Assistant Professor of Mathematics

Texas A&M University - San Antonio

Department of Mathematics

Classroom Hall, Room 314 U

Email: gdasilva@tamusa.eduResearch Interests

Analysis of PDE

I'm interested in nonlinear elliptic PDEs and elliptic systems, problems related to existence and regularity of solutions.

Hodge theory & Complex Algebraic Geometry

I'm interested in Algebraic cyles and its connections. Topics include cycle class maps, Hodge-D and Hodge conjectures, higher Chow groups, real regulators and related topics.

Current Teaching

- Real Analysis
- Fall 2024

- Intro to Proofs
- Fall 2024

Research Papers

- An elliptic equation with power nonlinearity and degenerate coercivity. Under peer review (Journal of Differential Equations)
- An inhomogeneous p-laplacian equation with a Hardy potential. Under peer review (Advanced Nonlinear Studies)
- Quasilinear elliptic equations with superlinear convection. Under peer review (Calculus of Variations and Partial Differential Equations)
- On a fully nonlinear k-Hessian system of Lane-Emden type. Under peer review (Potential Analysis)
- Radially symmetric solutions to a Lane-Emden type system. Under peer review (Nonlinear Analysis)
- Lyapunov exponents for G
_{2}variations of Hodge structures. Preprint - The Complexity of Higher Chow Groups,
*Canadian Math. Bull.*, 2023, DOI: 10.4153/S0008439522000509

with James Lewis - The Chow motive of a Fano variety of k-planes,
*Communications in Algebra*, 2023, DOI: 10.1080/00927872.2023.2252078

with James Lewis - Known cases of the Hodge conjecture (arXiv)
- On the topology of Fano smoothings
*Interactions with Lattice Polytopes*, Springer, 2022, DOI: 978-3-030-98327-7

with Tom Coates and Alessio Corti - On the monodromy of elliptic surfaces,
*Israel Journal of Mathematics, 2022,*DOI: s11856-022-2458-4 - Notes on the Hodge Conjecture for Fermat Varieties, Experimental Results , Volume 2 , 2021 , e22, DOI: 10.1017/exp.2021.14
- On the arithmetic of Landau-Ginzburg model of a certain class of threefolds,
*CNTP Vol. 13, No. 1, 2019,*DOI: 10.4310/cntp.2019.v13.n1.a5 - Arithmetic of degenerating principal variations of Hodge structure: examples arising from mirror symmetry and middle convolution,

*Canad. J. Math. 68, 2014,*DOI: 10.4153/CJM-2015-020-4

with Matt Kerr and Greg Pearstein

Books

- Real Analysis: Functions of a real variable. (In talks with Springer UTM)

Notes

Selected Past Teaching

- Differential Equations
- Spring 2023

- Calculus I
- Spring 2022

- Algebraic Geometry
- Spring 2019

- Manifolds
- Autumn 2017

Talks

- The Hodge Conjecture: A million dollar problem
- [slides]
- Remarks on the Hodge Conjecture for Fermat Varieties
- [slides]
- Ph.D. thesis
- [slides]
- On the arithmetic of Landau-Ginzburg model of a certain class of 3-folds (My Ph.D. major oral)
- [slides]

Computer Code

- Complete Intersections Hodge numbers calculator

Type the multidegree (as list of numbers):

Type the dimension (as a unique number):

Result:

A rank of my 10 favorite textbooks

With respect to the didatics (how easy it is to follow) of the author not the content of the book.

- Riemannian Geometry, M. do Carmo
- Differential Geometry of Curves and Surfaces, M. do Carmo
- Introduction to Real Analysis, E. Lages Lima
- Principles of Algebraic Geometry, P. Griffiths
- Algebraic Geometry, R. Hartshorne
- Fourier Analysis, E. Stein
- Complex Analysis, L. Ahlfors
- Partial Differential Equations, L. Evans
- Abstract Algebra, D. Dummit