael.seidl@physik.uni-regensburg.de / augusto.gerolin@jyu.fi.

[Tiivistä opetusryhmien aika- ja paikkatiedot]

Luento [ryhmien tarkat tiedot ja ilmoittautuminen]

Luento 1 [ryhmätiedot ja ilmoittautuminen]; ilmoittautuneita 7, maksimi 25
ilm.aika: 1.10.2017 00:00 - 15.1.2018 23:59
 PaikkaViikkoPäiväPvmKloOhjaajaLisätietojaURITapahtuman tiedot
1MaD 3552ma8.1.201812:15-14:00-Tapahtuman tiedot
2MaD 3553ma15.1.201812:15-14:00-Tapahtuman tiedot
3MaD 3554ma22.1.201812:15-14:00-Tapahtuman tiedot
4MaD 3555ma29.1.201812:15-14:00Gerolin GaveaTapahtuman tiedot
5MaD 3556ma5.2.201812:15-14:00Gerolin GaveaTapahtuman tiedot
6MaD 3557ma12.2.201812:15-14:00Gerolin GaveaTapahtuman tiedot
7MaD 3558ma19.2.201812:15-14:00Gerolin GaveaTapahtuman tiedot
8MaD 3559ma26.2.201812:15-14:00Gerolin GaveaTapahtuman tiedot
Luento 2 [ryhmätiedot ja ilmoittautuminen]; ilmoittautuneita 7, maksimi 25
ilm.aika: 1.10.2017 00:00 - 15.1.2018 23:59
 PaikkaViikkoPäiväPvmKloOhjaajaLisätietojaURITapahtuman tiedot
1MaD 3553ke17.1.201816:15-18:00-Tapahtuman tiedot
2MaD 3554ke24.1.201816:15-18:00-Tapahtuman tiedot
Luento 3 [ryhmätiedot ja ilmoittautuminen]; ilmoittautuneita 7, maksimi 25
ilm.aika: 1.10.2017 00:00 - 15.1.2018 23:59
 PaikkaViikkoPäiväPvmKloOhjaajaLisätietojaURITapahtuman tiedot
1MaD 3552to11.1.201814:15-16:00-Tapahtuman tiedot
2MaD 3553to18.1.201814:15-16:00-Tapahtuman tiedot
3MaD 3554to25.1.201814:15-16:00-Tapahtuman tiedot
4MaD 3555to1.2.201814:15-16:00Gerolin GaveaTapahtuman tiedot
5MaD 3556to8.2.201814:15-16:00Gerolin GaveaTapahtuman tiedot
6MaD 3557to15.2.201814:15-16:00Gerolin GaveaTapahtuman tiedot
7MaD 3558to22.2.201814:15-16:00Gerolin GaveaTapahtuman tiedot
8MaD 3559to1.3.201814:15-16:00Gerolin GaveaTapahtuman tiedot

Tentti [ryhmien tarkat tiedot ja ilmoittautuminen]

Tentti 14.3.2018 [tenttiin ilmoittautuminen]; ilmoittautuneita 2, maksimi 200
ilm.aika: 15.12.2017 06:30 - 8.3.2018 16:00
 PaikkaViikkoPäiväPvmKloOhjaajaLisätietojaURITapahtuman tiedot
1-11ke14.3.201808:00-12:00Gerolin GaveaTapahtuman tiedot