Welcome!
I’m an Applied Math Instructor at MIT Mathematics. My research interests are in numerical analysis and scientific computing with an emphasis on fast numerical linear algebra, approximation theory, and all things eigenvalue-related!
Education
I completed my Ph.D. in applied mathematics at Cornell’s Center for Applied Math, where I worked closely with my supervisor, Alex Townsend in the math department. My research committee included David Bindel and Anil Damle from the computer science department.
Before that, I recieved a dual B.Sc in physics and mathematics from Rensselaer Polytechnic Institute (RPI), where I conducted research in cellular motors (math modeling), nanomaterials (computational physics), and ultrathin films (experimental physics). My interests in math and physics accelerated after high school while working as an R&D technician at Praxis Technology, a small biomedical firm in Glens Falls, NY.
Research
The spectral properties of infinite-dimensional operators often provide visual and intuitive insights into the behavior of complex interactions. For instance, the stability of a fluid flow, the scattering cross-section of a nucleus, and the influence of decentralized hubs in large networks are intimately connected to the spectral properties of such operators. I design algorithms that accurately and efficiently extract eigenvalues, eigenfunctions, and spectral measures of differential and integral operators. The algorithms use a mix of functional analysis and approximation theory to preserve and exploit the structure of the continuous operator, regardless of the underlying discretization used for numerical computation. In practice, this means
- resolving inherently infinite-dimensional phenomenon related to continuous spectrum
- avoiding spectral pollution from the underlying finite-dimensional discretizations
- black-box approaches to a broad class of infinite-dimensional spectral problems
- enhanced flexibility when selecting discretization schemes
Software implementations exploiting fast linear algebra are available at https://github.com/SpecSolve.
Papers
- J. Zvonek, A. Horning, and A. Townsend, “ContHutch++: Stochastic trace estimation for implicit integral operators.” Submitted 2023.
- M.J. Colbrook, A. Horning, K. Thicke, and A.B. Watson, “Computing spectral properties of topological insulators without artificial truncation or supercell approximation.” IMA Journal of Applied Mathematics 88.1 (2023): 1-42.
- A. Horning and Y. Nakatsukasa, “Twice is enough for dangerous eigenvalues.” SIAM Journal on Matrix Analysis and Applications 43.1 (2022): 68-93.
- M.J. Colbrook, A. Horning, and A. Townsend, “Computing spectral measures of self-adjoint operators.” SIAM Review 63.3 (2021): 489-524.
- A. Horning, and A. Townsend, “FEAST for differential eigenvalue problems.” SIAM Journal on Numerical Analysis 58.2 (2020): 1239-1262.
- Y.P. Timalsina, et al. “Effects of nanoscale surface roughness on the resistivity of ultrathin epitaxial copper films.” Nanotechnology 26.7 (2015): 075704.
- C. Daniels, et al. “Elastic, plastic, and fracture mechanisms in graphene materials.” Journal of Physics: Condensed Matter 27.37 (2015): 373002.
Conference Proceedings
- M.J. Colbrook and A. Horning, “SpecSolve: Spectral methods for spectral measures” ICOSAHOM 2020+1. Lecture Notes in Computational Science and Engineering, Springer vol. 137 (2023).
- A. Horning, R. Morgan, and E. Nielson, “Minimum number of observations for exoplanet orbit determination.” Techniques and Instrumentation for Detection of Exoplanets IX. 4. Vol. 11117. SPIE (2018).
Invited and contributed talks
- December 2022, “Differentiating functionals of data-sparse matrix factors,” SCAN Seminar, Cornell University, Ithaca, NY
- July 2022, “Computing spectral properties of topological insulators with disorder,” WAVES 2022, Paris, France
- July 2022, “Computing spectral properties of topological insulators with disorder,” SIAM AN22, Pittsburgh, PA
- November 2021, “Computing spectra of infinite-dimensional operators,” NMPDE Seminar, MIT, Cambridge, MA.
- January 2021, “Computing spectral properties of infinite-dimensional operators,” Applied Math Seminar, UC Berkeley / Lawrence Berkeley Laboratory, Berkeley, CA.
- December 2020, “Twice is enough for dangerous eigenvalues,” CMS Winter Meeting, Montreal, CA (virtual conference platform).
- November 2020, “Twice is enough for dangerous eigenvalues,” NLA Seminar, University of Manchester, Manchester, UK.
- Jul. 2020, “How to diagonalize differential and integral operators with continuous spectrum,” SIAM Annual Meeting, Toronto, CA (virtual conference platform). Slides.
- Dec. 2019, “Computing spectral measures of differential and integral operators,” Complex Analysis Workshop, Isaac Newton Institute, Cambridge, UK
- Jul. 2019, “Computing the spectrum of a differential operator: a resolvent-based approach,” CAKE, University of Cambridge, Cambridge, UK
- Jun. 2019, “Computing the spectrum of a differential operator: a resolvent-based approach,” The 28th BNAC, University of Strathclyde, Glasgow, Scotland
- Sept. 2018, “A continuous analogue of FEAST for differential eigenvalue problems,” SCAN Seminar, Cornell University, Ithaca, NY
- Jul. 2018, “A continuous analogue of FEAST for differential eigenvalue problems,” ICOSAHOM, Imperial College, London, UK
- April 2018, “A continuous analogue of FEAST for differential eigenvalue problems,” Applied Math Days, RPI, Troy, NY
- Aug. 2015, “Mathematical model for steric disinhibition,” Mathematical Biosciences Institute Capstone Event, Ohio State, Columbus OH
- Oct. 2014, “Electrical resistivity in thin copper films,” American Vacuum Society, RPI, Troy, NY