Research & Publications

Research Interests

My work has evolved from a focus on Algebraic Geometry and Scheme Theory to the foundational intersections of Algebraic Logic and Universal Algebra. Throughout my research, I have sought to identify the structural laws that govern both geometric transformations and logical systems.

In the earlier phase of my career, I focused on birational geometry, specifically the factorization of morphisms, the theory of ramification, and the Kodaira dimension of algebraic varieties. My more recent work utilizes Clone Theory as a framework to algebraize first-order logic and the lambda calculus, aiming to provide a rigorous algebraic treatment of quantifier theories and completeness theorems.

Selected Publications & Preprints

1987 Kodaira dimension of algebraic function fields American Journal of Mathematics 109, no. 4, 669–693

PDF

1988 An invariant approach to the theory of logarithmic Kodaira dimension of algebraic varieties Bulletin of the American Mathematical Society (N.S.) 19, no. 1, 319–323

PDF

1988 Factorization of birational morphisms with fiber dimension bounded by 1 (with Tie Luo) Mathematische Annalen 282, no. 4, 529–534

1992 Ramification divisor of regular schemes Algebraic geometry and algebraic number theory (Tianjin, 1989--1990), 77–91

1993 Factorization of birational morphisms of regular schemes Mathematische Zeitschrift 212, no. 4, 505–509

1994 On the ramification theory of regular schemes Proceedings of the American Mathematical Society 120, no. 2, 339–343

1995 Formal Kähler different and uniform bound for morphisms of regular local rings Communications in Algebra 23, no. 1, 165–176

1995 On the geometry of metric sites Journal of Algebra 176, no. 1, 210–229

2007 Clones and Genoids in Lambda Calculus and First Order Logic

arXiv:0712.3088 [math.LO]

2008 Clone Theory: Its Syntax and Semantics, Applications to Universal Algebra, Lambda Calculus and Algebraic Logic

arXiv:0810.3162 [cs.LO]

2009 Clone Theory and Algebraic Logic

arXiv:0907.4531 [cs.LO]

2010 A New Approach to Abstract Machines - Introduction to the Theory of Configuration Machines

arXiv:1007.3303 [cs.LO]

2011 Universal Algebra and Mathematical Logic

arXiv:1104.4606 [math.LO]

2013 Algebraic Logic, I: Quantifier Theories and Completeness Theorems

arXiv:1301.0667 [math.LO]