Ralph R. Gomez
Professor of Mathematics
Swarthmore College

500 College Avenue, Swarthmore PA 19081. Office: Science Center 152. Phone: (610)690-5727. email: rgomez1(at)swarthmore(dot)edu

Research Interests

My area of research is in differential geometry. I am interested in geometric structures that are related to ideas from physics. In particular, my focus is in Sasaki-Einstein geometry as well as generalized contact (and complex) geometry. More recently, I have become interested in the possible interplay between BHK mirror symmetry and Sasaki-Einstein geometry.

Other academic interests

Since 2021, I have been particularly captivated by the topic of UAP (Unidentified Anomalous Phenomenon) and I have been following the subject closely. A lot of the available evidence seems to suggest the phenomenon is indeed real and has been observed virtually all over the world over the past several decades, if not longer. But it is unclear what the phenomenon actually is. I am a proponent of treating the topic of UAP as a genuine scientific question in need of more data. Whatever the full explanation of UAP turns out to be, without a doubt it will be fascinating.

Publications and Preprints

*=undergraduate collaboration

(with *Nana Brobbey and *Xiuming Zhang) Berglund-Hübsch Duals and Sasaki-Einstein 5-Manifolds, in progress
(with Jaime Cuadros Valle and Joe Lope Vicente) Non-existence of extremal Sasaki metrics and the Berglund-Hübsch transpose, J.Geom.Phys, 214, (2025).
(with Jaime Cuadros Valle and Joe Lope Vicente) Berglund-Hübsch Transpose Rule and Sasaki-Einstein Rational Homology 7-spheres, Commun. Math. Phys. 405, 199 (2024)
Berglund-Hübsch Transpose Rule and Sasakian Geometry, Ann Glob Anal Geom 65, 2 (2024).
Sasaki-Einstein 7-Manifolds, Orlik Polynomials And Homology. Symmetry. 11 (7) 947 (2019).
A Note on Smale Manifolds and Lorentzian Sasaki-Einstein Geometry, Bull. Math. Soc. Sci. Math. Roumanie, (107) no. 2, (2016), 151-158.
(with Janet Talvacchia) Generalized CoK\”ahler Geometry and an Application to Generalized K\”ahler Structures, J. Geom. Phys. 98 (2015).
(with Janet Talvacchia) On Products of Generalized Geometries, Geom. Dedicata 175 (2015).
Lorentzian Sasaki-Einstein Metrics on Connected Sums of S² ×S³, Geom. Dedicata 150 (2011).
Thesis: On Lorentzian Sasaki-Einstein Geometry. Advisor: Charles P. Boyer.

Resource: Homology of links This resource calculates the homology of links coming from hypersurfaces in weighted projective space.

Teaching

Spring 2025: Math 105 Geometry II Seminar (Riemannian Geometry), Math 35 Multivariable Calculus with Theory
Fall 2024: Math 28 Honors Linear Algebra, Math 65 Topics in Geometry: Differential Geometry of Curves and Surfaces
Spring 2024: Math 35 Honors Multivariable Calculus, Math 67 Introduction to Modern Algebra
Fall 2023: Math 15 Single-Variable Calculus, Math 39 Discrete Mathematics with an Introduction to Proof
Sabbatical 2022-2023
Spring 2022: Math 101 Real Analysis II Seminar
Fall 2021: Math 15 Single-Variable Calculus, Math 15SP STEM Scholars Program
Spring 2021: Math 34 Several-Variable Calculus
Fall 2020: Math 63 Real Analysis, Math 27 Linear Algebra

about me

Here is an interview I did with the Swarthmore College newspaper “The Phoenix”. In the first part of the interview, I discuss my mathematical research and in the second part of the interview, I share my journey in becoming a mathematician.
I was also featured in Lathisms.
There is a much more detailed description of my journey in becoming a mathematician which is available here. It is also available in the book “Testimonios: Stories of Latinx and Hispanic Mathematicians“.