Zhiyu Zhang's Homepage
I am a Szegö Assistant Professor at Stanford University since Fall 2023. I work with Richard Taylor and Xinwen Zhu .
Emails: zyuzhang[at]stanford.edu / zhiyuzhangmath[at]gmail.com .
Office: Room 382-C, Sloan Mathematical Center.
Postdoctoral researcher (Spring 2023) at MSRI, the Euler System program.
PHD at MIT (2018-2022) under the supervision of Wei Zhang (and Zhiwei Yun).
Undergraduate at Tsinghua Univeristy (2014-2018), visiting student at École normale supérieure (Spring 2018).
My full CV and Research Statement are available by request.
Teaching
Teaching at Fall 2024 (Course TBD).
MATH 145 Algebraic Geometry, Spring 2024, MWF 9:30 AM -- 10:20 AM. Website (under construction).
MATH 263C Topics in Representation Theory, Spring 2024, MWF 10:30 -- 11:20 PM. Website (under construction).
Previous teaching at MIT: Recitation Instructor for 18.06 Linear Algebra (Fall 2021). Teaching Assistant for 18.726 Algebraic Geometry II, 18.706 Algebra II (Spring 2022), 18.786 Number Theory II, 18.737 Algebraic Groups (Spring 2021), 18.785 Number Theory I (Fall 2020), 18.102 Introduction to Functional Analysis (Spring 2020), 18.705 Commutative Algebra (Fall 2019).
Research
I am broadly interested in number theory, representation theory, arithmetic geometry and Hodge theory. I study them from both the absolute and relative perspective. I study cycles (e.g. curves) on moduli spaces and the local-global compatibility (e.g. fundamental lemmas).
Topics: L-functions, Diophantine equations, period integrals, algebraic cycles, Shimura varieties, Iwasawa theory, mod p / p-adic / Arakelov geometry, BSD conjecture, Langlands program, trace formulas, theta correspondences and Kudla program (quantum arithmetic geometry). For people interested in related research, we could begin a discussion by appointment or e-mail.
A detailed description of my research
I study L-functions from multiple perspectives and arithmetic invariants of motives ("heights'' e.g. Chow groups counting sizes of solutions to Diophantine equations), a central topic in number theory with many exciting things to be further explored e.g. Hasse-Weil conjecture, Riemann hypothesis, Deligne conjecture, Braverman-Kazhdan program and Beilinson-Bloch-Kato conjecture. Using cycles on moduli spaces and automorphic representation theory, towards these conjectures I study period integrals, Gross-Zagier type formulas, local arithmetic invariants, Euler systems, and non-vanishing of analytic invariants.
I study Langlands program and related representation theory. A fundamental topic in mathematics is to show certain maps are surjective / injective with matching invariants. I am interested in the automorphic spectrum, period integrals and L-functions. Period integrals are closely related to Langlands functoriality. I am interested in the relative Langlands duality and categorical Langlands, and their relations to topology, geometrization on curves and mathematical physics (TQFT, spin structures, and symplectic geometry). I also compute local invariant integrals at unramified and ramified primes, with applications to representations of p-adic groups via spectral method. Ramification is unavoidable in practice and gives understandings of the full L function e.g. the functional equation. Specific examples include theta correspondences and Gan-Gross-Prasad conjectures. A specific topic is construction of integral representations of L-functions and regularization e.g. Kronecker limit formula, and the use of several Eisenstein series.
I study moduli spaces and special cycles / functions on them. I study their geometry (with translational symmetry), arithmetic, motives / cohomology and related trace distributions. These moduli spaces provide stages for Langlands program and (geometric and automorphic) representation theory, where these cycles enhance the whole story (e.g. via modularity or arithmetic intersections). These cycles produce interesting arithmetic invariants of moduli spaces (e.g. degrees and volumes). I study (relative) trace formulas which have many applications in number theory e.g. endoscopic classifications / Langlands-Kottwitz-Scholze method. I study magic connections between cycles and analytic invariants e.g. arithmetic fundamental lemmas and arithmetic Siegel-Weil formulas. I am also interested in related complex / p-adic / Arakelov geometry, derived enhancements (objects over a family), singular terms and boundary degenerations. Specific topics: Kudla program and modularity of arithmetic theta series for general test functions; construction of new moduli spaces relative to pure Shimura varieties; applications of compactifications; explicit descriptions of the mod p geometry at Bruhat-Tits levels (e.g. when do they contain projective spaces) with applications.
With applications to cycles on moduli spaces, p-adic L functions and automorphic forms, I am also interested in the recent development of p-adic / mod p geometry (p-adic analogs of notions in complex analytic geometry). Sometimes, the correct geometric stages shall be analytic. Local geometric results may lead to global arithmetic results via local uniformizations and local-global compatibility. I am interested in the powerful coherent theory and arithmetic Langlands program by Zhu and Fargues-Scholze. Specific topics: Bun_G, spectral action and geometric Satake with semi-global applications; relative geometry of integral models with levels; analytic geometry of (overconvergent) ordinary locus with semi-global applications; new definitions of Shimura varieties.
I am also interested in the arithmetic geometry of motives / curves / polynomial equations that are not directly related to L-functions. Topics include arithmetic of quadratic lattices (e.g. ), arithmetic reduction modulo primes, arithmetic finiteness / boundedness and arithmetic statistics. Firstly, transcendental uniformization maps and (complex and p-adic) Hodge theory, big monodromy results for cycles on moduli spaces are powerful to related Shafarevich and unlikely intersections questions. Specific topics inlcude the use of mixed Shimura varieties, the relation between arithmetic intersection and exceptional arithmetic reductions. Secondly, arithmetic geometry of motives (and maps between motives) are closely related to their heights (at least in an average sense), e.g. Faltings theorem and Bogomolov conjecture. A specific topic is the use of Mumford-Tate groups of abelian varieties. Thirdly, it's interesting to classify motives using stratification on related moduli spaces and study their different behaviors of arithmetics e.g. supersingular elliptic curves.
Applications : Diophantine equations e.g. twisted Fermat equations, analytic number theory, unlikely intersections and transcendence, equidistribution and dynamics, counting problems / enumerative geometry, Tate conjecture and Hodge conjecture, explicit class field theory e.g. Kronecker’s Jugendtraum, arithmetic finiteness e.g. class number one, local and global Langlands functoriality, automorphic spectrum, Galois representations e.g. level raisings, (p-adic) L-functions e.g. constructions, algebraic curves, abelian varieties e.g. Faltings heights, moduli of curves, automorphic motives, arithmetic of quadratic forms e.g. Bhargava-Hanke 290 Theorem, inverse Galois problem, computation of Pi by Chudnovsky brothers, algebraic topology, geometric topology e.g. affine springer fibers and knots, measure theory e.g. the Ruziewicz Problem, combinatorics e.g. Ramanujan graph, logic e.g. Hilbert's Tenth Problem, spectral theory and L^2-functions, mathematical physics e.g. partition functions and Monstrous moonshine, multiple zeta values and Feynman amplitudes...