Educational Background

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

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

Current Courses

MTH 3102 Intro to Linear Algebra

MTH 2001 Calculus 3

Selected Publications

Here is a list of my publications and preprints.

1. K. Beard and A. Welters, Matrix monotonicity and concavity of the principal pivot transform, arXiv:2302.04293 [math-fa] (preprint, 2023). doi: 10.48550/arXiv.2302.04293
2. B. Alshammari and A. Welters, On the spectral theory of linear differential-algebraic equations with periodic coefficients, arXiv:2211.02134 [math-sp] (preprint, 2022). doi: 10.48550/arXiv.2211.02134
3. 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, 2021). doi: 10.48550/arXiv.2112.14287
4. 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
5. 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
6. 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
7. 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
8. 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
9. 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
10. 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
11. 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
12. 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
13. S. P. Shipman and A. Welters, Resonant electromagnetic scattering in anisotropic layered media, J. Math. Phys. 54, 103511 (2013). doi: 10.1063/1.4824686
14. 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
15. 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
16. 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
17. 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), 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: MP-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:

• 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
  • Dissertation title: On the Spectral Theory of Linear Differential-Algebraic Operators with Periodic Coefficients
  • http://hdl.handle.net/11141/3571

Current/Past MS students (with Thesis):

• Kenneth Beard - Graduation Date: May 2022
  • Thesis title: Relaxation of Variational Principles for Z-problems in Effective Media Theory
  • http://hdl.handle.net/11141/3497
• 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