Research Interests
My research interests lie in number theory, in particular in the following three different areas:
- Analytic number theory: bounded gaps between primes, sieve methods, zeros of \( L \)-functions.
- Iwasawa theory and Galois representations: Taylor–Wiles method and beyond, Birch and Swinnerton-Dyer conjecture, Bloch–Kato conjectures.
- Automorphic forms and arithmetic geometry: heights in Diophantine geometry, Langlands correspondence, \( p \)-adic cohomology theories, motives.
I have also recently been interested in Logic, in particular in doing mathematics with computers, formalized mathematics and Homotopy Type Theory.
Seminar Talks
I am thankful for the faculties in the UBC Mathematics Department to let me speak in a few graduated-level seminar classes when I was an undergraduate, below are the slides I created for all of my talks (newest first):
- 2015-10-07 Selberg Trace Formula for Compact Quotients, for MATH 592 at UBC [seminar course on Arthur's Trace Formula], co-supervised by Professors L. Silberman, J. Gordon, and W. Casselman.
- 2015-10-05 Group and Galois Cohomology: An introduction, for MATH 613E at UBC [seminar course on local class field theory], supervised by Professor J. Gordon.
- 2015-03-03 The Arthur-Selberg trace formula, for MATH 592 at UBC [seminar course on trace formula for \( {\operatorname{GL}}_{2}({\mathbb{Q}}_{p}) \) ], co-supervised by Professors L. Silberman, J. Gordon, and W. Casselman.
- 2015-02-11 Basics of Hodge-de Rham Theory IV: Hodge Theory for the Fundamental Groupoid and Drinfeld Associators, for MATH 620D at UBC [seminar course on Étale Fundamental Groups and Hodge-de Rham Theory], supervied by Professor R. Sujatha.
- 2015-02-04 Basics of Hodge-de Rham Theory III: Iterated Integrals and Chen's \( \pi_1 \) de Rham Theorem, for MATH 620D at UBC [seminar course on Étale Fundamental Groups and Hodge-de Rham Theory], supervied by Professor R. Sujatha.
- 2015-01-28 Basics of Hodge-de Rham Theory II: Hodge-de Rham Spectral Sequences and Periods of Mixed Hodge-Tate Structures, for MATH 620D at UBC [seminar course on Étale fundamental groups and Hodge-de Rham theory], supervised by Professor R. Sujatha.
- 2015-01-21 Basics of Hodge-de Rham Theory I: Motivations and Definitions, for MATH 620D at UBC [seminar course on Étale Fundamental Groups and Hodge-de Rham Theory], supervied by R. Sujatha.
- 2014-06-02 What is the Riemann Hypothesis and Why Do We Care?, for the 11th PIMS Young Researchers Conference at UBC, Vancouver, B.C.
- 2012-04-24 The Tate Module: An Introduction, MATH 620E at UBC [seminar course on elliptic curves], supervised by Professor M. Bennett.
- 2012-01-31 Mordell's Theorem, Part I, for Math 620E at UBC [seminar course on elliptic curves], supervised by Professor M. Bennett
- 2011-09-20 Weil Representations for \({\operatorname{GL}}_{2}({\mathbb{F}}_{p})\) , for MATH 529A at UBC [seminar course on representation theory of \({\operatorname{GL}}_{2}({\mathbb{Q}}_{p})\)], co-supervised by Professors L. Silberman, J. Gordon, and W. Casselman.
- 2011-09-13 Representations of \({\operatorname{GL}}_{2}({\mathbb{F}}_{p})\), for MATH 529A at UBC [seminar course on representation theory of \({\operatorname{GL}}_{2}({\mathbb{Q}}_{p})\) , co-supervised by Professors L. Silberman, J. Gordon, and W. Casselman.
- 2011-08-26 A Feast of \(L\)-functions, for MATH 529 at UBC [seminar course on Tate's thesis], co-supervised by Professors L. Silberman, J. Gordon, and W. Casselman.
- 2011-03-28 Non-vanishing Properties of \(L\)-functions at the Critical Point, for MATH 613E at UBC [seminar course in analytic number theory], supervised by Professor G. Martin.
- 2011-02-13 Selberg's Proof of \(\zeta(s)\) with a Finite Proportion of Zeros on the Critical Line, for MATH 613E at UBC [seminar course in analytic number theory], supervised by Professor G. Martin.
- 2011-02-11 Hardy's Proof of \(\zeta(s)\) with Infinitely Many Zeros on the Critical Line, for MATH 613E at UBC [seminar course in analytic number theory], supervised by Professor G. Martin.
- 2011-01-19 Topic Introduction: the Maier Matrix Method, for MATH 613E at UBC [seminar course in analytic number theory], supervised by Professor G. Martin.
Publications
I attended the workshop "Counting Arithmetic Objects (Ranks of Elliptic Curves)" at the Centre de Recherches Mathématiques (CRM) in mid-November, 2014, and I typed up the notes with Stanley Yao Xiao at Waterloo at that time (we also intended to add references to each lecture):
I am particularly grateful to Professor Greg Martin, for his amazing teaching skills in Math 223, for allowing me to give my first talk in his seminar class, and for collaborating with me to write our first reserch-level survey paper:
Original (mostly Mathematical) Research Discoveries and Novel (both Notational and Patented) Inventions.
Recently I have been exploring new (undiscovered to other people) mathematical fields, including (newest first):
- 2024-11-13 Hybrid Cohomological Theory: Integrating Linear and Non-Linear Algebraic Structures.
- 2024-11-13 Epita-Tetratica Theory: Transformations, Explicit Formulas, and General Structures.
- 2024-11-05 Intricate Relationships in Higher Infinite Hierarchical Epita-Tetra Structures within the Yang Program.
- 2024-11-05 Higher Infinite Hierarchical Yang-Langlands Program.
- 2024-11-05 Higher BSD-Bloch-Kato-Deligne-Beilinson Analogous Conjectures in Epita-Tetratica Theory.
- 2024-11-05 Higher Analogous Functions in Epita-Tetratica Theory via Knuth's Higher Arrows.
- 2024-11-05 Epita-Tetratica Theory: Prime Races and Oscillation Phenomena.
- 2024-11-04 Epita-Tetratica Theory and Higher Analogs of Zeta Functions.
- 2024-11-03 Rigorous Framework for Higher Order Quantizations.
- 2024-11-03 Generalized Non-Abelian Quantum Hall Effect and Extensions.
- 2024-11-03 Foundations of \(\mathbb{Y}_{3}(\mathbb{Q}_{p})\) Analysis.
- 2024-11-03 Foundations of \(\mathbb{Y}_{3}(\mathbb{C})\) Analysis.
- 2024-11-03 Foundational Development of \(\mathbb{Y}_{3}(\mathbb{R})\) Analysis.
- 2024-11-03 Applications of Generalized Spectral Sequences in Multi-Disciplinary Mathematical Computations.
- 2024-11-03 A Hybrid Zeta Function with \(p\)-adic and Complex Properties.
- 2024-11-03 Galois Representations for Infinite-Dimensional Modular Groups.
- 2024-11-02 Yang Number Systems Extended.
- 2024-11-02 Foundations of Kunth's Arrows Number Theory.
- 2024-11-02 Exponential Number Theory.
- 2024-11-01 Multi-Layered Yang Framework- Rigorous Structure and Properties.
- 2024-10-31 On the Existence of Irrational Zeros for the Riemann Zeta Function and General \(L\)-Functions.
- 2024-10-31 On Efficient Algorithms for Computing \(L\)-functions on the Critical Line.
- 2024-10-31 Motivic Sieves and Number Theory.
- 2024-10-31 A Generalized Definition of Analytic Conductor for \(L\)-Functions.
- 2024-10-30 Extending and Rigorously Developing \(\mathbb{Y}_{3}(\mathbb{F})\) Number Systems.
- 2024-10-28 Outline of Combinatorial Omni-Disciplinary Theory (CODT).
- 2024-10-22 Generalization of Prime Element Races in \(\mathbb{Y}_{n}(F)\) Number Systems.
- 2024-10-12 Automated Exploration and Expansion of Large Cardinal Axioms .
- 2024-09-18 The Necessity of AI in Future Mathematical Discoveries.
- 2024-09-13 Indefinite Extensions and Expansions of \(R\)-Motivic Stable Homotopy Theory.
- 2024-09-12 Indefinite Expansion of Quantum Motive-Stacks and Automorphic Forms- Integrating Derived Quantum Galois Cohomology and Noncommutative Quantum Cobordism.
- 2024-09-12 Mathematical Expansion on Macroscopic and Microscopic Scales .
- 2024-09-12 Fields Constructed from Automorphic Forms and Motives- Expansion into Quantum Motive-Stacks with Derived Lie Algebras, Automorphic Derived Noncommutative Modules, and Non-Archimedean Quantum Galois Cohomology .
- 2024-09-07 Further Development of Exotic Fields, Automorphic Forms, and Derived Motives .
- 2024-09-07 Mathematical Innovations and Computational Enhancements for Exotic Fields .
- 2024-09-07 Exploration of Exotic Fields Derived from \(\mathbb{Q}\) III .
- 2024-09-07 Exploration of Exotic Fields Derived from \(\mathbb{Q}\) II .
- 2024-09-07 Exploration of Exotic Fields Derived from \(\mathbb{Q}\) I .
- 2024-08-17 Non-Associative Galois Theory.
- 2024-08-05 Ultimate Generalization and Abstraction Frameworks.
- 2024-08-05 Field Extensions, Completions, and Galois Groups in Yang Number Systems.
- 2024-08-05 Enhancements to the Langlands Program through Galois Groups of Field Extensions.
- 2024-08-03 The Theory of General Infinitensors.
- 2024-08-03 Convoluted Mathematics: A Comprehensive Study.
- 2024-08-03 Algorithm for Enumerating Pairwise Disjoint Mathematical Foundations.
- 2024-08-03 Advancements in Fractoluminics: Theoretical Foundations and Practical Applications.
- 2024-08-03 Developed Fields in Arithmetic and Geometry: Ultimate Questions and Tools.
- 2024-07-31 Factorwebs- A New Construct in Number Theory.
- 2024-07-31 Advanced Study of Yang \(\mathbb{Y}_{\alpha}(F)\) Number Systems- New Tools, Techniques, and Operations.
- 2024-07-31 Advancements in Non-Associative Magmas- Theory, Applications, and Computational Methods.
- 2024-07-31 Algorithm for Enumerating Pairwise Disjoint Mathematical Foundations.
- 2024-07-31 Criteria for Evaluating Adherence to Peace Commitments in Political Parties and Religious Group.
- 2024-07-31 Developments in Homotopy Type Metric \(\infty\)-Theory and Homotopy Type Metric \(\infty\)-Homotopy Type Theory.
- 2024-07-31 Generalized Hierarchical Mathematical Structures- Infinite, Finite, Discrete, Continuous, Probabilistic, and More.
- 2024-07-31 Hierarchical Infinite Topology, Number Systems, and Number Theory- Extensions of Traditional Mathematical Structures.
- 2024-07-30 Theoretical Development of \({\mathbb{Y}}_{3}\) and Higher-Dimensional Generalizations
- 2024-07-30 Lunithor: A New Framework in Computational Astronomy and Celestial Mechanics.
- 2024-07-27 Xylorics: A New Mathematical Theory.
- 2024-07-27 Zebrythm: Investigating Rhythmic Properties and Periodic Behaviors of Numbers.
- 2024-07-27 Aurantis: A New Mathematical Framework.
- 2024-07-27 Advancements in Mathematical Tools and Theories.
- 2024-07-27 Iterated Projective Limits of meta-Categories: A Comprehensive Framework.
- 2024-07-26 Developed Fields in Arithmetic and Geometry: Ultimate Questions and Tools.
- 2024-07-26 A Rigorous Development of Cosmic Mathematics.
- 2024-07-26 Ultimate Generalization, Abstraction, and Refinement of Compressed Sensing (CS).
- 2024-07-25 Advancements in Fractoluminics: Theoretical Foundations and Practical Applications.
- 2024-07-25 Development of Vivarus: A Mathematical Field.
- 2024-07-25 Exalon: A Rigorous Development.
- 2024-07-25 Fusaris: The Study of Fusion-like Processes in Abstract Mathematics.
- 2024-07-25 Astralon: A Mathematical Exploration of Constellation-Like Structures.
- 2024-07-25 Terralon: Investigating Earth-Like Foundational Principles in Mathematical Models.
- 2024-07-25 Rigorous Development of Sylvaris: Mathematical Study of Tree-like Branching Patterns.
- 2024-07-25 Heliara: The Study of Helix-Like, Spiral Structures in Mathematics.
- 2024-07-25 Galionis: A Study of Galaxy-like Structures in Abstract Mathematical Contexts.
- 2024-07-25 Fractomagnetoneurodynamics: A Comprehensive Studys.
- 2024-07-25 Luminara: A Study of Luminous Properties in Abstract Mathematical Contexts.
- 2024-07-25 Nocturnis: Mathematical Exploration of Night-like, Dark Structures.
- 2024-07-25 Celestara: Mathematical Study of Celestial-Like Structures.
- 2024-07-25 Flarionis: Mathematical Study of Flare-Like Bursts.
- 2024-07-25 Development of Cylaris: A New Mathematical Field.
- 2024-07-25 Quasarias: The Study of Quasar-Like, Intense Energy Bursts in Mathematics.
- 2024-07-25 Thalasson: Mathematical Exploration of Ocean-like Wave Phenomena.
- 2024-07-25 Development of Ignarion: A Mathematical Framework for Combustion-like Processes.
- 2024-07-25 Rigorous Development of Ventara: A Mathematical Field for Wind-like Dispersive Behaviors.
- 2024-07-25 Foundations of Serenith.
- 2024-07-25 Mythonomics: Foundations and Applications.
- 2024-07-25 Development of Cylaris: A New Mathematical Field.
- 2024-07-24 Anomaly Detection and Chaos Theory: An In-Depth Analysis.
- 2024-07-24 The Field of Brontax: A Comprehensive Study.
- 2024-07-24 Glythorith: A Comprehensive Study.
- 2024-07-23 Introduction to Algeotrix: A Novel Mathematical Theory.
- 2024-07-23 Introduction to UnicodeLang.
- 2024-07-23 Varnomatics: The Study of Variable Norms in Abstract Algebraic Structures.
- 2024-07-23 Foundations of Jynorion Entities.
- 2024-07-22 Neurotropics.
- 2024-07-20 Theoretical Development of \(\mathbb{Y}_{3}\) and Higher-Dimensional Generalizations.
- 2024-07-18 Quantimorph Theory.
- 2024-07-18 Xylograph: A New Field in Number Theory.
- 2024-07-12 Novel Extensions and Integrations in Galois Topological Theory.
- 2024-07-05 The Eternal Truth of Mathematics: A Multifaceted Exploration.
- 2024-07-03 Periodic Table for Mathematical Structures.
- 2014-06-30 Consciousness.
- 2024-06-29 Phasorics: Advanced Studies in Abstract Phase.
- 2014-06-29 Iterative Processes of Algebraic Closures, Completions, and Yang \(\mathbb{Y}_{m}\) Number Systems: An Infinite Landscape of Fields and Rings.
- 2014-06-28 Generalized Infinite-Dimensional Weighted Non-Linear Homotopy-Invariant Boundary and Coboundary Maps.
- 2014-06-27 Exploration and Expansion of Yang \(\mathbb{Y}_{m}\) Number Systems.
- 2014-06-26 Fluxient Algebroids: Foundations and Basic Properties.
- 2024-06-29 Enigmagraphy: The Study of Paradoxical Spaces.
- 2024-06-22 Anomalotropy: Theory and Applications in Interdisciplinary Research.
- 2024-06-22 Development and Application of Cryptometrics,
- 2024-06-11 Yang Number Systems: An introduction.
- 2024-06-10 Counting Distinct 4-Dimensional Number Systems.
- 2024-06-02 Fluxient Algebroids: Foundations and Basic Properties.