Zhiyu Zhang's Homepage

I am a Szegö Assistant Professor at Stanford University since Fall 2023. I work with Xinwen Zhu . Office: Room 382-C, Sloan Mathematical Center.

Emails: zyuzhang[at]stanford.edu / zhiyuzhangmath[at]gmail.com .
Postdoctoral researcher (Spring 2023, Simons Laufer Mathematical Sciences Institute).
PHD (2018-2022, MIT) under the supervision of Wei Zhang (and Zhiwei Yun).
Undergraduate (2014-2018, Tsinghua), visiting École normale supérieure (Spring 2018).

Research

I am interested in number theory, representation theory, arithmetic geometry / topology, and Hodge theory, from absolute and relative perspectives. I study moduli spaces (e.g. moduli of motives and Hitchin stacks), tautological classes / correspondences and special functions / cycles / sheaves on them, giving better understandings of moduli e.g. their cohomology. I am interested in enumerative invariants e.g. volumes and L-functions, and arithmetic applications. I also study automorphic forms and relative duality in Langlands program. I have been studying Gross-Zagier formulas, Weil representations, Fourier coefficients, explicit mod p geometry, arithmetic fundamental lemmas and theta correspondences over p-adic numbers, with ramifications (necessary for applications: no level 1 modular form of weight 2 exists).

Currently I enjoy thinking about
  • discovering more cycles and q-series e.g. mirabolic cycles. Applications: arithmetic volumes, Kudla program, Euler systems, Tate and Bloch-Kato conjecture.
  • automorphic functions: L-functions, Fourier coefficients, explicit constructions and transfers, and how to think geometrically.
  • period integrals, relative trace formulas, orbital integrals: p-adic / motivic variants, fundamental lemmas, germs, spectral analogs, spherical and ramified countings.
  • exploring new geometry of cycles and moduli spaces beyond classical algebraic geometry, and combine it with p-adic and archimedean geometry.
  • generalizing geometric pictures, local and global duality [BZSV] to tame and wild ramifications, to non-split groups and beyond de Rham settings.

    I hope to formulate new conjectures and geometric objects (based on new examples) and prove new theorems (there are still many exciting things to be further explored). I enjoy the compatibility of pictures over p-adic numbers, real numbers, rational numbers and topology (e.g. from local systems on 3-manifolds and physics).

    Automorphic forms, L-functions, periods and relative Langlands duality
    1. On general twisted Gan-Gross-Prasad conjectures and twisted Asai L-functions (with Weixiao Lu and Danielle Wang). In preparation.
    Structures and invariants of Hamiltonian spaces and quantization
    Cycles on moduli spaces and arithmetic Langlands duality
    2. Vector-valued local arithmetic Siegel--Weil formulas for quadratic forms with levels. (with Qiao He and Baiqing Zhu). In preparation.
    3. Non-reductive special cycles and twisted arithmetic fudamental lemmas . Arxiv:2406.00986. Submitted.
    4. Kudla-Rapoport conjecture at unramified primes with maximal parahoric level (with Sungyoon Cho and Qiao He). Arxiv:2312.16906. Submitted.
    5. Arithmetic transfers, modularity of arithmetic theta series and geometry of local--global Shimura varieties at parahoric levels. PHD thesis at MIT, 2022.
    6. Maximal parahoric arithmetic transfers, resolutions and modularity. Arxiv:2112.11994 . Duke Mathematical Journal , to appear.
    Taylor expansion, Shtukas, Hitchin moduli and higher Langlands duality
    7. Relative Fundamental Lemmas for Spherical Hecke Algebras and Multiplicative Hitchin Fibrations: the Jacquet--Rallis Case (with X. Griffin Wang). Arxiv:2408.15155. Submitted.
    Applications of categorical Langlands and spectral actions on moduli spaces
    Geometry, decomposition and invariants of moduli spaces and cycles
    8. On unitary Shimura varieties at ramified primes. (With Yu Luo and Andreas Mihatsch). In preparation.

    Teaching

    --MATH 145, Algebraic Geometry, Winter 2025.
    --MATH 216B, Graduate Algebraic Geometry, Winter 2025.
    MATH 216A Graduate Algebraic Geometry, Fall 2024, TuTh 12:00-01:20 PM, Room 381T. Website .
    MATH 263A Topics in Representation Theory, Function Field Arithmetic, Fall 2024, TuTh 10:30-11:50 AM, Room 381T. Website (under construction).
    MATH 145 Algebraic Geometry, Spring 2024, MWF 9:30-10:20 AM. Website .
    MATH 263C Topics in Representation Theory, Relative Langlands duality, Spring 2024, MWF 10:30-11:20 PM. Website .

    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).

    Seminars, Service and Outreach

    I welcome review invitations and have been reviewers for several mathematical journals.
    I enjoy working with students from all backgrounds interested in number theory, algebraic geometry and representation theory. We could begin a discussion by e-mail.
    Stanford Number Theory Seminar (Monday 2-3 pm, with lunch).
    Stanford Topology Seminar (Tuesday 4 pm).
    Stanford Algebraic Geometry Seminar (Friday 11:30 am).
    Joint Berkeley-Stanford Algebraic Number Theory (Tuesday 2:30-4:30 pm).
    Math Seminars at UC Berkeley.
    Stanford Tea time: 3:30 pm Weekdays (+ 5:30 pm Friday).
    Math 269 TuTh (1:30-2:50 pm), Spring 2024. MIT Number Theory Seminar
    MIT Lie Groups Seminar

    I co-organized 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 .
    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.

    Time Schedule

    2023: Fall (Sep 26-Dec 8).
    2024: Winter (Jan 8-Mar 15), Spring (Apr 1-June 5), Fall (Sep 23 - Dec 6).
    2025: Winter (Jan 6-Mar 14), Spring (Mar 31-June 4), Fall (Sep 22 - Dec 12).
    2026: Winter (Jan 5-Mar 22), Spring (Mar 30-June 10).

    News / Travels

  • [July 23-30 2026] International Congress of Mathematicians (ICM) 2026 , Philadelphia, USA.
  • During the year of 2025, I will be on the job market.
  • [Sep 2025-April 2026] Special Year on Arithmetic Geometry, Hodge Theory, and o-minimality , IAS, Princeton.
  • [June 23-July 4, 2025] Enumerative Geometry and Geometric Representation Theory , Università di Pisa, Italy.
  • [April 7-11, 2025] Hot Topics: Interactions between Harmonic Analysis, Homogeneous Dynamics, and Number Theory , SLMSI, Berkeley.
  • [March 8-12, 2025] Arizona Winter School 2025: p-adic groups , Tucson, USA.
  • [Jan 27-31, 2025] Motives and mapping class groups , Pasadena, California.
  • [Dec 9-13, 2024] Hot Topics: Life after the Telescope Conjecture , SLMSI, Berkeley.
  • [Nov 9-11, 2024] Representation Theory Days (in honor of George Lusztig) , MIT, Room 2-190.
  • [Oct 28–Nov 1, 2024] Mathematics and Machine Learning Program, Harvard.
  • [Oct 20-25, 2024] Oberwolfach Seminar: Reduction of Arithmetic Varieties Germany.
  • [10/8/2024] Seminar talk on irrationality of zeta(3) via Eichler integrals and modular forms on Gamma_0(6) (Apery, Beukers) and Zagier’s sporadic sequence A and C at Joint Berkeley-Stanford Algebraic Number Theory .
  • [Sep 30-Oct 4, 2024] New Advances in the Langlands Program: Geometry and Arithmetic Clay Mathematical Institute, University of Oxford
  • [Sep 9-13, 2024] Visit and Seminar talks at Number Theory / Representation Theory Seminar , University of Wisconsin-Madison.
  • [Aug 2024] A new paper with X. Griffin Wang on spherical relative Fundamental Lemmas .
  • [July 22-26, 2024] Conference on "Arithmetic Geometry" in Honour of Gerd Faltings' 70th Birthday , Bonn, Germany.
  • [July 7-12, 2024] Conference talk at BIRS workshop Branching Problems for Representations of Real, p-Adic and Adelic Groups , Kelowna, Canada.
  • [July 1-5, 2024] Conference on Arithmetic Geometry and Applications , ICTP, Italy.
  • [June 24-28, 2024] Workshop on Number Theory and Physics , ICTP, Italy.
  • [June 18-21, 2024] Modular Forms, L-functions, and Eigenvarieties , Paris.
  • [June 9-15, 2024] Conference Talk at Simons Symposium: Periods and L-values of Motives (2024) , Krün, Germany.
  • [June 3-8, 2024] Summer School and Workshop on Relative Langlands Duality , University of Minnesota.
  • [May 20-24, 2024] 36th Automorphic Forms Workshop , Oklahoma State University.
  • [April 27-28, 2024] Western Algebraic Geometry Symposium (WAGS), UC Davis.
  • [4/22/2024] Seminar talk at Stanford Number Theory seminar.
  • [4/19/2024] Seminar talk at UCSB Number Theory seminar, Santa Barbara.
  • [April 6-7, 2024] Conference talk on the AMS Sectional Meetings , Howard University, Washington, DC.
  • [3/27/2024] Seminar talk at Johns Hopkins Number Theory Seminar .
  • [March 26-29, 2024] Arithmetic Quantum Field Theory Conference , Harvard.
  • [March 11-15, 2024] Workshop on p-adic Arithmetic Geometry , Princeton.
  • [3/12/2024] Seminar talk on Hasse invariants, canonical subgroups, overconveregent anticanonical towers at Joint Berkeley-Stanford Algebraic Number Theory .
  • [March 2-6, 2024] Senior participant (feel free to ask me for help) of Arizona Winter School 2024: Abelian Varieties .
  • [2/27/2024] Seminar talk at BIMSA-YMSC Tsinghua Number Theory Seminar.
  • [2/22/2024] Seminar talk at Number Theory / Representation Theory Seminar, University of Wisconsin - Madison .
  • [Jan 29-Feb 2, 2024] AIM workshop analytic, arithmetic, and geometric aspects of automorphic forms , Pasadena, California.
  • [1/26/2024] Seminar talk at Automorphic Forms and Arithmetic , Columbia University.
  • [1/18/2024] Seminar talk at Arithmetic Geometry Preprint Seminar , Stanford.
  • [Jan 8-12, 2024] AIM workshop arithmetic intersection theory on Shimura varieties , Pasadena, California.
  • [Jan 3-6, 2024] Joint Mathematics Meetings 2024 , San Francisco.
  • [December 2023] Monodromy and Its Applications , Princeton.
  • [11/30/2023] Seminar talk at Caltech Number Theory Seminar .
  • [November 2023] Workshop on p-adic Arithmetic Geometry , Princeton.
  • [10/24/2023] Seminar talk on deformations of absolute Hodge cycles at Joint Berkeley-Stanford Algebraic Number Theory .
  • [October 2023] Conference talk at Arithmetic Algebraic Geometry, a conference on the occasion of Michael Rapoport's 75th birthday at Münster.
  • [September 2023] Conference Arithmétique des formes automorphes , Paris.
  • [August 07-11, 2023] Confernece Global Langlands, Shimura varieties, and shtukas , Bonn.
  • [July 2023] Lecture series on Arithmetic geometry, hermitian lattices and theta correspondences , Peking University.
  • [July 2023] Conference talk at Tianyuan Mathematics Research Center, Kunming.
  • [June 2023] Conference talk at Arithmetic Algebraic Geometry conference, the Chinese University of Hong Kong, Hong Kong.
  • [May 2023] Algebraic geometry and cohomology in mixed characteristic , Northwestern University.
  • [March 2023] Seminar talk at ES Special Seminar , MSRI, Berkeley.
  • [March 2023] Conference Shimura Varieties and L-functions (in honor of Shouwu Zhang) , MSRI, Berkeley.
  • [March 2023] Arizona Winter School 2023: Unlikely Intersections , Arizona.
  • [03/01/2023] Seminar talk on Sharifi's Conjecture III: the Results of Fukaya-Kato at ES Learning Seminar, MSRI, Berkeley.
  • [01/04/2023] Conference talk on my thesis at Joint Mathematics Meetings 2023 , Boston.
  • During Fall 2022, I was on the job market.
  • [Oct-Dec 2022] Scheduled visit to Max Planck Institute for Mathematics (canceled due to visa and Covid-19).
  • [August 2022] Conference talk at Community-building in the Langlands Program (CLAP) , Bonn.
  • [July 2022] 2022 IHES Summer School on the Langlands program , Paris.
  • [July 6-14, 2022] (Virtual) International Congress of Mathematicians 2022 .
  • [July 1-15, 2022] (Virtual) Representations and Characters: Revisiting the Works of Harish-Chandra and André Weil .
  • [May 2022] Seminar talk at Lie Groups and Representation Theory , University of Maryland.
  • [April 2022] Seminar talk at Algebra and Number Theory Seminar , University of Arizona.
  • [April 2022] Seminar talk at Algebra and Number Theory Seminar , Yale University.
  • [April 2022] Seminar talk at Boston College NT/AG Seminar .
  • [March 2022] Arizona Winter School 2022: Automorphic Forms Beyond GL2, Arizona.
  • [March 2022] CURRENT DEVELOPMENTS IN MATHEMATICS 2021-22 , Harvard.
  • [November 2021] (virtual) Basic Functions, Orbital Integrals, and Beyond Endoscopy (Online) (21w5228) , BIRS workshop.
  • [July 2021] (virtual) Theta Series: Representation Theory, Geometry, and Arithmetic , The Fields Institute, Toronto.
  • [April 2021] (virtual) Arbeitsgemeinschaft: Derived Galois Deformation Rings and Cohomology of Arithmetic Groups , Oberwolfach, Germany.
  • [04/24/2021] (virtual) Seminar talk at Séminaire Mathjeunes , Paris.
  • [Jan 4-8, 2021] (virtual) Arithmetic intersection theory on Shimura varieties, American Institute of Mathematics, San Jose, California.
  • [March 7-11, 2020] Arizona Winter School 2020: Nonabelian Chabauty.

    Friends and Collaborators

    John R. L. Anderson , Sebastian Bartling , Chenjing Bu , Evan Chen , Lin Chen , Sungyoon Cho , Tony Feng , Andrew Graham , Linus Hamann , Qiao He , Pol van Hoften , Yuanyang Jiang , Rahul Krishna , Spencer Leslie , Qirui Li , Siyan Daniel Li-Huerta , Weixiao Lu , Joey Yu LUO , Lucas Mason-Brown , Andreas Mihatsch , Gyujin Oh , Congling Qiu , Yousheng Shi , Naomi Sweeting , Matteo Tamiozzo , Longke Tang , David Urbanik , Danielle Wang , Griffin Wang , Boya Wen , Zhixiang Wu , Jianqiao Xia, Kai Xu , Liyang Yang , Ziquan Yang , Fan Ye , Murilo Corato Zanarella , Mingjia Zhang , Roy Zhao , Baiqing Zhu , Jialiang Zou ,

    Talk / Lecture Notes

    Dehn twists, and full monodromy of Kodaira–Parshin families (2022), MIT STAGE seminar.
    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).

    Other Useful Info

    Links Arxiv
    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, D. Loeffler 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 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
    Between electric-magnetic duality and the Langlands program 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
    Towards a Jacquet-Langlands correspondence for unitary Shimura varieties David Helm
    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
    The arithmetic and topology of differential equations by Don Zagier
    Elliptic Modular Forms and Their Applications by Don Zagier
    Lecture notes on perfectoid Shimura varieties by Ana Caraiani
    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
    Derivatives of Eisenstein series and Faltings heights by S. Kudla, M. Rapoport, T. Yang. Modular Functions and Special Cycles by Maryna Viazovska
    Selected talks by A. Okounkov
    Survey on Some perspectives on Eisenstein series by Erez Lapid
    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 Cohomology of Arithmetic Groups by Akshay Venkatesh
    ICM address Statistics of Number Fields and Function Fields by Akshay Venkatesh and Jordan S. Ellenberg
    ICM address Hecke algebras and harmonic analysis by Eric Opdam
    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

    Other Projects

    Links Google Scholar Page
    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.