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Welters, Aaron

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Aaron Welters

Associate Professor | College of Engineering and Science: Department of Mathematics and Systems Engineering

Contact Information

awelters@fit.edu
(321) 674-7202
Frederick C. Crawford Bldg, 319
Google Scholar
ResearchGate

Expertise

Mathematical Physics (Electromagnetism, Photonic Crystals, Composites); Linear algebra; Complex Analysis; Functional Analysis; Operator Theory

Personal Overview

I am an Associate Professor in the Department of Mathematics and Systems Engineering (MSE) at Florida Tech (FIT). Before coming to FIT, I was held a two-year postdoctoral position (from 2012-2014) as an Applied Mathematics Instructor in the Department of Mathematics at Massachusetts Institute of Technology (MIT). In 2011, I completed my Ph.D. in Mathematics at the University of California, Irvine (UCI). During 2011-2012, I was a VIGRE Postdoctoral Researcher in the Department of Mathematics at Louisiana State University (LSU).

Educational Background

Ph.D., Mathematics, University of California-Irvine, Irvine, CA

B.A., Mathematics, St. Cloud State University, St. Cloud, MN

Selected Publications

Here is a list of my publications and preprints.

  1. G. W. Milton and A. Welters, Complete characterization of symmetric Kubo-Ando operator means satisfying Molnar's weak associativity, arXiv:2405.20108 [math.FA] (preprint). doi: 10.48550/arXiv.2405.20108
  2. A. Stefan and A. Welters, Continuity of the roots of a nonmonic polynomial and applications in multivariate stability theory, arXiv:2112.14287 [math.CA] (preprint). doi: 10.48550/arXiv.2112.14287
  3. K. Beard and A. Welters, Matrix monotonicity and concavity of the principal pivot transform, Linear Algebra Appl., 628, 323-350 (Feb. 1, 2024). doi: 10.1016/j.laa.2023.11.016
  4. B. Alshammari and A. Welters, On the spectral theory of linear differential-algebraic equations with periodic coefficients, Journal of Analysis and Mathematical Physics, 13(94) (2023). doi: 10.1007/s13324-023-00856-0
  5. K. Beard, A. Stefan, R. Viator, and A. Welters, Effective operators and their variational principles for discrete electrical network problems, J. Math. Phys., 64(7), 073501 (2023). doi: 10.1063/5.0130429
  6. A. Stefan and A., Welters, Extension of the Bessmertnyĭ Realization Theorem for Rational Functions of Several Complex Variables. Complex Anal. Oper. Theory 15, 115 (2021). doi: 10.1007/s11785-021-01150-2
  7. A. Stefan and A. Welters, A short proof of the symmetric determinantal representation of polynomials, Linear Algebra Appl., 627, 80-93 (2021). doi: 10.1016/j.laa.2021.06.007
  8. M. Cassier, A. Welters, and G. W. Milton, A rigorous approach to the field recursion method for two-component composites with isotropic phases, Chap. 10 in: G. W. Milton (editor), Extending the Theory of Composites to Other Areas of Science. Milton-Patterson Publishing, Salt Lake City, UT, 2016. ISBN: 978-1483569192. doi: 10.48550/arXiv.1601.01378
  9. M. Cassier, A. Welters, and G. W. Milton, Analyticity of the Dirichlet-to-Neumann map for the time-harmonic Maxwell's equations, Chap. 4 in: G. W. Milton (editor), Extending the Theory of Composites to Other Areas of Science. Milton-Patterson Publishing, Salt Lake City, UT, 2016. ISBN: 978-1483569192. doi: 10.48550/arXiv.1512.05838
  10. A. Figotin and A. Welters, On overdamping phenomena in gyroscopic systems composed of high-loss and lossless components, J. Math. Phys. 57, 042902 (2016) doi: 10.1063/1.4944721
  11. S. P. Shipman and A. Welters, Pathological scattering by a defect in a slow-light periodic layered medium, J. Math. Phys. 57, 022902 (2016). doi: 10.1063/1.4941137
  12. A. Figotin and A. Welters, Lagrangian framework for systems composed of high-loss and lossless components, J. Math. Phys. 55, 062902 (2014). doi: 10.1063/1.4884298
  13. A. Welters, Y. Avniel, and S. G. Johnson, Speed-of-light limitations in passive linear media, Phys. Rev. A 90, 023847 (2014). doi: 10.1103/PhysRevA.90.023847
  14. S. P. Shipman and A. Welters, Resonant electromagnetic scattering in anisotropic layered media, J. Math. Phys. 54, 103511 (2013). doi: 10.1063/1.4824686
  15. S. P. Shipman and A. Welters, Resonance in anisotropic layered media, 2012 International Conference on Mathematical Methods in Electromagnetic Theory, 2012, pp. 227-232, doi: 10.1109/MMET.2012.6331235
  16. A. Figotin and A. Welters, Dissipative properties of systems composed of high-loss and lossless components, J. Math. Phys. 53, 123508 (2012). doi: 10.1063/1.4761819
  17. A. Welters, On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a Jordan Block, SIAM J. Matrix Anal. Appl., 32:1, 1-22 (2011). doi: 10.1137/090761215
  18. A. Welters, On the Mathematics of Slow Light. Thesis (Ph.D.)-Univ. of Calif., Irvine. ProQuest LLC, Ann Arbor, MI, 2011.

Recognition & Awards

Aaron Welters (PI) and Xianqi Li (Co-PI), Collaborative Research: Data-driven Realization of State-space Dynamical Systems via Low-complexity Algorithms, National Science Foundation (NSF), $125,000, Aug. 1, 2024-July 31, 2027, Grant no.: DMS 2410678.

Aaron Welters (PI), Variational principles, bounds, and realizability of effective operators for metamaterial synthesis using multphase composites, Simons Foundation, Travel Support for Mathematicians, $42,000 ($8,400/year), Sept. 1, 2023-Aug. 31, 2028, Gift ID: MPS-TSM-00002799.

Aaron Welters (PI), YIP early career award, On a Theory of Broadband Absorption Suppression in Magnetic Composites, Air Force Office of Science Research (AFOSR), $264,199.41, Apr. 1, 2015-Mar. 31, 2018, AFOSR Grant no.: FA9550-15-1-0086. AFOSR Program Officer: Dr. Arje Nachman, Electromagnetics. Technical report @ https://apps.dtic.mil/sti/pdfs/AD1058297.pdf

Research

Research Interests:

  • Applied mathematics
  • Mathematical physics

Focus Areas:

  • Electromagnetics
  • Material science (composites and effective media) 
  • Dissipative systems

Mathematical Specializations:

  • Linear algebra
  • Functional analysis
  • Spectral and scattering theory
  • Perturbation theory
  • Passive linear systems theory
  • Herglotz-Nevanlinna functions

Current/Past Research Topics:

  • Positive operator means (e.g., Kubo-Ando means)
  • Monotonicity and concavity/convexity properties of matrix functions
  • Spectral theory of linear differential-algebraic equations (DAEs)
  • Realizability theory of multivariate functions (e.g., Bessmertnyi realizations, effective operator representations) 
  • Theory of composites and its extensions
  • Herglotz-Nevanlinna functions and their applications
  • Broadband absorption suppression in magnetic composites
  • Slow-light enhancement of light-matter interactions
  • Speed-of-light limitations in complex media
  • Wave propagation in complex and periodic media [e.g., metamaterials, composites, photonic crystals, materials with defects, slow and fast light, guided modes (i.e., embedded eigenvalues), resonance phenomena]

Current/Past Ph.D students: 

  • Anthony Stefan - Expected Graduation Date: May 2025
    • Dissertation title (tentative): Bessmertny ̆ı realizations of effective tensors with symmetries in multiphase composites for metamaterial synthesis.
  • Bader Alshammari - Graduation Date: July 2022

Current/Past MS students (with Thesis):

  • Kenneth Beard - Graduation Date: May 2022
  • Anthony Stefan - Graduation Date: May 2021
    • Thesis title: Schur Complement Algebra and Operations with Applications in Multivariate Functions, Realizations, and Representations
    • http://hdl.handle.net/11141/3260
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