Spectral Theory, Semi Classical Analysis and Condensed Matter Physics
Mathematics and physics have, for centuries now, been in a symbiotic relationship which resulted in some of the most remarkable edifices of the human spirit grappling, as it naturally does, with a range of physical phenomena (itself included!). Under the label of mathematical physics, the study of challenging problems at the foundations of theoretical physics has exploited, informed and extended formal mathematical developments, while at the same time allowing for predictions that are driving cutting edge experimental research. Einstein's theory of general relativity comes to mind, having seen the light more than a century ago, then spawned a series of predictions, and prodigious mathematical elaborations, with rigorous singularity theorems and the observational “confirmation" of predicted singularities (aka black holes) being honored by this year's Nobel Prize in Physics.
The present thematic program is both a continuation and an extension of CAMS long-standing involvement with mathematical physics. It focuses on state of the art mathematical machinery [mainly in spectral theory, and semi-classical (microlocal) analysis, but also calculus of variations, and geometric measure theory] devised to handle thorny problems in quantum mechanics, quantum field theory and exotic phases of solids and liquids (e.g. superconductivity and superfluidity), then others dealing with nonlinear dispersive waves, and turbulence [Important to note here that pioneering theoretical physics work (on superconductors and superfluids), which has fueled rigorous mathematical research explored in our program, was recognized by the 2003 Nobel Prize awarded to Abrikosov, Ginzburg and Legett].
A series of mini-courses (all delivered online) prepares the ground for an international conference on the “Mathematics of Condensed Matter and Beyond" [in February, 2021], and continues with related lectures on semi-classical analysis and spectral asymptotics soon after. Seminars on a broader spectrum of mathematical physics concerns will punctuate this progression. The program aims to enrich mathematical research and collaboration, feed the intellectual drive of gifted ambitious students of mathematics and mathematical physics, feature the work of a group of our colleagues and their collaborators, and encourage the involvement of physicists and applied mathematicians with program themes and activities.
Organized by: Wafaa Assaad (Laboratory of Mathematics, Doctoral School of Science and Technology (DSST), Lebanese University) & Ayman Kachmar (Department of Mathematics, Faculty of Science, Lebanese University) in close collaboration with CAMS and DSST-LU.
Program Activities:
Mini-courses
In addition to regular seminars and workshops, CAMS, in partnership with the mathematics laboratory of the Lebanese University doctoral school for science and technology (DSST), hosts a series of five mini-courses within the mathematical physics thematic semester. The first two courses, on the magnetic Laplace operator, serve as an introduction to the spectral asymptotics of differential operators of mathematical physics and are related to the semi-classical analysis field, which has important applications in PDE, quantum mechanics physics and condensed matter.
A third course, on the discrete Laplacian, serves as an introduction to the recently growing field of discrete geometry. It introduces a discrete analogue of harmonic morphisms between Riemannian manifolds. The audience should be familiar with some basic definitions of graph theory, have a little background of differential geometry, as tangent vectors, differential forms and harmonic functions.
Another course, on spectral measures, deepens the theoretical background on the spectrum of unbounded operators along with handful applications and some original proofs on the subject.
The last two courses in the series, on semi-classical analysis, feature plenty of examples on the splitting of the eigenvalues in symmetric settings, starting from early breakthrough results in the 80's up to recent results induced by the domain's geometry, and provide an introduction to the domain of microlocal analysis along with the necessary background from symplectic geometric and pseudo-differential operators.
The prerequisites are elementary functional analysis: Hilbert spaces, bounded and unbounded operators, Fourier transform, Lebesgue spaces, and basic tools to handle the Lebesgue integral.
The audience can be quite broad: graduate students, young and senior researchers in mathematics, mathematical physics, and theoretical physics.
Students at the Lebanese University doctoral school DSST who attend the mini-courses are eligible of ECTS credits.
Graduate students will be elligible to a certificate, issued by CAMS, in recognition of attendance of program courses.
November 2020-March 2021