The author of over 100 research articles (many with undergraduate coauthors) and five books, Garcia works in operator theory, complex variables, matrix analysis, number theory, analytic combinatorics and discrete geometry. He has served on the editorial board of several well-known journals and has received five National Science Foundation (NSF) grants as principal investigator. He was the inaugural winner of the American Mathematical Society's Dolciani Prize for Excellence in Research (2019) and is a Fellow of the American Mathematical Society.
- Operator Theory
- Complex Analysis
- Matrix Analysis
- Number Theory
- Analytic Combinatorics
- Discrete Geometry
Operator Theory by Example with Javad Mashreghi and William T. Ross, Oxford University Press, 2023.
Matrix Mathematics: A Second Course in Linear Algebra with Roger Horn, Cambridge University Press, 2023 (second edition of A Second Course in Linear Algebra)
100 Years of Math Milestones: The Pi Mu Epsilon Centennial Collection (with Steven J. Miller). American Mathematical Society, 2019.
Finite Blaschke Products and Their Connections (with Javad Mashreghi and William Ross), Spinger, 2018.
A Second Course in Linear Algebra with Roger Horn. Cambridge University Press, 2017.
Introduction to Model Spaces and Their Operators with Javad Mashreghi, William Ross. Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2016.
Frobenius-Rieffel norms on finite-dimensional C*-algebras (with Konrad Aguilar, Elena Kim ’21) Oper. Matrices 16 (2022), no. 3, 733-758.
Norms on complex matrices induced by complete homogeneous symmetric polynomials (with Konrad Aguilar, Ángel Chávez, Jurij Volćić) Bull. London Math. Soc. 54 (2022), no. 6, 2045-2584.
Factorization length distribution for affine semigroups IV: a geometric approach to weighted factorization lengths in three-generator numerical semigroups (with Christopher O'Neill, Gabe Udell ’21) Comm. Algebra 8 (2022), no. 50, 3481-3497.
Explicit estimates for Artin L-functions: Duke's short-sum theorem and Dedekind zeta residues (with Ethan Simpson Lee) J. Number Theory (2022), no. 238, 920-943.
Powers of rational matrices (with Roger A. Horn, Florian Luca, Kuldeep Sarma) Amer. Math. Monthly 129 (2022), no. 2, 177.
Unconditional explicit Mertens' theorems for number fields and Dedekind zeta residue bounds (with Ethan Simpson Lee) Ramanujan J. 57 (2022), 1169–1191.
Factorization length distribution for affine semigroups III: modular equidistribution for numerical semigroups with arbitrarily many generators (with Mohamed Omar, Christopher O'Neill, Tim Wesley ’21) J. Austral. Math. Soc 113 (2022), no. 1, 21-35.
Awards & Honors
Awards & Honors
ILAS Hans Schneider Lectureship (2023)
Nominating Committee, Association for Women in Mathematics 2022-2023
Member-at-Large, American Mathematical Society Council (2020-)
National Science Foundation (NSF) Research Grant (principal investigator) DMS-1800123: “Opportunities in Operator Theory,” $179,998, (2019-2021)
Elected to Pi Mu Epsilon (National Mathematics Honor Society), (2019)
Elected to Sigma Xi (Scientific Research Honor Society) (2019)
Fellow of the American Mathematical Society (2019)
Member of the editorial board of Notices of the American Mathematical Society (2019)
Senior participant: Institute for Pure and Applied Mathematics (IPAM) program on Quantitative Linear Algebra (2018)
Member of the editorial board of the American Mathematical Monthly (2017-2022)
David L. Hirsch III and Susan H. Hirsch Research Initiation Grant, $10,000 (2017-2019 )
Director’s Mathematician in Residence (DMiR), Budapest Semesters in Mathematics (DMS) Program (2017)
Member of the editorial board of Proceedings of the American Mathematical Society (2016-2020)
Member of the editorial board of Involve (2011-)
National Science Foundation (NSF) Research Grant (principal investigator) DMS-1265973: "Operators on Hilbert Space," 2013-2016.
Member of the Research Experiences for Undergraduate Faculty (REUF) Advisory Board (2013-)
Member of the editorial board of Annals of Functional Analysis (2013-2020)
Pomona College Nominee, CASE US Professors of the Year Award (2012)
Pomona College Nominee, CASE US Professors of the Year Award (2011)
National Science Foundation (NSF) Research Grant (principal investigator) DMS-1001614: "Complex symmetric operators - theory and applications," 2010-14
Pomona College, Wig Distinguished Professorship Award for Excellence in Teaching, 2009, 2019
American Institute of Mathematics (AIM), Member, Human Resources Board, Palo Alto, CA, 2008-present
National Science Foundation (NSF) Research Grant (principal investigator) DMS-0638789: "Complex symmetric operators and function theory," 2006-10.
University of California, Santa Barbara; Mochizuki Memorial Fund Award, 2005; Outstanding Educator, 2004
University of California, Berkeley; Nikki Kose Memorial Teaching Prize, 2003; Outstanding Graduate Student Instructor, 1999