Research interests:
Spectral theory, analysis on graphs, abstract and applied harmonic analysis, representation theory of Lie groups, mathematical physics
Publications and Preprints:
- Data-driven discretizations and machine learning for Fredholm integral equations, In preparation.
- Some Plancherel-type formulas for Lie groups, In preparation.
- A proof of the Weyl integral formula for SU(2) by the equivariant localization formula of Atiyah–Bott, Submitted.
- From uniform boundedness to the boundary between convergence and divergence, (with O. Khanmohamadi), Accepted for publication in Mathematics Magazine
In this article we introduce a dual of the uniform boundedness principle which does not require completeness. We also indicate a connection between the dual principle and a question in spirit of du Bois-Reymond regarding the boundary between convergence and divergence of sequences.
- On the positivity of Kirillov’s character formula, Mathematical Physics, Analysis and Geometry, 23, 13 (2020). [url][pdf]
We give a direct proof for the positivity of Kirillov’s character on the convolution algebra of smooth, compactly supported functions on a connected, simply connected nilpotent Lie group G. Then we use this positivity result to construct a representation of G×G and establish a G × G-equivariant isometric isomorphism between our representation and the Hilbert–Schmidt operators on the underlying representation of G. In fact, we provide a framework in which we establish the positivity of Kirillov’s character for coadjoint orbits of groups such as SL(2, R) under additional hypotheses that are automatically satisfied in the nilpotent case. These hypotheses include the existence of a real polarization and the Pukanzsky condition.
- A structured inverse spectrum problem for infinite graphs and unbounded operators, Bull. Austral. Math. Soc., 98 (2018), pp. 363–371. [url][pdf]
Given an infinite graph G on countably many vertices, and a closed, infinite set Λ of real numbers, we prove the existence of an unbounded self-adjoint operator whose graph is G and whose spectrum is Λ.
In an earlier work (with K. Hassani Monfared), we studied the problem above assuming that Λ was a compact set. In that setting we show that the set of limit points of Λ equals the essential spectrum of the constructed operator A, and the isolated points of Λ are eigenvalues of A with multiplicity one. Moreover, we prove that any two such operators constructed by our method are approximately unitarily equivalent.
- A structured inverse spectrum problem for infinite graphs (with
K. Hassani Monfared), Linear Algebra Appl., 539 (2018), pp. 28–43. [url][pdf]
An excerpt from a review of the article in LAA: “This [work] enhances the connection between the (discrete) inverse eigenvalue problem and the (continuous) inverse Sturm–Liouville problem.
Based on my understanding, this paper is the first paper that considers the inverse eigenvalue problem on an infinite graph. This creates a new research area for the original problem, and it is foreseeable that many future results will build on the top of this paper.”
- Four fundamental spaces of numerical analysis (with O. Khanmohamadi), Mathematics Magazine, 91:4 (2018), 243–253. [url][pdf]
The precise statement of the equivalence of stability and convergence of a consistent approximation method used for solving an exact problem has been called the fundamental theorem of numerical analysis (see, e.g., Gowers et al. [PCM, § IV.21, by L.N. Trefethen]). In this article, we first motivate the importance of stability by looking at an example. We then give a picture which in a unified way introduces the reader to the fundamental theorem of numerical analysis and the operators and spaces associated with it.
- On the existence of nowhere-zero vectors for linear transformations (with S. Akbari, K. H. Monfared, M. Jamaali, and D. Kiani) Bull. Austral. Math. Soc., 82 (2010), pp. 480–487. [url][pdf]
Inspired by a conjecture of Noga Alon and his combinatorial Nullstellensatz, this paper studies the existence of vectors with no zero entries in the kernel and image of linear transformations.
- On teaching mathematics, Journal of Science and Society, 88 (2009), pp. 19–21. (In Persian)