My research papers in mathematics;

                                             1. The Geometry of 1-Based Minimal Types  (dvi) or (pdf) (with Byunghan Kim)
                                             2. Constructing the Hyperdefinable Group from the Group Configuration (dvi) or (pdf) (with Byunghan Kim and Jessica Young)
                                             3. Infinitesimals in a Recursively Enumerable Prime Model  (dvi) or (pdf)
                                             4. Zariski Structures and Algebraic Curvesj(dvi) or (pdf) or ohjgg
                                             5. A Non-Standard Bezout Theorem for Curves (dvi) or (pdf) (a version to be published in
Birational Geometry, Kaehler-Einstein Metrics
                                                 and Degenerations, Springer Proceedings in Mathematics & Statistics)                                            
                                             6. A Theory of Branches for Algebraic Curves  (dvi) or (pdf)  (older version)
                                             7. A Theory of Divisors for Algebraic Curves  (dvi) or (new version) (in progress)
                                             8. Some Geometry of Nodal Curves  (dvi)  or (pdf)
                                             9. Some Notes on Non-Standard Differentiation and Integration (dvi) or (pdf)
                                            10. An Interpretation of Newton's Work in Calculus (dvi) or (pdf) , with attached diagrams; (1, 2, 3, 4, 5, 6)
                                            11. A Theory of Duality for Algebraic Curves  (dvi) or (pdf)
                                            12. Flash Geometry of Algebraic Curves (dvi) or (pdf)  (A Proof of Severi's Conjecture)
                                            13. Severi's Conjecture and Single Node Curves (dvi) or (pdf)
                                            14. A Theory of Harmonic Variations (dvi) or (pdf) 
                                            15. The Geometry of Linear Regular Types (pdf)
                                            16. Some Notes on Quantifier Elimination and Model Completeness (dvi) or (pdf)
                                            17.  A Nonstandard Approach to the Theory of Algebraic Curves (pdf)
                                            18.  Applications of Nonstandard Analysis to Probability Theory (pdf)
                                            19.  A Simple Proof of the Fourier Inversion Theorem Using Nonstandard Analysis  (pdf)
                                            20. A Simple Proof of the Uniform Convergence of Fourier Series Using Nonstandard Analysis (pdf)
                                            21.  A Simple Proof of the Martingale Representation Theorem using Nonstandard Analysis (pdf)
                                            22.  Solving the Heat Equation Using Nonstandard Analysis  (pdf) 0 1 2 3 4 5 6 7  (attached files for option
                                                   pricing; run spectrum.m in MATLAB) (in progress)
                                            23. Decay Rates for Cusp Functions (pdf)
                                            24. A Note on Inflexions of Curves (pdf) (in progress)
                                            25. Non Standard Analysis and Physics (pdf) (in progress)
                                            26. A Simple Proof of the Uniform Convergence of Fourier Series in Solutions to the Wave Equation (pdf)
                                            27. A Note on Convergence of Fourier Series (pdf)
                                            28. A Lemma on Polynomial Roots (pdf)
                                            29. An Inversion Theorem for Laplace Transforms
                                            30. A Topological Note (pdf, required in 14)
                                            31. Electron Bunching  (pdf) (in progress)
                                            32. Solving Schrodinger's Equation Using Nonstandard Analysis (pdf) (in progress)
                                            33. Bounding the Number of Maximal Torsion Cosets on Algebraic Varieties (pdf)
                                            34. Historical Research into Plucker and Laplace (pdf)
                                            35. Historical Research into Laplace (contd) (pdf)
                                            36. Historical Research into Newton (pdf)
                                            37. Historical Research into Piero della Francesca (pdf)
                                            38. A Nonstandard Approach to Solving N'th Order, Linear, Inhomogeneous ODE's with Smooth Function Coefficients (pdf)
                                            39. A Proof of the Ergodic Theorem using Nonstandard Analysis (pdf)
                                            40. Applications of Nonstandard Analysis to Riemann Sums (pdf)
                                            41. Riemann Sums for Returning Points (pdf) (in progress)
                                            42. Notes on the Weil Conjectures for Curves (pdf)
                                            43. A Nonstandard Approach to Equidistribution (pdf)
                                            44.  A Nonstandard Approach to Equidistribution in Ergodic Theory (pdf) (in progress)
                                            45. Results on the Nonstandard Laplacian (pdf, required in 32) (in progress)
                                            46. A Nonstandard Version of the Fokker-Planck Equation (pdf) (in progress)
                                            47. Nonstandard Martingales, Markov Chains and the Heat Equation (pdf)
                                            48. Nonstandard Methods for Solving the Heat Equation (pdf)
                                            49. Nonstandard Methods for Solving Schrodinger's Equation (pdf) (in progress)
                                            50. Oscillatory Integrals (pdf) (in progress)
                                            51  Simple Proofs of the Riemann-Lebesgue Lemmas using Nonstandard Analysis (pdf)
                                            52. Computing the Distribution of Velocities of Some Solutions to the Nonstandard Diffusion Equation (pdf) (in progress)
                                            53. Schrodinger's Equation and Related Charge Density (pdf)
                                            54. A Nonstandard Version of Dirichlet's Theorem (pdf)
                                            55. Antiderivatives of Inverse Functions (pdf)
                                            56. A Nonstandard Solution to the Wave Equation (pdf)
                                            57. A Nonstandard Poisson Summation Formula (pdf)
                                            58. Some Arguments for the Wave Equation in Quantum Theory (pdf), published in Open Journal of Mathematical Sciences, (2021)
                                            59. Some Arguments for the Wave Equation in Quantum Theory 2 (pdf), published in Open Journal of Mathematical Sciences, (2022)
                                            60. Equilibria in Electrochemistry and Maximal Rates of Reaction (pdf), (submitted to Open Journal of Mathematical Sciences), (2022)
                                            61. The Continuity Equation and Particle Motion (pdf) (in progress)

A book I wrote on nonstandard analysis.

Some powerpoint presentations for the Newton Project at Culverhay;

                                             1. Integration
                                             2. The Fundamental Theorem of Calculus

  My PhD adviser, at M.I.T, was Professor Byunghan Kim, who is an expert in the area of simple theories. A simple theory is a structure in which certain amalgamation
properties hold. You can find, here, some geometric pictures of 1-amalgamation and 3-amalgamation (1>2>3), used, particularly, in the second paper. My PhD thesis
was mainly concerned with developing properties of minimal structures in such theories. A minimal structure is, very roughly speaking, an abstract version of an algebraic
curve. Since then, I have been a research fellow at Edinburgh University, The University of Camerino and The University of Exeter, specialising in the geometry of
such curves.

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