RAHMATI, MOHAMMAD REZA

Mohammad Reza Rahmati is a mathematician and interdisciplinary researcher whose work lies at the intersection of pure mathematics, applied mathematics, mathematical physics, control theory, nonlinear dynamics, optics, and scientific machine learning. He received his Ph.D. in Mathematics from the Centro de Investigación en Matemáticas (CIMAT), Mexico, in 2015. He is currently a postdoctoral researcher at the Centro de Investigaciones en Óptica (CIO) in León, Guanajuato, Mexico, and is a member of Mexico’s National System of Researchers (SNII), Level I.

His research in pure mathematics is centered on algebraic and complex geometry, Hodge theory, singularity theory, motives, moduli spaces, representation theory, and related aspects of mathematical physics. A recurring theme in his work is the study of how geometric and cohomological structures behave under deformation and degeneration. This includes mixed Hodge structures, variations of Hodge structure, limiting mixed Hodge structures, generalized theta divisors, compactified Jacobians, graph motives, Calabi–Yau varieties, and geometric constructions associated with singular spaces. He is particularly interested in the interaction between local singularity data and global geometric invariants, as well as in the role of filtrations, spectral sequences, logarithmic complexes, and motivic decompositions in organizing this information.

Another component of his mathematical research concerns moduli and classification problems. His work investigates spaces of geometric or dynamical objects modulo appropriate notions of equivalence, using tools from algebraic geometry, Lie theory, flag varieties, Schubert geometry, and invariant theory. This perspective also appears in his studies of jet differentials, tautological cycles, higher Abel–Jacobi invariants, and Schur-type decompositions of motives. These topics connect classical questions in geometry with modern developments in topology, mathematical physics, and the theory of moduli spaces.

On the applied side, his research includes automatic control, nonlinear systems, dynamical systems, stability theory, photonics, fiber lasers, adaptive optics, wavefront sensing, and computational imaging. In control theory, he has studied controllability and observability invariants, Brunovský indices, nonlinear multi-input multi-output systems, feedback equivalence, and the geometric structure of classification spaces. His approach combines classical control-theoretic ideas with methods from differential geometry, algebraic geometry, topology, and representation theory. A central objective is to understand not only whether a system is controllable or observable, but also how these properties vary across parameter spaces, degenerate at singular configurations, and can be encoded by geometric and discrete invariants.

In optics and photonics, his work focuses particularly on the modeling and analysis of erbium-doped fiber lasers. He has investigated passive and active Q-switching, saturable absorbers, pump modulation, pulse formation, reservoir timescales, relaxation mechanisms, and the dependence of pulse width and repetition frequency on physical parameters. These studies use coupled nonlinear rate equations, asymptotic reasoning, stability analysis, and numerical simulations to clarify the physical mechanisms responsible for different laser regimes. A major aim of this work is to develop models that remain physically interpretable while providing quantitative predictions useful for experiments and device design.

His research in adaptive optics and computational imaging explores wavefront reconstruction from indirect and potentially noisy optical measurements. This includes curvature wavefront sensing, phase diversity, Zernike-mode reconstruction, inverse problems, and the comparison of classical regularized methods with neural-network-based approaches. He is interested in learning-based models that preserve information about the underlying physics rather than treating reconstruction solely as a black-box prediction problem. Examples include multi-target neural architectures that estimate both optical phase and local wavefront curvature, as well as hybrid reconstruction strategies that combine learned quantities with Poisson solvers, regularization, or physically motivated consistency conditions.

Scientific machine learning forms a bridge between his mathematical and experimental interests. His work uses neural networks, physics-informed objectives, optimization methods, and synthetic-data generation to address nonlinear inverse problems and parameter estimation. Particular attention is given to robustness, reproducibility, noise sensitivity, identifiability, and the mathematical interpretation of learned representations. He is interested in how geometric, topological, and differential constraints can be incorporated into data-driven models to improve their reliability and generalization.