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.

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. 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 (August 5, 2026).
2. International Workshop on Operator Theory and its Applications 2026: Lower Bounds for Dyadic Square Functions of Indicator Functions of Sets (August 4, 2026).
3. International Workshop on Operator Theory and its Applications 2026: Generalized Fock-Space and Fractional Derivatives: Uniqueness of Sampling and Interpolation Sets (August 3, 2026).
4. Southern California Applied Mathematics Symposium 2026: On the Complex SGD with Application to Kernel Regression in Reproducing Kernel Hilbert Spaces and Directional Bias (April 25, 2026).
5. Workshop on Applications of Hypercomplex Analysis 2026: Fractional Supershifts and Their Associated Cauchy Evolution Problems (January 16, 2026).
6. Joint Mathematics Meetings 2026: Lower Bounds for Dyadic Square Functions of Indicator Functions of Sets (January 6, 2026).
7. Joint Mathematics Meetings 2026: The Mittag-Leffler Bargmann Transform, and Its Extension to the Hypercomplex Setting (January 4, 2026).
8. International Workshop on Operator Theory and its Applications 2025: Thermal States on Mittag-Leffler Fock Space of the Slitted Plane (July 18, 2025).
9. International Workshop on Operator Theory and its Applications 2025: The Mittag-Leffler Bargmann Transform, and Its Extension to the Quaternionic Setting (July 18, 2025).
10. Southern California Applied Mathematics Symposium 2025: Special Functions as Solutions to the Least Square Problems (April 26, 2025).
11. 22nd Annual Workshop on Applications and Generalizations of Complex Analysis 2025: Lower Bounds for Dyadic Square Functions of Indicator Functions of Sets (March 29, 2025).
12. Mathematics of Data, Dynamics, and Life Sciences (MDDLS) 2025: On the Complex Representer Theorem and Applications to Minimization Problems (March 20, 2025).
13. Advances in Operator Theory with Applications to Mathematical Physics 2024: On the Complex Representer Theorem and Applications (November 19, 2024).
14. 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 (January 11, 2024).
15. Joint Mathematics Meetings 2024: Thermal States on Mittag-Leffler Fock Space of the Slitted Plane (January 4, 2024).
16. Heat Flow, Two-Point Inequalities, and Beyond 2023: On Complex Hypercontractivity after S. Janson (October 31, 2023).
17. Advances in Operator Theory with Applications to Mathematical Physics 2022: On the Mittag-Leffler Bargmann (MLB) Transform (November 15, 2022).
18. Workshop on Operator Theory with an Eye on Linear Systems and Hypercomplex Analysis 2022: On the Mittag-Leffler Bargmann (MLB) Transform (May 31, 2022).
19. SCCUR 2021: Generalized Fock Space Using Fractional Derivatives with Applications to Uniqueness of Sampling and Interpolation Sets (November 20, 2021).
20. International Workshop on Operator Theory and its Applications 2021: Generalized Fock Space and Fractional Derivative (Special Session 3, abstract p. 22) (August 11, 2021).
21. RAMiCS 2020: Commutative Doubly-Idempotent Semirings Determined by Chains and by Preorder Forests (conference proceedings) (October 29, 2020).
22. Capstone Presentation 2020: The Structure of Distributive Idempotent Weakly Conservative Lattice-Ordered Magmas (May 9, 2020).
23. Capstone Presentation 2020: Minimal Surfaces (May 9, 2020).
24. Joint Mathematics Meetings 2020: Commutative Doubly-Idempotent Semirings Determined by Chains and by Preorder Forests [abstract] (January 17, 2020).
25. AMS Fall Western Sectional Meeting 2019: Some Structural Results about Commutative Doubly-Idempotent Semirings [abstract] (November 2019).

Posters

1. Integrating Machine Learning for Pharmacovigilance Adverse Event Signal Detection compared to Classical Disproportionality
2. In Depth Analysis between accident causes and vehicle passenger count
3. Lattices visualized as 3D-objects in a browser and in virtual reality
4. Machine Learning and Neural Networks: The theory behind implementations
5. The Poincaré Duality Theorem and its Applications
6. Results about commutative doubly-idempotent semirings
7. The structure of distributive idempotent lattice-ordered magmas
8. The Mathematical Basis of General Relativity

Other Projects

In addition to math research I have also looked and worked on projects in different topics

1. Abstract Algebra and Virtual Reality
2. Machine Learning