The relation between mathematics and physics is very deep, both historically and philosophically. It continues to evolve and there is no fixed frontier.(M.F. Atiyah)
I am interested in Geometry and Mathematical Physics (see my short introduction to twistor theory pdf). Most of my articles are on arXiv, so do not forget to check out my preprints section. I have also recently started a blog to discuss research topics, so if interested, kindly check it out. I currently have a few ongoing projects, in various stages of completion:
Continuous Maps from Spheres converging to Boundaries of Convex Hulls, which is joint work with Peter J. Olver and which was accepted in Forum of Mathematics, Sigma. Given n distinct points in R^d, we construct an explicitly defined family of continuous maps from the d-1 dimensional sphere to R^d, with the property that this family converges, as a set-valued mapping, to the boundary of the convex hull of the given n points. We also identify the limiting map as a kind of inverse of the Gauss map of the boundary of the convex hull, suitably defined. The proof makes use of set-valued homology.
In Weights, Weyl-equivariant maps and a rank conjecture, which was published online in Experimental Mathematics (2020), I generalize the Atiyah-Sutcliffe problem on configurations of points to a Lie-theoretic setting.
I have reduced my symplectic versions of the Atiyah-Sutcliffe conjectures to the original (unitary) Atiyah-Sutcliffe conjectures: Root Systems and the Atiyah-Sutcliffe Problem. This work is now published in the Journal of Mathematical Physics (2019).
One of my articles is actually in combinatorics, but inspired by both Rota's basis conjecture, and the Atiyah-Sutcliffe problem on configurations of points: "Determinants, Choices and Combinatorics", which is published in Discrete Mathematics (2018).
Together with Sir M.F. Atiyah, we wrote an article: "The relativistic Geometry and Dynamics of Electrons", which is published in Foundations of Physics (2018). It can be viewed as part of Sir Michael's geometrization program of the foundations of Physics.
Jointly with Prof. Ernst Huijer from AUB, we have been working on modeling the magnetic potential inside and outside a polygonal shaped magnet under an applied magnetic field, using conformal mappings. If interested, please check out Toby Driscoll's SC Toolbox for computing Schwarz-Christoffel mappings. I did my best to try to port it to Octave 5.0. If interested, please check out the repository: sc_toolbox_octave.
In collaboration with D. Calderbank and R. Bielawski, we worked on the problem of describing quaternionic manifolds with symmetry. This work is almost complete, but frozen for lack of time.
Together with Prof. Nazih Nahlus (at the American University of Beirut), we worked on finding nice proofs for some known and less known facts concerning Lie algebras. The work is entitled: Commutators and Cartan subalgebras in Lie Algebras of Compact Semisimple Lie Groups, published in the Journal of Lie Theory (2017).
I found a generalization of the Atiyah-Sutcliffe problems where the target space consists of flags in spaces of polynomials in several variables: Configuration spaces of points, symmetric groups and polynomials of several variables.
I made progress on the hyperbolic version of the Atiyah problem on configurations of points, in the case of four points: On the Atiyah problem on hyperbolic configurations of four points. This work is published in Geometriae Dedicata.
I am interested in the Berry-Robbins problem, which is related to a proof of the spin-statistics theorem that does not involve Quantum Field Theory, and the related Atiyah and Sutcliffe conjectures on configurations of n distinct points in R^3. I came across a version of the Atiyah and Sutcliffe problem for the groups Sp(n), instead of the groups U(n). It is published in SIGMA 10 (2014): Configurations of points and the Symplectic Berry-Robbins Problem.
UPDATE: I recently reduced my symplectic version of the Atiyah-Sutcliffe conjectures to the original unitary Atiyah-Sutcliffe conjectures, as mentioned above.
As explained in the paper, I also wrote a small program in Python in order to
test conjecture 2 numerically. The Python program can be found here: atiyah_spn.txt.
Just rename it to a .py extension instead of a .txt extension. From a Python shell, first go to the folder where
you have downloaded the Python file using the command cd, then import atiyah_spn.py using
import atiyah_spn as at
A sample test of the conjecture would be
[x,MyD] = at.test(4,500, 10, 0.1)
val_min, ind_min = at.find_min_real(MyD)
val_max, ind_max = at.find_max_real(MyD)
fig1 = at.plot_conf(x[ind_min])
fig2 = at.plot_conf(x[ind_max])
The first line generates 500 configurations of 4 points (the 10 and 0.1 are parameters, explained in the
comments in the python file) and computes their determinants. The second line finds the minimum of the real
parts of those determinants (which I conjecture is greater or equal to 1, for all n>2), as well as its index
position, and the third line finds the maximum of the real parts of the determinants, as well as its index
position. Then the program plots the configurations corresponding to the minimum value and the maximum value
of the real parts of the determinants, for this sample of 500 configurations, in figures 1 and 2 respectively.
To close the two figures, simply use
An example of a minitwistor correspondence in higher dimension, between 3-Grassmannians and Quadrics, as well as solutions to some generalized Bogomolny equations on 3-Grassmannians, and holomorphic vector bundles on the quadrics, which are trivial on the twistor lines. The plan is to include it in a larger paper with R. Bielawski and D. Calderbank, but the project seems frozen for now. I need to check my proof and my equations more carefully, and come up with some examples.
My PhD thesis at Stony Brook, written under Prof. C. LeBrun's supervision can be found here:
Hyperkahler 4n-manifolds with n commuting quaternionic Killing fields (PhD thesis). My approach there was to use local holomorphic Darboux coordinates on hyperkahler manifolds, and simplify the local expressions of the quaternionic Killing fields, as much as possible. My approach, and interest at the time, was in the local geometry of such data.