Short CV

Employment:

September 1, 2004 - June 6, 2007: Ph.D. in Mathematics, Université Paris XI, France. Advisor: Bernard Helffer. 

September 1, 2007 - August 31, 2008: ATER (Teaching and research associate),  Université Paris XI, France. 

September 1, 2008 - August 31, 2010: Post-doc, University of Århus, Denmark. Advisor: Søren Fournais. 

September 1, 2010 - August 31, 2014: Associate Professor, Lebanese University, Lebanon.

September 1, 2014 - August 31, 2023: Professor, Lebanese University, Lebanon.

September 1, 2023-: Professor, The Chinese University of Hong Kong, Shenzhen.

Service:

October 1, 2017 - September 30, 2019: Chairperson of the mathematics department, Lebanese University, Lebanon.

February 2009: Search committee, Lebanese University, Lebanon.

October 1, 2020 - August 31, 2023: Steering committee, Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, Lebanon. 

Events:

Organizer (with W. Assaad) of the Thematic Program in Mathematical Physics, American University of Beirut, Lebanon. Nov 2020 - March 2021.

 Organizer (with M. Correggi) Mathematics of Condensed Matter and Beyond (online conference), American University of Beirut. Feb 22-25, 2021.

Ph.D. students:

R. Fahs, University of Angers (defended July 2023). Co-advised with N. Raymond .

W. Assad,  Lund University (defended September 2019).  Co-advised with  M. P.-Sundqvist. 

K. Attar, Lebanese University and Université Paris XI (defended June 2015). Co-advised with B. Helffer. 

M. Nasrallah. Lebanese University and University of Århus (defended November 2013). Co-advised with Søren Fournais.

Selected works

Magnetic Laplacian:

(Pure magnetic iso-perimetric inequality):

A. Kachmar, V. Lotoreichik. On the isoperimetric inequality for the magnetic Robin Laplacian with negative boundary parameter. J. Geometric Anal. Vol. 32, No. 6, Paper No. 182, 20 pp. (2022).

(Spectral asymptotics, semi-classical regime):

W. Assaad, A. Kachmar, B. Helffer. Semi-classical eigenvalue estimates under magnetic steps. To appear in Analysis & PDE. Preprint on arXiv. 

B. Helffer, S. Fournais, A. Kachmar, N. Raymond. Effective operators on an attractive magnetic edge. To appear in J. Éc. Polytech., Math. Preprint on arXiv. 

(Sum of eigenvalues, semi-classical regime):

S. Fournais, A. Kachmar. On the energy of bound states for magnetic Schrödinger operators. J. London Math. Soc. Vol. 80 (1) 233-255 (2009).

(Accumulation of eigenvalues below the essential spectrum):

M. Goffeng, A. Kachmar, M. P. Sundqvist. Clusters of eigenvalues of the magnetic Laplacian with Robin condition. J. Math. Phys. Vol. 57 (6) article number  063510 (2016).

Robin Laplacian (strong coupling regime):

B. Helffer, A. Kachmar. Eigenvalues  for  the Robin Laplacian in domains with variable curvature.

Transactions of AMS, Vol. 369 (5) 3253-3287 (2017).

B. Helffer, A. Kachmar, N. Raymond. Tunneling for the Robin Laplacian in smooth planar domains. Commmun. Contemp. Math. Vol. 19 (1)  1650030, 38 pp. (2017).

Phase transitions:

(Proof of oscillations generated by Aharononv-Bohm potential, non-simple connectivity of domains and Robin condition; results consistent with the Little-Parks experiment).

A. Kachmar, X.B. Pan. Oscillatory patterns in the Ginzburg-Landau model driven by the  Aharonov-Bohm potential. J. Funct. Anal. Vol. 279, No. 10, Article ID 108718, 37 p. (2020).

B. Helffer, A. Kachmar. Thin domain limit and counterexamples to strong diamagnetism.  Rev. Math. Phys. Vol. 33, No. 2, Article ID 2150003, 35 p. (2021).

A. Kachmar, M. P.-Sundqvist. Counterexample to strong diamagnetism for the magnetic Robin Laplacian. Mathematical Physics, Analysis and Geometry. Vol. 23, art. no. 27 (2020).

(Transition from surface to bulk superconductivity; Abrikosov lattices)

S. Fournais, A. Kachmar. Nucleation of bulk superconductivity close to critical magnetic field.  Advances in Mathematics. 226 1213 - 1258 (2011).

Non-linear effective models, density of superconductivity:

(Distribution of density/order parameter)

A. Kachmar. The Ginzburg-Landau order parameter near the second critical field.  SIAM J. Math. Anal. 46 (1) 572-587 (2014).

B. Helffer, A. Kachmar. The density of superconductivity in the bulk regime. Indiana Univ. Math. J. 67 (6) pp. 2181-2198 (2018).

(New effective models and applications)

B. Helffer, A. Kachmar. The Ginzburg-Landau functional with vanishing magnetic field. Arch.  Rational Mech.  Anal. 218 (1) 55-122 (2015).

B. Helffer, A. Kachmar. From constant to non-degenerately vanishing magnetic fields in superconductivity. Annales de l'Institut Henri Poincaré - Analyse non-linèaire Vol. 34, 423-438 (2017). 

3D Ginzburg-Landau functional:

(Vortex energy in 3D)

A. Kachmar. The ground state energy of the three dimensional Ginzburg-Landau model in the mixed phase.  J. Funct. Anal. 261 (11) 3328 - 3344 (2011).

(Surface superconductivity in 3D; Abrikosov energy)

A. Kachmar, S. Fournais, M. Persson The ground state energy of the three dimensional Ginzburg-Landau functional. Part II: Surface regime. J. Math. Pures Appl. 99  343-374 (2013).

Landau-DeGennes model (liquid crystals):

S. Fournais, A. Kachmar, X.B. Pan. Existence of surface smectic states in liquid crystals.  J. Funct. Anal.  274, 900-958 (2018).

Selected recent works

Magnetic Laplacian in a sector:

(Proof of existence of discrete spectrum in a sector with Neumann boundary condition and under constant magnetic field).

V. Bonnaillie-Noël, S. Fournais, A. Kachmar, N. Raymond. Discrete spectrum for the magnetic Laplacian on perturbed  half-spaces. Accepted in  Bull. Lond. Math. Soc. arXiv:2208.13646. 

Electro-magnetic quantum tunneling:

(Accurate calculation of quantum tunneling under constant magnetic field and radial symmetric potentials)

B. Helffer, A. Kachmar. Quantum tunneling in deep potential wells and strong magnetic field revisited. Accepted in Pure Appl. Anal. arXiv:2208.13030. 

B. Helffer; A. Kachmar; M. Sundqvist. Flux and symmetry effects on quantum tunneling. Accepted in Mathematische Annalen. arXiv:2307.06712. 

Counting magnetic eigenvalues:

S. Fournais, R.L. Frank, M. Goffeng, A. Kachmar, M. Sundqvist. Counting Negative Eigenvalues for the Magnetic Pauli Operator. Accepted in Duke Mathematical Journal. arXiv:2307.16079.  

Teaching

Graduate Course: 

MEDP514 (28 hours) Spectral Theory and Applications to PDE. 

(Master 2) Lectures on the following topics: Unbounded operators, Kato-Rellich Theorem, Friedrichs' (Extension) Theorem, Spectrum, Compact Resolvent, Min-Max principle, Weyl's Theorem, Harmonic Oscillator, Harmonic Approximation. 

Undergraduate courses:

M3301 (60 hours) & M3306 (30 hours): 

Introduction to measure theory, Lebesgue integral, convergence theorems, product measures, Fubini-Tonelli Theorem, convolution product and Fourier transform in R.

M3311 (30 hours):

Introduction to Bounded Operators:  Adjoints, Compact Operators and Ascoli's Theorem. 

M3320 (30 hours):

Introduction to PDE: Separation of Variables, Integral Methods, Heat, Wave and Laplace equations, Uniqueness of solution, harmonic functions.

M2206 (60 hours):

Normed Vector spaces, Riesz Compactness Theorem,  bounded operators, Hilbert spaces, projection Theorem, Riesz representation theorem, Hilbert basis, Fourier Series.

Math202 (33 hours):

Line integrals, vector fields, Green's theorem, surface integrals, divergence and Stoke's theorems, 1st order ODE, 2nd order ODE (reduction of order, characteristic equations, undetermined coefficients, etc), power series solutions of ODE, systems of ODE.

Visiting Positions

Visiting Professor, Lund University, Sweden (6 months: September 1, 2022 -- February 28, 2023).

Visiting Professor, University of Angers, France (1 month: January 16 -- February 17, 2022). 

Selected Invitations

Nantes Université, France. (45 days: November 1, 2021 -- December 15, 2021).

Politecnico di Milano, Italy. Three Months of Quantum Mathematics in Milano. (May 16-27, 2022).

Institute Mittag-Leffler, Sweden. Spectral Methods in Mathematical Physics. (Two weeks: Jan 21-Feb 02, 2019).

New York University at ECNU Shanghai, China. (Two weeks: November 5-15, 2018). 

Banff International Research Station, Canada. Phase transition models. (One week: May 01-05, 2017).