Graduate students Option 1:
need to have background in real, complex, functional, and harmonic analysis, measure theory.
You will also need to know general theory of linear operators on Hilbert spaces along with spectral theory. To review finite-dimensional spectral theory, see Chapters 3 and 4 of my MCTP lectures. To learn infinite-dimensional spectral theory and functional analysis via a simplified approach, see the book Erwin Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1st edition, 1989, ISBN-10: 0471504599. Option 2:
need to have a solid background in advanced calculus and advanced linear algebra (linear operators on finite dimensional Hilbert spaces).
Any other background is most welcome since the more mathematics you know, the better for your research.
One more thing: I require a good knowledge of academic English.
Undergraduate students need to have background in linear algebra and advanced calculus. You will also need
to learn Chapters 3, 4, 11 of my MCTP lectures.
Here is my REU project (scroll to the bottom). Basic knowledge of mathematics software (e.g., MATLAB) can be helpful in the project.
Research Supervised/Supported
PostDoc.:
Arup Chattopadhyay (08/2016 - 07/2017):
Fulbright-Nehru scholar visiting UNM.
Next occupation: Assistant Professor at Indian Institute of Technology Guwahati.
Ph.D. Students:
Roman Sverdlov (09/2017 - ):
Papers "A model of Berezin integral that can be reduced to the limit of a Riemann sum",
"Electromagnetic Lagrangian on a causal set that resides on edges rather than points",
"Restoring locality of scalar fields on a causal set by avoiding the use of d'Alembertians".
Supported by NSF DMS-1554456 CAREER.
Anastassiya Semenova (Fall 2017):
Supported by NSF DMS-1500704.
Bishnu Sedai (04/2014 - 07/2017):
PhD dissertation.
Paper "Trace formulas for perturbations of operators with Hilbert-Schmidt resolvents", Opuscula Mathematica, 38 (2018), no. 2, 253-260.
Educational talk at Albuquerque Math Teacher's Circle, at Bosque School, Albuquerque, NM, 2016. Supported by NSF DMS-1301717, 1500704, 1554456 CAREER.
Next occupation: Instructor at Virginia Polytechnic Institute and State University.
Clément Coine (Spring 2017):
Research project. PhD student at Université de Franche-Comté visiting UNM.
Paper "Higher order S^{2}-differentiability and application to Koplienko trace formula."
Travel supported by NSF DMS-1554456 CAREER.
Abdel Mohamed (Fall 2016):
PhD research project.
Supported by NSF DMS-1554456 CAREER.
B.S. Students:
Sarka E. Blahnik (08/2016 - 05/2017):
Undergraduate Honors thesis "Perturbations with zero spectral shift functions".
Research talk at the Society of Physics Students, UNM chapter.
Outreach presentation at Wilson Middle School, Albuquerque, NM, 2017.
Supported by NSF DMS-1148801 MCTP, 1554456 CAREER.
Outstanding undergraduate student award.
Next occupation: astrophysics student at UNM.
Kirstin Harriger (03/2013 - 04/2014):
Undergraduate thesis "First and second order spectral shift functions for pairs of self-adjoint matrices".
Research talk at SUnMaRC, Phoenix, AZ, 2014. Outreach presentation at
South Valley Academy, Albuquerque, NM, 2014.
Supported by NSF DMS-1148801 MCTP.
Next occupation: graduate student at UNM.
Clément Coine, "Continuous linear and bilinear Schur multipliers and applications to perturbation theory", Université de Franche-Comté, France, PhD, 2017.
David Weirich, "Weighted inequalities for dyadic operators over spaces of homogeneous type",
UNM, PhD, 2017.
Fred Kaul, ''Jost functions for Jacobi operators with super-exponentially decaying parameters'',
UNM, MS, 2016.
Nuriye Atasever, ''The Hilbert transform as an average of dyadic shift operators'',
UNM, MS, 2014.
Yohei Tanaka, "A topological approach to spectral flow'', Flinders University, Australia, MS, 2014.
Undergraduate Honors Thesis Reader
Sam Mundy, ''The Riemann-Roch theorem for number fields'',
UNM, BS, 2015.
Juan Pablo Madrigal Cianci, ''Study of modifications of the option pricing theory: a practical approach'',
UNM, BS, 2015.
Miranda Mundt, ''Mixing in fluids'',
UNM, BS, 2015.
Academic Adviser
Academic adviser for about 6 undergraduate and 3 graduate mathematics majors, 08/2012 -.
Mentor at the Association for Women in Mathematics Mentor Network, 08/2011 - 05/2012.
Educational Talks
"Research in an undergraduate school'', Math Club, University of New Mexico, 09/2012
"Operator functions and their derivatives", lecture for REU mathematics students, Texas A&M University, 06/2010
"Careers with PhD in Mathematics", panel discussion, Texas A&M University, 03/2010