Educational Background

Ph.D., Mathematics, University of California, Irvine, Irvine, CA, 2011

B.A., Mathematics, magna cum laude, St. Cloud State University, St. Cloud, MN, 2004

Selected Publications

Selected publications and current preprints are listed below.

1. S. S. Babu and A. Welters, Symmetric Bessmertnyĭ Realizations and Field Extension Problems in Characteristic 2 – A Differential Algebra Approach, arXiv:2605.04910 [math.RA] (preprint, 2026). doi: 10.48550/arXiv.2605.04910

2. M. Cassier, G. W. Milton, and A. Welters, Broadband quasistatic passive cloaking: bounds and limitations in the near-field regime, arXiv:2601.02169 [math-ph] (preprint, 2026). doi: 10.48550/arXiv.2601.02169

3. A. Welters, Effective Operators in the Mathematical Theory of Composite Materials: The Hilbert Space Framework. In: D. Alpay, F. Colombo, and I. Sabadini (eds.), Operator Theory, Springer, Cham, 2026, pp. 3555–3578. doi: 10.1007/978-3-032-16356-1_81

4. J. Elsinger, I. Orzel, and A. Welters, Bessmertnyĭ realizations of symmetric multivariate rational matrix functions over any field, Linear Algebra and its Applications 737, 80–118 (2026). doi: 10.1016/j.laa.2026.02.014

5. Y. Grabovsky, G. W. Milton, and A. Welters, Complete characterization of symmetric Kubo–Ando operator means satisfying Molnár’s weak associativity, Linear Algebra and its Applications 719, 158–182 (2025). doi: 10.1016/j.laa.2025.04.015

6. A. Stefan, A. Welters, and J. Elsinger, Kharitonov’s Theorem with Degree Drop: a Wronskian Approach, Complex Analysis and Operator Theory 19, Article 74 (2025). doi: 10.1007/s11785-025-01696-5

7. K. Beard and A. Welters, Matrix monotonicity and concavity of the principal pivot transform, Linear Algebra and its Applications 682, 323–350 (2024). doi: 10.1016/j.laa.2023.11.016

8. B. Alshammari and A. Welters, On the spectral theory of linear differential-algebraic equations with periodic coefficients, Analysis and Mathematical Physics 13, Article 94 (2023). doi: 10.1007/s13324-023-00856-0

9. K. Beard, A. Stefan, R. Viator, Jr., and A. Welters, Effective operators and their variational principles for discrete electrical network problems, Journal of Mathematical Physics 64(7), 073501 (2023). doi: 10.1063/5.0130429

10. 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

11. A. Stefan and A. Welters, Extension of the Bessmertnyĭ Realization Theorem for Rational Functions of Several Complex Variables, Complex Analysis and Operator Theory 15, Article 115 (2021). doi: 10.1007/s11785-021-01150-2

12. A. Stefan and A. Welters, A short proof of the symmetric determinantal representation of polynomials, Linear Algebra and its Applications 627, 80–93 (2021). doi: 10.1016/j.laa.2021.06.007

13. M. Cassier, A. Welters, and G. W. Milton, A rigorous approach to the field recursion method for two-component composites with isotropic phases. Chapter 10 in G. W. Milton (ed.), Extending the Theory of Composites to Other Areas of Science, Milton–Patton Publishers, Salt Lake City, UT, 2016, pp. 287–308. ISBN: 978-1-4835-6919-2. doi: 10.48550/arXiv.1601.01378

14. M. Cassier, A. Welters, and G. W. Milton, Analyticity of the Dirichlet-to-Neumann map for the time-harmonic Maxwell’s equations. Chapter 4 in G. W. Milton (ed.), Extending the Theory of Composites to Other Areas of Science, Milton–Patton Publishers, Salt Lake City, UT, 2016, pp. 95–122. ISBN: 978-1-4835-6919-2. doi: 10.48550/arXiv.1512.05838

15. A. Figotin and A. Welters, On overdamping phenomena in gyroscopic systems composed of high-loss and lossless components, Journal of Mathematical Physics 57(4), 042902 (2016). doi: 10.1063/1.4944721

16. S. P. Shipman and A. Welters, Pathological scattering by a defect in a slow-light periodic layered medium, Journal of Mathematical Physics 57(2), 022902 (2016). doi: 10.1063/1.4941137

17. A. Figotin and A. Welters, Lagrangian framework for systems composed of high-loss and lossless components, Journal of Mathematical Physics 55(6), 062902 (2014). doi: 10.1063/1.4884298

18. A. Welters, Y. Avniel, and S. G. Johnson, Speed-of-light limitations in passive linear media, Physical Review A 90(2), 023847 (2014). doi: 10.1103/PhysRevA.90.023847

19. S. P. Shipman and A. Welters, Resonant electromagnetic scattering in anisotropic layered media, Journal of Mathematical Physics 54(10), 103511 (2013). doi: 10.1063/1.4824686

20. S. P. Shipman and A. Welters, Resonance in anisotropic layered media, 2012 International Conference on Mathematical Methods in Electromagnetic Theory, pp. 227–232 (2012). doi: 10.1109/MMET.2012.6331235

21. A. Figotin and A. Welters, Dissipative properties of systems composed of high-loss and lossless components, Journal of Mathematical Physics 53(12), 123508 (2012). doi: 10.1063/1.4761819

22. A. Welters, On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a Jordan Block, SIAM Journal on Matrix Analysis and Applications 32(1), 1–22 (2011). doi: 10.1137/090761215

23. A. Welters, On the Mathematics of Slow Light. Ph.D. dissertation, University of California, Irvine, 2011. ProQuest LLC, Ann Arbor, MI.

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 multiphase 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), Air Force Young Investigator Research Program (YIP) award, On a Theory of Broadband Absorption Suppression in Magnetic Composites, U.S. Air Force Office of Scientific Research (AFOSR), final award amount $264,199.41, Apr. 1, 2015–Mar. 31, 2018, grant no. FA9550-15-1-0086. Program officer: Dr. Arje Nachman, Electromagnetics. Technical report: Defense Technical Information Center record.

Research

RESEARCH DIRECTIONS

My research comprises three interconnected directions. They overlap substantially: realization theory provides structured descriptions of effective response; effective-media problems generate questions about operator and matrix functions; and layered electromagnetic media lead to spectral problems for differential-algebraic equations. Across these directions, I study how analytic, algebraic, variational, and spectral structures constrain mathematically and physically meaningful systems.

Realization Theory and Structured Representations

I study when multivariable scalar- and matrix-valued functions, polynomials, and operators admit structured representations by matrix pencils, Schur complements, state-space systems, or passive electrical networks. Central objects include Bessmertnyĭ, positive-real, and Herglotz–Nevanlinna functions, together with symmetric determinantal representations and network synthesis. I am interested both in constructive representation theorems and in obstructions that clarify the limits of such theories. Together, these questions form part of the mathematical theory of realizability.

Effective Operators, Composite Materials, and Operator and Matrix Functions

I study effective operators for heterogeneous materials and networks using Hilbert-space Z-problems, generalized Schur complements, and variational principles. This connects composite-material theory directly with realization theory and gives rise to questions of independent interest in linear algebra and operator theory.

For normalized scalar effective operators of isotropic two-phase composites, perspective functions connect effective-medium theory with Herglotz–Nevanlinna and operator-monotone functions and, through Kubo–Ando theory, with operator means. Questions from the same constrained-system framework have also led to conditions for monotonicity and concavity of the principal pivot transform. My current interests include variational principles, bounds, and realizability for multiphase composites and metamaterial synthesis.

Spectral Theory, Wave Propagation, and Periodic Media

I develop spectral and operator-theoretic methods for wave propagation in periodic and complex media. For passive lossless one-dimensional photonic crystals, passage to time-harmonic fields and separation of variables reduce Maxwell’s equations to periodic differential-algebraic equations in which a scaled frequency serves as the spectral parameter. This connects self-adjoint operator theory and Floquet–Bloch analysis with questions about dispersion, resonance, scattering, and slow light, continuing and broadening a research program that began with my doctoral work. Related interests include speed-of-light limitations in passive media, passive cloaking, wave propagation in materials with defects, and dissipative and high-loss systems.

RESEARCH AREAS AND KEYWORDS

I work broadly in applied mathematics and mathematical physics, with particular emphasis on the following interconnected areas:

• Realization theory and passive systems: multivariable functions, Bessmertnyĭ realizations, structured matrix representations, Herglotz–Nevanlinna functions, passive linear systems, and electrical-network synthesis.
• Effective operators and composite materials: effective media, multiphase composites, metamaterial synthesis, perspective functions, Kubo–Ando operator means, variational principles, and monotonicity and convexity/concavity properties of matrix and operator functions.
• Spectral theory and wave propagation: periodic differential-algebraic equations arising from Maxwell’s equations, photonic crystals, scattering and resonance, guided modes and embedded eigenvalues, slow light, speed-of-light limitations, passive cloaking, passive and dissipative systems, and materials with defects.
• Mathematical foundations and methods: linear algebra, complex analysis, functional analysis, operator theory, spectral and scattering theory, and perturbation theory.

RESEARCH OPPORTUNITIES FOR STUDENTS

I welcome research inquiries from Florida Tech undergraduates and from current or prospective M.S. and Ph.D. students in Applied Mathematics. Information about Florida Tech’s Applied Mathematics B.S., M.S., and Ph.D. programs is available through the university’s program pages.

Student projects may address questions in pure mathematics or develop rigorous mathematics motivated by physics, engineering, electrical networks, electromagnetism, or composite materials. The projects are primarily theoretical and analytical rather than computational, but they can begin with concrete questions and be developed to suit a student’s preparation and interests.

Possible starting questions include:

• When can a function or operator be represented by a structured matrix system or passive electrical network?

• How does the internal structure of a composite material constrain its effective behavior?

• How can spectral theory explain wave propagation, resonances, or slow light in layered media?

A project can begin with a manageable special case and grow into deeper mathematical research as the student develops the necessary background. Depending on the topic, useful preparation may include linear algebra, differential equations, real or complex analysis, functional analysis, or mathematical physics. Students are not expected to know all these subjects before beginning, and prior research experience is not required.

Funding for individual research projects is not always available at the outset. As students demonstrate sustained interest and progress, I work with them to identify and pursue appropriate research-funding opportunities when possible. General information about graduate funding and assistantships is available from Florida Tech.

Interested students are welcome to email me with a brief introduction describing their academic program, relevant coursework, and one or two topics that interest them. Prospective graduate students may also include a CV.

STUDENT RESEARCH AND MENTORING

I mentor students at the undergraduate, master’s, and doctoral levels, helping them move from an accessible starting question toward increasingly independent mathematical research. Student projects have led to theses, dissertations, presentations, and coauthored peer-reviewed publications.

Current Ph.D. Student

• Anthony Stefan — Expected completion: Summer 2027. Dissertation: Bessmertnyĭ Realizations of Effective Tensors with Symmetries in Multiphase Composites for Metamaterial Synthesis. His earlier M.S. research developed into continuing work and multiple joint publications in realization theory and electrical networks.

Former Graduate Students

• Bader Alshammari — Ph.D. in Applied Mathematics, July 2022. Dissertation: On the Spectral Theory of Linear Differential-Algebraic Operators with Periodic Coefficients. His research developed spectral theory for periodic differential-algebraic equations motivated by Maxwell’s equations in layered media and led to a joint peer-reviewed publication.

• Kenneth Beard — M.S. in Applied Mathematics, May 2022. Thesis: Relaxation of Variational Principles for Z-Problems in Effective Media Theory. His research began with effective-operator problems in composite materials and led to joint publications on effective operators, electrical networks, and the monotonicity and concavity of the principal pivot transform.

• Anthony Stefan — M.S. in Applied Mathematics, May 2021. Thesis: Schur Complement Algebra and Operations with Applications in Multivariate Functions, Realizations, and Representations.

Undergraduate Research

• Ian Orzel — Undergraduate researcher, Spring 2022. His research on symmetric Bessmertnyĭ realizations contributed to a coauthored 2026 publication.