Shou-Wu Zhang

{{Infobox scientist

| name = Shou-Wu Zhang

| native_name = 张寿武

| image = Shou-Wu Zhang.JPG

| image_size =

| caption = Shou-Wu Zhang in 2014

| birth_date = {{birth date and age|1962|10|09}}

| birth_place = Hexian, Anhui, China

| death_date =

| death_place =

| nationality = American

| fields = Mathematics

| workplaces = {{ubl|Princeton University|Columbia University|Institute for Advanced Study}}

| alma_mater = Sun Yat-sen University (BS)
Chinese Academy of Sciences (MS)
Columbia University (PhD)

| thesis_title = Positive Line Bundles on Arithmetic Surfaces

| thesis_url = https://www.jstor.org/stable/2946601

| thesis_year = 1991

| doctoral_advisor = Lucien Szpiro

| academic_advisors = Wang Yuan

| doctoral_students = {{ubl|Wei Zhang|Xinyi Yuan|Tian Ye|Yifeng Liu}}

| notable_students = {{ubl|Bhargav Bhatt}}

| known_for = {{ubl|Arakelov theory|Arithmetic dynamics|Bogomolov conjecture|Gross–Zagier theorem}}

| awards = {{collapsible list|title=List of Awards|Sloan Fellowship (1997)|Morningside Medal (1998)|Clay Prize Fellow (2003)|Guggenheim Fellow (2009)|American Academy of Arts and Sciences Fellow (2011)|American Mathematical Society Fellow (2016)}}

}}

Shou-Wu Zhang ({{zh|c= 张寿武|p=Zhāng Shòuwǔ}}; born October 9, 1962) is a Chinese-American mathematician known for his work in number theory and arithmetic geometry. He is currently a Professor of Mathematics at Princeton University.

Biography

=Early life=

Shou-Wu Zhang was born in Hexian, Ma'anshan, Anhui, China, on October 9, 1962.{{cite web|url=http://www.amss.ac.cn/xwdt/zhxw/2011/201111/t20111111_3394825.html |title=从放鸭娃到数学大师 |trans-title=From ducklings to mathematics master |publisher=Academy of Mathematics and Systems Science |date=11 November 2011 |accessdate= 5 May 2019 |language=zh}}{{cite web|url=https://news.sina.com.tw/article/20190503/31155020.html |title=專訪張壽武：在數學殿堂里，依然懷抱小學四年級的夢想 |trans-title=Interview with Zhang Shou-Wu: In the mathematics department, he still has his dream from fourth grade of elementary school |publisher=Beijing Sina Net |date=3 May 2019 |accessdate= 5 May 2019 |language=zh}}{{cite web|url=http://www.hx.gov.cn/content/detail/59fb250a9e8b43880f0056a8.html |title=旅美青年数学家张寿武 |trans-title=Zhang Shouwu, a young mathematician in the United States |publisher=He County Government |date=2 November 2017 |accessdate= 5 May 2019 |language=zh}} Zhang grew up in a poor farming household and could not attend school until eighth grade due to the Cultural Revolution. He spent most of his childhood raising ducks in the countryside and self-studying mathematics textbooks that he acquired from sent-down youth in trades for frogs. By the time he entered junior high school at the age of fourteen, he had taught himself calculus and had become interested in number theory after reading about Chen Jingrun's proof of Chen's theorem which made substantial progress on Goldbach's conjecture.{{cite web|url=http://edu.sina.com.cn/l/2019-05-04/doc-ihvhiqax6533865.shtml |title=专访数学家张寿武：要让别人解中国人出的数学题 |trans-title=Interview with mathematician Zhang Shouwu: Let others solve the math problems of Chinese people |publisher=Sina Education |date=4 May 2019 |accessdate= 5 May 2019 |language=zh}}

=Education=

Zhang was admitted to the Sun Yat-sen University chemistry department in 1980 after scoring poorly on his mathematics entrance examinations, but he later transferred to the mathematics department after feigning color blindness and received his bachelor's degree in mathematics in 1983.{{cite magazine|last=Leong |first=Y. K. |date=July–December 2018 |url=https://ims.nus.edu.sg/wp-content/uploads/2020/05/imprints-32-2018.pdf |title=Shou-Wu Zhang: Number Theory and Arithmetic Algebraic Geometry |issue=32 |magazine=Imprints |pages=32–36 |publisher=The Institute for Mathematical Sciences, National University of Singapore |access-date=5 May 2019}} He then studied under analytic number theorist Wang Yuan at the Chinese Academy of Sciences where he received his master's degree in 1986.{{cite web|url=https://zhuanlan.zhihu.com/p/31700224 |title=专访数学家张寿武：数学苍穹闪烁中国新星 |trans-title=Interview with mathematician Zhang Shouwu: A new Chinese star flashing in the mathematical sky |publisher=Zhishi Fenzi |date=4 December 2017 |accessdate= 5 May 2019 |language=zh}} In 1986, Zhang was brought to the United States to pursue his doctoral studies at Columbia University by Dorian M. Goldfeld. He initially studied under Goldfeld and then Hervé Jacquet, before deciding to work with Lucien Szpiro, a visiting professor at Columbia at the time, and Gerd Faltings at Princeton University. He received his PhD at Columbia University under the supervision of Szpiro in 1991.{{MathGenealogy|id=37832}}

=Career=

Zhang was a member of the Institute for Advanced Study and an assistant professor at Princeton University from 1991 to 1996. In 1996, Zhang moved back to Columbia University where he was a tenured professor until 2013. He has been a professor at Princeton University since 2011 was named the Henry Burchard Fine Professor from 2014 to 2015,{{cite web |url=https://www.princeton.edu/news/2014/04/07/faculty-chosen-endowed-professorships |website=Princeton University |title=Faculty chosen for endowed professorships |author=Office of Communications |accessdate=March 21, 2021}} and was named a Eugene Higgins Professor in 2021.{{cite web |url=https://www.princeton.edu/news/2021/05/26/faculty-members-named-endowed-professorships |website=Princeton University |title=Faculty members named to endowed professorships |author=Office of Communications |accessdate=May 26, 2021}}

Zhang is on the editorial boards of: Acta Mathematica Sinica, Algebra & Number Theory, Forum of Mathematics, Journal of Differential Geometry, National Science Review, Pure and Applied Mathematics Quarterly, Science in China, and Research in Number Theory. He has previously served on the editorial boards of: Journal of Number Theory, Journal of the American Mathematical Society, Journal of Algebraic Geometry, and International Journal of Number Theory.

Research

Zhang's doctoral thesis Positive line bundles on Arithmetic Surfaces {{harvs|last=Zhang |year=1992}} proved a Nakai–Moishezon type theorem in intersection theory using a result from differential geometry already proved in Tian Gang's doctoral thesis. In a series of subsequent papers ({{harvs|nb|last=Zhang |year1=1993 |year2=1995a |year3=1995b}}, {{harvs|nb|last1=Szpiro |last2=Ullmo |last3=Zhang |year=1997}}), he further developed his theory of 'positive line bundles' in Arakelov theory which culminated in a proof of the Bogomolov conjecture with Emmanuel Ullmo and a further generalization of the Bogomolov conjecture ({{harvs|nb|last=Zhang |year=1998}}).{{Citation |last=Ullmo |first=Emmanuel |title=Positivité et Discrétion des Points Algébriques des Courbes |journal=Annals of Mathematics |volume=147 |issue=1 |year=1998 |pages=167–179 |doi=10.2307/120987 |jstor=120987 | zbl=0934.14013 |arxiv=alg-geom/9606017 }}

In 2001, Zhang proved a generalization of the Gross–Zagier theorem to modular abelian varieties of GL(2) type ({{harvs|nb|last=Zhang |year1=2001}}). In particular, the result led him to a proof of the rank one Birch-Swinnerton-Dyer conjecture for modular abelian varieties of GL(2) type over totally real fields through his work relating the Néron–Tate height of Heegner points to special values of L-functions in {{harvs|last=Zhang |year1=1997 |year2=2001}}.{{cite journal|last=Zhang |first=Wei |title=The Birch–Swinnerton-Dyer conjecture and Heegner points: a survey |journal=Current Developments in Mathematics |volume=2013 |pages=169–203|doi=10.4310/CDM.2013.v2013.n1.a3 |year=2013 |doi-access=free }}. In 2013, Zhang and his former doctoral students Xinyi Yuan and Wei Zhang established a full generalization of the Gross–Zagier theorem to all Shimura curves ({{harvs|nb|last1=Yuan |last2=Zhang |last3=Zhang |first3=W. |year=2009}} {{harvs|nb|last1=Yuan |last2=Zhang |last3=Zhang |first3=W. |year=2013}}).

In arithmetic dynamics, {{harvs|txt|last=Zhang |year1=1995a |year2=2006}} posed conjectures on the Zariski density of non-fibered endomorphisms of quasi-projective varieties and {{harvs|txt|last1=Ghioca |last2=Tucker |last3=Zhang |year=2011}} proposed a dynamical analogue of the Manin–Mumford conjecture.{{cite journal|title=Current trends and open problems in arithmetic dynamics |journal=Bulletin of the American Mathematical Society |volume=56 |issue=4 |pages=611–685 |first1=Robert |last1=Benedetto |first2=Patrick |last2=Ingram |first3=Rafe |last3=Jones |first4= Michelle |last4=Manes |first5=Joseph H. |last5=Silverman |authorlink5=Joseph H. Silverman |first6=Thomas J. |last6=Tucker |doi=10.1090/bull/1665|year=2019 |arxiv=1806.04980 |s2cid=53550119 }}

In 2018, {{harvs|txt|last1=Yuan |last2=Zhang |year=2018}} proved the averaged Colmez conjecture which was shown to imply the André–Oort conjecture for Siegel modular varieties by Jacob Tsimerman.{{cite journal|journal=Notices of the American Mathematical Society |title=February 2018 |issn=1088-9477 |volume=65 |issue=2 |year=2018 |page=191}}

Awards

Zhang has received a Sloan Foundation Research Fellowship (1997) and a Morningside Gold Medal of Mathematics (1998). He is also a Clay Foundation Prize Fellow (2003), Guggenheim Foundation Fellow (2009), Fellow of the American Academy of Arts and Sciences (2011), and Fellow of the American Mathematical Society (2016).{{citation|url=http://www.ams.org/profession/ams-fellows/new-fellows|title=2016 Class of the Fellows of the AMS|publisher=American Mathematical Society|accessdate=2015-11-16}}{{cite web|url=https://www.gf.org/fellows/all-fellows/shou-wu-zhang/|title=Shou-Wu Zhang|publisher=John Simon Guggenheim Foundation|accessdate=31 January 2019}} He was an invited speaker at the International Congress of Mathematicians in 1998.{{cite web|url=https://www.mathunion.org/icm-plenary-and-invited-speakers?combine=&page=163 |title= ICM Plenary and Invited Speakers |accessdate=31 January 2019}}{{cite book|author=Zhang, Shou-Wu|chapter=Small points and Arakelov theory|title=Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II|year=1998|pages=217–225|chapter-url=https://www.elibm.org/ft/10011657000}}

Selected publications

=Arakelov theory=

{{Citation|last=Zhang |first=Shou-Wu |title=Positive line bundles on arithmetic surfaces |journal=Annals of Mathematics |volume=136 |issue=3 |year=1992 |pages=569–587 |doi=10.2307/2946601|jstor=2946601 }}.
{{Citation|last=Zhang |first=Shou-Wu |title=Admissible pairing on a curve |journal=Inventiones Mathematicae |volume=112 |issue=1 |year=1993 |pages=421–432|doi=10.1007/BF01232429 |bibcode=1993InMat.112..171Z |s2cid=120229374 }}.
{{Citation|last=Zhang |first=Shou-Wu |title=Small points and adelic metrics |journal=Journal of Algebraic Geometry |volume=8 |issue=1 |year=1995a |pages=281–300}}.
{{Citation|last=Zhang |first=Shou-Wu |title=Positive line bundles on arithmetic varieties |journal=Journal of the American Mathematical Society |volume=136 |issue=3 |year=1995b |pages=187–221|doi=10.1090/S0894-0347-1995-1254133-7 |doi-access=free }}.
{{Citation|last=Zhang |first=Shou-Wu |title=Heights and reductions of semi-stable varieties|journal=Compositio Mathematica|volume= 104|issue=1|year=1996|pages=77–105}}.
{{Citation|last=Zhang |first=Shou-Wu |title=Gross–Schoen cycles and Dualising sheaves|journal=Invent. Math.|volume=179|issue=1|year=2010|pages=1–73|doi=10.1007/s00222-009-0209-3 |bibcode=2010InMat.179....1Z |s2cid=5698835 }}.
{{Citation| last1=Yuan |first1=Xinyi |last2=Zhang |first2=Shou-Wu |title=The arithmetic Hodge index theorem for adelic line bundles|journal=Math. Ann.|volume=367|year=2017|issue=3–4 |pages=1123–1171|doi=10.1007/s00208-016-1414-1 |s2cid=2813125 }}.

==Bogomolov conjecture==

{{Citation|last1=Szpiro |first1=Lucien |authorlink1=Lucien Szpiro |last2=Ullmo |first2=Emmanuel |authorlink2=Emmanuel Ullmo |last3=Zhang |first3=Shou-Wu |title=Equirépartition des petits points |journal=Inventiones Mathematicae |volume=127 |issue=2 |year=1997 |pages=337–347|doi=10.1007/s002220050123 |bibcode=1997InMat.127..337S |s2cid=119668209 }}.
{{Citation|last=Zhang |first=Shou-Wu |title=Equidistribution of small points on abelian varieties |journal=Annals of Mathematics |volume=147 |issue=1 |year=1998 |pages=159–165 |doi=10.2307/120986|jstor=120986 }}.

=Gross–Zagier formulae=

{{Citation|last=Zhang |first=Shou-Wu |title=Heights of Heegner cycles and derivatives of L-series|journal=Inventiones Mathematicae |volume=130 |issue=1 |year=1997 |pages=99–152|doi=10.1007/s002220050179 |bibcode=1997InMat.130...99Z |s2cid=10537873 }}.
{{Citation|last=Zhang |first=Shou-Wu |title=Heights of Heegner points on Shimura curves |journal=Annals of Mathematics |volume=153 |issue=1 |year=2001|pages=27–147|doi=10.2307/2661372 |jstor=2661372 |arxiv=math/0101269 |s2cid=119624920 }}.
{{Citation|last1=Yuan |first1=Xinyi |authorlink1=Xinyi Yuan |last2=Zhang |first2=Shou-Wu |last3=Zhang |first3=Wei |authorlink3=Wei Zhang (mathematician) |title=The Gross–Kohnen–Zagier Theorem over Totally Real Fields |journal=Compositio Mathematica |volume=145 |issue=5 |year=2009 |pages=1147–1162|doi=10.1112/S0010437X08003734 |doi-access=free }}.
{{Citation|last1=Yuan |first1=Xinyi |last2=Zhang |first2=Shou-Wu |last3=Zhang |first3=Wei |title=The Gross–Zagier formula on Shimura curves |series=Annals of Mathematics Studies |volume=184 |year=2013 |isbn=9780691155913}}.
{{Citation|last1=Liu |first1=Yifeng |last2=Zhang |first2=Shou-Wu |last3=Zhang |first3=Wei |title=A p-adic Waldspurger formula|journal=Duke Mathematical Journal |volume=167 |issue=4 |year=2018a |pages=743–833|doi=10.1215/00127094-2017-0045 |arxiv=1511.08172 |s2cid=4867572 }}.
{{Citation| last1=Yuan |first1=Xinyi |last2=Zhang |first2=Shou-Wu |title=On the averaged Colmez conjecture |journal=Annals of Mathematics |volume=187 |issue=2 |year=2018b |pages=553–638|doi=10.4007/annals.2018.187.2.4 |arxiv=1507.06903 |s2cid=118916754 }}.

=Arithmetic dynamics=

{{Citation|first=Shou-Wu |last=Zhang |chapter=Distributions in algebraic dynamics |title=Essays in geometry in memory of S.S. Chern |editor-last=Yau |editor-first=Shing-Tung |series=Surveys in Differential Geometry |volume=10 |publisher=International Press |location=Somerville, MA |year=2006 |pages=381–430 |doi=10.4310/SDG.2005.v10.n1.a9 |mr=2408228|doi-access=free }}.
{{Citation|last1=Ghioca |first1=Dragos |last2=Tucker |first2=Thomas J. |last3=Zhang |first3=Shou-Wu |title=Towards a dynamical Manin-Mumford conjecture |journal=International Mathematics Research Notices |volume=22 |year=2011 |pages=5109–5122 |doi=10.1093/imrn/rnq283 |mr=2854724|citeseerx=10.1.1.186.7348 }}.