About me
I am a third year PhD student at Texas A and M University in the Department of Mathematics.
My research interests lie in geometry ---> geometric PDEs, geometric microlocal analysis, low-dimensional topology, gauge theory and string theory ---> I think of problems in the (many) Floer theories,
gauge theory PDEs (e.g., Yang-Mills, Seiberg-Witten, Bogomolny, Hitchin, Nahm's, Kapustin-Witten), and homological mirror symmetry.
My advisors are Prof. Dean Baskin and Prof. Chris Pope.
Office: Blocker 619
Email: jlopez1234@tamu.edu
Motivation
God is a mathematician of a very high order and He used advanced mathematics in constructing the universe. - Paul Dirac
The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. - Henri Poincare
Math is hard. But you are not alone. - Anonymous
Publications
In preparation: Mazur manifolds
In preparation: Compactification of moduli space of monopoles
In preparation: On symplectic homotopy theory
Statement about teaching
One of my goals in life is to make high-level math reachable and understandable to a wider group of people than the one today. I want to teach these ideas in such a way that anyone, willingly, can understand them. It is common to hear that mathematics is very hard, or among the least liked subjects in school. However, this could be due to inadequate teaching, or lack of knowledge about the material. Hence, the history of mathematics can also be changed through teaching. I have been surrounded with academia for the most part of my life. Thus, I am a firm believer that education is a door where students can start pursuing great things, actually enjoying going to class, and regain hope in the things to come.
Education
2019-2020: Mast in Mathematics from Part III University of Cambridge. Wrote a thesis on Atiyah-Singer Index Theorem under the supervision of Dr. Alexei Kovalev
2018-2019: Postbac certificate in Mathematics from Iowa State University
2014-2018: BS degree in Mathematics from University of Texas Rio Grande Valley
2017: Studied abroad at ETH Zurich ETH Zurich. Changed my life
Some advice I follow:
Math is not a spectator sport. - Prof. Alessandro Carlotto
Value of books and pictures. - Prof. Guangbo Xu
Examples! - My advisor (as well as Richard Thomas, Peter Hintz)
Positions held
2024: 152 Calculus (with python) TA, TAMU
2023: 151 Calculus (with python) TA, TAMU
Summer 2023: 412 PDEs grader, TAMU
Spring 2023: Directed Reading Program, TAMU
2022: 150 Precalculus (with python) TA, TAMU
2022: 409 Advanced Calculus grader, TAMU
Spring 2022: Directed Reading Program, TAMU
2021: 602 PDEs grader, TAMU
2021: Teacher for Physics, Precalculus and Calculus, SJA
2021: Graduate teacher assistant, ISU
2020: Math tutor for Learning Center, UTRGV
2020: Resident Assistant for Upward Bound Program, UTRGV
Expository articles
Current: Main tools in geometric analysis and PDEs
Summer 2021: Principal bundles, Kobayashi and Nomizu Ch. 1,2
2020: Notes in Differential Geometry
2020: Primer on Lefschetz fibrations
Spring 2020: Atiyah-Singer Index Theorem , Master thesis
Sets of notes with exercises on my journey through graduate school
Topology: Floer theory
Analysis: Geometric PDEs
Physics: Scattering theory and string theory
Aspect of 3-dim manifolds, 4-dim manifolds, and Calabi-Yau manifolds
Class notes and learning seminars
Ongoing: Topology and Geometry Working Seminar, TAMU
Ongoing: Fourier Interal Operators, TAMU
Spring 2024: Spectral theory
Fall 2023: Representation theory
Fall 2022 - Spring 2023: K-theory and homotopy theory following Atiyah, Barnes-Rotzheim, McCearly and others
Fall 2022-Spring 2023: PDEs I and II following Evans and Lieb-Loss
Fall 2022: String theory, Prof. Katrin Becker
Spring 2022: Differential forms in algebraic topology follow Warner following Bott-Tu
Spring 2022: Morse theory
Spring 2022: Homological mirror symnetry, Prof. Chris Pope
Spring 2022: 4-dim topology, Scorpan
Fall 2021: Hodge theory mini-course following Warner
Fall 2021: Topology from a differentiable viewpoint, Prof. Guangbo Xu
Fall 2021: Topology
Fall 2021: Real Analysis
Summer 2021: GW/DT invariants , notes from the class by Prof. Sheshmani
Fall 2019: Part III Differential Geometry
Spring 2020: Part III Symplectic Geometry
Fall 2019: Part III Algebraic Topology
Fall 2019: Part III Algebraic Geometry
Spring 2020: Part III Complex Geometry
Problem solving in math
In my Master's, I was given some interesting problems. I believe is against the University policy to upload them on the internet. But they follow closely the books above. I have solutions to some selected exercises from my time in ISU: ODEs by Perko, PMA by Rudin, Linear algebra by Prof. Hogben, and Intro Algebra by Hungerford. I am pending to upload them.
I took the following Qualyfing exams:Summer 2022: Exercises for Top/DG Qual.
Summer 2023: String theory Qual. However, here are exercises from classical physics. The material for the Qual is above.
Exercises for topology and geometry, also PDEs.
Notes for analysis.
Seminars
PDEs working seminar, TAMU
Harmonic analysis and mathematical physics seminar, TAMU
Topology seminar, TAMU
Taubes secret seminar, online
Gauge theory virtual, online
Western Hemisphere Virtual Symplectic Seminar, online
Western Hemisphere Colloquium on Geometry and Physics, online
Symplectic Zoominar, online
Topology seminar and microlocal analysis seminar, Tsinghua online
Conferences and workshops
Spring 2024: TexAmP, TAMU
Spring 2024: Soliton dynamics workshop, TAMU
Spring 2024: TexAmP, TAMU
Fall 2023: SIAM, Lousiana
Fall 2023: Spectral theory conference for Prof. Peter Kuchment, TAMU
Summer 2023: Calabi-Yau manifolds mini-school, Stony Brook
Spring 2023: Geometry Fest, Princeton
Spring 2023: Mini-workshop with Peter Hintz, UNC
Spring 2023: AMS sectional meeting, Georgia Tech
Spring 2023: STCT II, Texas State
October 2022: Gauge theory, low dimensional topology workshop, and Floer homotopy theory workshop, MSRI
Summer 2022 (part 1): Hiro's notes for spectra, which are super amazing
Summer 2022 (part 2): Floer homotopy theory notes at UBC, exercises and some solutions
2022: Georgia Tech low-dimensional topology conference
2021: SIAM conference
Fall 2021: STCT I, Texas State
2021: Advances in Symplectic Topology
2020: AGITTOC
2019: MSRI H-Principle
2019: PCMI
2018: SAMMS at OSU
2018: Field of Dreams
2018: MSRI Topology and dynamical systems
Talks
Spring 2024: Eleventh Discrete Geometry and Algebraic Combinatorics Conference, UTRGV
Spring 2024: Graduate Student Seminar, TAMU
Fall 2022: Graduate Student Seminar, TAMU
Fall 2021: Graduate Student Seminar, TAMU
2021: Fitchburg State University talk promoting math
2020: Kuranishi structures, AGITTOC
2020: Stable curves, AGITTOC
These are my favorite math books
Salamon SW invariants
Buhler-Salamon
Donaldson-Kronheimer
Salamon-McDuff
Evans
Scorpan
Dyatlov-Zworski
Becker-Becker-Schwarz
Geometric scattering theory Melrose
Topological solitons
Dispersive PDEs Tao
Math methods Arkfen-Weber
Useful resources
Hiro Lee Tanaka
Qiangru Kuang
Terence Tao
Jonny Evans
Alexander Ritter
Denis Auroux
Richard Thomas
Keith Conrad
Ravi Vakil
Laura Fredrickson
Jason Lotay
Andras Gathmann
Arun Debray
Dexter Chua
Paul Minter
3Blue1Brown
Differential geometry videos
Measure theory videos
Khan Academy
Hobbies
British literature
Love to go out and eat with family
Hang out with friends
Play soccer
Art museums: after all, I painted for 12 years of my life
Music: guitar, bass, piano, classics, folk, country, classical music, 80s rock, jazz
Go on long walks
Hiking, trails
Boardgames
Meet new people organically
HP, LOTR, Narnia, Marvel, Star Wars, The Office, Parks and Rec, etc.. Yes, I am a big nerd