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.

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).

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.

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.

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I enjoy working with students from all backgrounds interested in number theory and representation theory. If you would like to write a thesis or do a research/directed reading project, we could begin a discussion by e-mail.

I co-organize the Arithmetic Geometry Preprint Seminar (Thursday 2:00-3:30 pm).

I maintain the problem list for the AIM workshop arithmetic intersection theory on Shimura varieties .

Stanford Number Theory Seminar (Monday 2:30-3:30 pm, with lunch).

Joint Berkeley-Stanford Algebraic Number Theory (Tuesday 2:30-4:30 pm).

Stanford Topology Seminar (Tuesday 4 pm).

Stanford Algebraic Geometry Seminar (Friday 11:30 am).

Math Seminars at UC Berkeley

Stanford Tea time: 3:30 pm Weekdays (+ 5:30 pm Friday).

Some topic courses in Stanford: Math 269 TuTh (1:30-2:50 pm), Spring 2024.

I co-organized Euler System Program Special Seminar with Ben Howard , Spring 2023.

I co-organized MIT STAGE Seminar (Spring 2020 -- Fall 2021) .

I organized a reading seminar on Rapoport--Zink spaces , Fall 2021.

I mentered several MIT undergraduates to learn and explore mathematics e.g. applied category theory via MIT directed reading program (2019-2021).

I was a math coach of Tsinghua Mathcamp for high school students in Summer 2019.

2024: Winter quarter (Jan 8-Mar 15), Spring quarter (Apr 1-June 5), Fall (Sep 23 - Dec 13).

2025: Winter quarter (Jan 6-Mar 21), Spring quarter (Mar 31-June 11), Fall (???).

I could do zoom meetings during 8 am - 11:30 pm (Pacific Standard Time). I usually arrive at the departmant around 9 am.

Spherical varieties and L-functions (2021). MIT spherical variety learning seminar.

Derived category of mixed complexes and Weil II (2021). Michigan BBDG seminar.

Perfectoidization and perfect prismatic complex (2021). MIT STAGE seminar.

The Drinfeld half plane (2021).

Galois category and Riemann existence theorem (2021). Princeton Exodromy seminar.

Frobenius on p-adic modular forms and the theta operator (2020). MIT STAGE seminar.

Weight spectral sequence and Weil conjecture (2020). MIT STAGE seminar.

p-divisible groups and Hodge structures (2019). MIT STAGE seminar.

A rough introduction to Lubin-Tate spaces (2018).

Overleaf

MacTutor

Mathscinet (AMS remote access available)

Stacks Project

The Automorphic Project

Kerodon, an online resource for homotopy-coherent mathematics

Mathematics Genealogy Project

Mathoverflow (see e.g. helpful answers of P. Scholze, W. Sawin and other people)

Mathjobs

Conferences in arithmetic geometry

quiver: a modern commutative diagram editor

LMFDB - The L-functions and modular forms database

Numdam, the French digital mathematics library

American Mathematical Society

European Mathematical Society

Mathematisches Forschungsinstitut Oberwolfach

African Institute for Mathematical Sciences

IAS/Park City Mathematics Institute

Institute of the Mathematical Sciences of the Americas

International Mathematical Olympiad

The University of Chicago Mathematics REU

Essential Number Theory

ICM 2026 in Philadelphia, USA

Virtual ICM 2022

Equipe Formes Automorphes, IMJ-PRG

PU/IAS Number Theory Seminar

Harvard Number Theory Seminar

Number Theory Web Seminar

UCSD Number Theory Seminar

Johns Hopkins Number Theory Seminar

Johns Hopkins University & University of Maryland Algebra and Number Theory Day

MIT Number Theory Seminar

MIT Lie Groups Seminar

MIT Juvitop Seminar

Boston College NT/RT Seminar

Boston University Number Theory Seminar

Weekly seminars at Columbia University

Events and Activities | Institute for Advanced Study

M2 "Mathématiques fondamentales"

Caltech Number Theory Seminar

UCLA Number Theory Seminar

Number Theory / Representation Theory Seminar, University of Wisconsin - Madison

Mathematical Events in Bonn

Arithmetic Geometry and Representation Theory in Münster

Seminar "Algebra and Number Theory" at University of Vienna

International Seminar on Automorphic Forms, TU Darmstadt

Upcoming conferences in algebraic geometry by Ravi Vakil

Conferences in arithmetic geometry by Kiran Kedlaya

Geometric Langlands page by David Ben-Zvi

Suggestions for mathematical writing and speaking by Bjorn Poonen

Finding your path by Rahul Pandharipande

Selected Articles on mathematicians by Allyn Jackson

The work of Robert Langlands by James G. Arthur

Cohomology of Arithmetic Groups by Günter Harder

Towards a theory of local Shimura varieties by Michael Rapoport, Eva Viehmann

On the Shafarevich and Tate conjectures for hyperkähler varieties by Yves André

Maass cusp forms with integer coefficients by Peter Sarnak

Lecture notes on Hodge theory by Phillip Griffiths

Lecture notes on derived algebraic geometry / algebraic stacks by Adeel A. Khan

Lecture notes on modularity lifting theorems by Toby Gee

Lecture notes on stacks and moduli by Jarod Alper

Arizona Winter School 2022 Lectures by Akshay Venkatesh

RAMpAGe Seminar

Kolyvagin's work on modular elliptic curves by Benedict H Gross

Foliations in Moduli Spaces of Abelian Varieties and Dimension of Leaves by Frans Oort

Values of Zeta Functions and Their Applications by Don Zagier

Lattices and L-functions from nothing by Andrew V. Sutherland

Tannaka Reconstruction and Quasi-Coherent Stacks, Representability Theorems in Spectral Algebraic Geometry by Jacob Lurie

Model Theory and Differential Equations by Joel Nagloo

Modular Functions and Special Cycles by Maryna Viazovska

Survey on Derived Symplectic Geometry by Damien Calaque

Survey on Algebraic Dilatations by Adrien Dubouloz, Arnaud Mayeux and João Pedro dos Santos

Survey on Affine Hecke Algebras and their representations by Maarten Solleveld

Survey on p-adic L-functions by Joaquín Rodrigues Jacinto and Chris Williams

ICM address On the Brumer-Stark Conjecture and Refinements by Samit Dasgupta and Mahesh Kakde

ICM address Topological Field Theory, Higher Categories, and Their Applications by Anton Kapustin

ICM address Categorification: tangle invariants and TQFTs by Catharina Stroppel

ICM address The distribution of values of zeta and L-functions by Kannan Soundararajan

ICM address No Where to Go But High: A Perspective on High Dimensional Expanders by Roy Gotlib and Tali Kaufman

ICM address Statistics of Number Fields and Function Fields by Akshay Venkatesh and Jordan S. Ellenberg

Youtube Math Channel Institut des Hautes Études Scientifiques (IHÉS)

Youtube Math Channel 3Blue1Brown for everyone

A TED talk on procrastination

A page on Most cited mathematicians

I believe that computers are able to help humans with proofs, computations and finding mathematical patterns. See also the talk by Michael R. Douglas at Western Hemisphere Colloquium on Geometry and Physics.

I am interested in increasing accessibility and descreasing language barriers in mathematics. I helped with Chinese translation in Visual Mathematical Dictionary project. I could also help translate math papers in Japanese and French.

The Banana Space is an online mathematics wiki in Chinese, similar to the nLab .

I am interested in connections between mathematics and arts. See works of Maurits Escher and Journal of Mathematics and the Arts . I did some mathematical paintings.

I am interested in serious recreational mathematics. I enjoy logic puzzles and puzzle games. With some people I made a free translation of the puzzle game Bean and Nothingness , designed by math PHDs from University of Michigan. I also enjoy math jokes, see for instance My Favorite Math Jokes by Tanya Khovanova.

I was the main editor of the student journal He Si during 2017-2018.

I won the gold medal in the overall part of Yau College Student Mathematics Contest in 2017.

I was a volunteer for Strings 2016 Conference in Beijing.

For me, doing math is a life-long Odyssey (see also A Mathematical Odyssey ). I also enjoy traveling (thinking and doing math along the way). I have been to more than 20 countries.