Natanael S. Alpay

Mathematics · Quantitative Research · Machine Learning

Research Interests

I am a Ph.D. candidate in mathematics at the University of California, Irvine. My research spans probability, harmonic analysis, and complex analysis, with connections to optimization and machine learning. I combine rigorous mathematical reasoning with algorithm development and computational experiments.

My work includes kernel methods and complex-valued stochastic optimization, alongside applied machine learning for prioritizing drug and vaccine safety signals for expert review. In ongoing research on portfolio compression and deep hedging, I study how representation and learning errors affect financial risk.

I use AI and large language models as conjecture engines to generate candidate formulas, identify patterns, and explore possible approaches. I then examine the assumptions, search for counterexamples, and test ideas through reproducible computational experiments. Mathematical claims are established through rigorous proof, with attention to when the results apply and where their limitations lie.

PublicationsResearch talksPostersComputational projectsStudent Research and Mentoring

Research Publications and Preprints

  1. N. Alpay. Sharp discrete Hardy constants in low dimensions. Preprint, arXiv:2610.05421 (2026).
  2. N. Alpay. The sharp discrete Hardy inequality on ℤ3. Preprint, arXiv:2608.25262 (2026).
  3. N. Alpay, P. Cerejeiras, and U. Kähler. Generalized Fock-space and fractional derivatives: Uniqueness of sampling and interpolation sets. Results in Applied Mathematics 31 (2026), article 100737.
  4. N. Alpay. Sharp Ternary Martingale Isoperimetry and n-adic Takagi-Type Lower Bounds. Preprint, arXiv:2607.11069 (2026).
  5. N. Alpay and E. Battaglia. Complex Stochastic Gradient Descent and Directional Bias in Reproducing Kernel Hilbert Spaces. Preprint, arXiv:2604.23017 (2026).
  6. N. Alpay, A. De Martino, and K. Diki. Representer Theorem in Complex Reproducing Kernel Hilbert Spaces with Applications to Fock and Hardy Spaces and Superoscillations. Preprint, arXiv:2604.22165 (2026).
  7. N. Alpay. Fractional Supershifts and their associated Cauchy Evolution problems. Preprint, arXiv:2601.11829 (2026).
  8. N. Alpay, A. De Martino, and K. Diki. The Fueter Mittag-Leffler Bargmann transform. Journal of Mathematical Analysis and Applications 549 (2025), no. 2, article 129480.
  9. N. Alpay and T. C. John. Thermal states on Mittag-Leffler Fock space of the slitted plane. Journal of Mathematical Analysis and Applications 547 (2025), no. 1, article 129314.
  10. N. Alpay and P. Ivanisvili. Sharp Lower Bounds for Dyadic Square Functions of indicator functions of sets. Preprint, arXiv:2502.16045 (2025; revised 2026).
  11. N. Alpay, P. Jipsen, and M. Sugimoto. Varieties of unary-determined distributive ℓ-magmas and bunched implication algebras. Logical Methods in Computer Science 20 (2024), no. 1, article 12.
  12. N. Alpay and K. Diki. On the Mittag Leffler Bargmann (MLB) transform. Journal of Mathematical Analysis and Applications 527 (2023), no. 2, article 127458.
  13. N. Alpay. A new characterization of the Hardy space and of other Hilbert spaces of analytic functions. Istanbul Journal of Mathematics 1 (2023), no. 1, pp. 1–11.
  14. N. Alpay, P. Jipsen, and M. Sugimoto. Unary-Determined Distributive ℓ-magmas and Bunched Implication Algebras. In Relational and Algebraic Methods in Computer Science (RAMiCS 2021), Lecture Notes in Computer Science 13027, Springer, 2021, pp. 19–36. Conference version (2021) · Title links to the expanded journal version (2024).
  15. N. Alpay and P. Jipsen. Commutative Doubly-Idempotent Semirings Determined by Chains and by Preorder Forests. In Relational and Algebraic Methods in Computer Science (RAMiCS 2020), Lecture Notes in Computer Science 12062, Springer, 2020, pp. 1–14.

Research Talks and Conferences

  1. International Workshop on Operator Theory and its Applications 2026 — On the Complex Representer Theorem and Applications to Minimization Problems Over Reproducing Kernel Hilbert Spaces .
  2. International Workshop on Operator Theory and its Applications 2026 — Lower Bounds for Dyadic Square Functions of Indicator Functions of Sets .
  3. International Workshop on Operator Theory and its Applications 2026 — Generalized Fock-Space and Fractional Derivatives: Uniqueness of Sampling and Interpolation Sets .
  4. CHOC Pediatric and Lifespan Data Science Conference — Integrating Machine Learning for VAERS Vaccine Safety Signal Triage vs. Classical Disproportionality: A Pediatric and Lifespan Evaluation .
  5. Southern California Applied Mathematics Symposium 2026 — On the Complex SGD with Application to Kernel Regression in Reproducing Kernel Hilbert Spaces and Directional Bias .
  6. Biostatistics Symposium of Southern California 2026 — Integrating Machine Learning for Pharmacovigilance Adverse Event Signal Detection Compared to Classical Disproportionality . Poster PDF
  7. Workshop on Applications of Hypercomplex Analysis 2026 — Fractional Supershifts and Their Associated Cauchy Evolution Problems .
  8. Joint Mathematics Meetings 2026 — Lower Bounds for Dyadic Square Functions of Indicator Functions of Sets .
  9. Joint Mathematics Meetings 2026 — The Mittag-Leffler Bargmann Transform, and Its Extension to the Hypercomplex Setting .
  10. International Workshop on Operator Theory and its Applications 2025 — Thermal States on Mittag-Leffler Fock Space of the Slitted Plane .
  11. International Workshop on Operator Theory and its Applications 2025 — The Mittag-Leffler Bargmann Transform, and Its Extension to the Quaternionic Setting .
  12. Southern California Applied Mathematics Symposium 2025 — Special Functions as Solutions to the Least Square Problems .
  13. 22nd Annual Workshop on Applications and Generalizations of Complex Analysis 2025 — Lower Bounds for Dyadic Square Functions of Indicator Functions of Sets .
  14. Mathematics of Data, Dynamics, and Life Sciences (MDDLS) 2025 — On the Complex Representer Theorem and Applications to Minimization Problems .
  15. Advances in Operator Theory with Applications to Mathematical Physics 2024 — On the Complex Representer Theorem and Applications .
  16. Workshop on Schur Analysis and Applications to Hypercomplex Analysis, Neural Networks and Linear Systems 2024 — Thermal States on Mittag-Leffler Fock Space of the Slitted Plane .
  17. Joint Mathematics Meetings 2024 — Thermal States on Mittag-Leffler Fock Space of the Slitted Plane .
  18. Heat Flow, Two-Point Inequalities, and Beyond 2023 — On Complex Hypercontractivity after S. Janson .
  19. Advances in Operator Theory with Applications to Mathematical Physics 2022 — On the Mittag-Leffler Bargmann (MLB) Transform .
  20. Workshop on Operator Theory with an Eye on Linear Systems and Hypercomplex Analysis 2022 — On the Mittag-Leffler Bargmann (MLB) Transform .
  21. SCCUR 2021 — Generalized Fock Space Using Fractional Derivatives with Applications to Uniqueness of Sampling and Interpolation Sets .
  22. International Workshop on Operator Theory and its Applications 2021 — Generalized Fock Space and Fractional Derivative (Special Session 3, abstract p. 22) .
  23. RAMiCS 2020 — Commutative Doubly-Idempotent Semirings Determined by Chains and by Preorder Forests (conference proceedings) .
  24. Capstone Presentation 2020 — The Structure of Distributive Idempotent Weakly Conservative Lattice-Ordered Magmas .
  25. Capstone Presentation 2020 — Minimal Surfaces .
  26. Joint Mathematics Meetings 2020 — Commutative Doubly-Idempotent Semirings Determined by Chains and by Preorder Forests [abstract] .
  27. AMS Fall Western Sectional Meeting 2019 — Some Structural Results about Commutative Doubly-Idempotent Semirings [abstract] .

Posters

Computational Projects

My applied work connects mathematical methods with data analysis, implementation, and reproducible experiments.

Explore all computational projects →