Brief on current research
Ayman Kachmar investigates questions at the interface of spectral theory, quantum mechanics and magnetic phase transitions, which share in common the Schroedinger operator with a magnetic field. Being hard to express in general, the eigenvalues of the Schroedinger operator are approximated in various relevant asymptotic regimes, which display a nice interaction between the geometry of the space and the regularity properties of the magnetic field under consideration.
As a visiting Professor at CAMS, from
July 1 to July 30, 2022, Ayman Kachmar will conduct research and deliver a series of lectures, while teaching a summer course at the AUB Mathematics department too.
The lectures will discuss various topics in real and functional analysis, counting eigenvalues, phase transitions in the presence of a magnetic field, and isoperimetric inequalities. They will be split into two parts, with the first intended exclusively to students and focusing on revisiting some classical problems in the field of mathematical analysis, while the second, intended to researchers and students, is devoted to topics related to the research of Ayman Kachmar.
During his stay at CAMS, Ayman Kachmar will have time to discuss with colleagues and students from the mathematics, physics, and engineering departments.
Independently, CAMS and the Department of Mathematics at AUB host a summer research camp for the mathematics students, who are encouraged to attend the lectures to be delivered by Ayman Kachmar. Students from the engineering and physics departments will be invited too.
Schedule of lectures
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Lecture 1: July 7, 2022 at 3pm - CAMS Seminar room 416 (addressed to students)
Title: From Pythagoras to Newton and beyond
Abstract: The proof of the celebrated Pythagorean theorem has some measure theoretic aspects, while the theorem leads to the observation of square roots which are generally irrational numbers. Approximating square roots is a special case of approximating the roots of a polynomial, where the celebrated Newton’s method provides an efficient algorithm, and in particular a sequence of rational numbers that approximates the square root of 2. Based on these grounds, I’ll speak about topics in modern real analysis, like density and duality.
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Lecture 2: July 13, 2022 at 3pm - CAMS Seminar room 416 (addressed to students)
Title: From Matrices to Operators
Abstract: The diagonalization of Hermitian matrices is a quiet classical theorem in linear algebra, which leads to the min-max variational expressions of the eigenvalues. When formulated in the language of operators, the outcome provides a smooth transition from linear algebra to functional analysis.
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Lecture 3: July 20, 2022 at 3pm - CAMS Seminar room 416 (addressed to students and researchers)
Title: Counting eigenvalues: an example of a Weyl law
Abstract: I’ll prove the celebrated Weyl theorem, on the number of eigenvalues for the Schroedinger operator, in a one dimensional setting, which will give me the occasion to explain the Dirichlet-Neumann bracketing technique. I’ll end up with two recent results related to my research. (I’ll use material from Lecture 2).
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Lecture 4: July 26, 2022 at 3pm - CAMS Seminar room 416 (adressed to students)
Title: Magnetic periodic conditions
Abstract: Many problems in geometry/physics lead to function spaces with periodic boundary condition. In the presence of a magnetic field, periodicity has to agree with ``gauge invariance’’, thereby leading to magnetic periodic conditions. I’ll explain this concept, and its relation to the Abrikosov energy. I’ll end up with examples illustrating how one can detect a phase transition through an accurate calculation of the energy of a system. (I’ll use material from Lecture 2).
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Seminar: July 27, 2022 at 3pm - College Hall, Auditorium B1 [Poster]
Title: A new magnetic isoperimetric inequality
Abstract: I’ll present a recent work I did with V. Lotoreichik which provides a rare example where one can prove the spectral isoperimetric inequality for the Laplaican with a magnetic field. (I’ll use the material discussed in Lectures 2).