Richard Thomas (mathematician)

Thomas obtained his PhD on gauge theory on Calabi–Yau manifolds in 1997 under the supervision of Simon Donaldson at the University of Oxford. In his dissertation research with Donaldson, he defined the Donaldson–Thomas invariants of Calabi–Yau 3-folds, a major topic in geometry and the mathematics of string theory.

Before joining Imperial College, he was member of the Institute for Advanced Study in Princeton, New Jersey,{{cite web | url=https://www.ias.edu/scholars/richard-thomas | title=Richard Thomas - Scholars | Institute for Advanced Study | date=9 December 2019 }} and affiliated with Harvard University and the University of Oxford. He was made professor of pure mathematics in 2005.{{cite web|url=https://www.imperial.ac.uk/pictures/imperial_acad_proms_05.pdf|archive-url=https://web.archive.org/web/20181002180850/https://www.imperial.ac.uk/pictures/imperial_acad_proms_05.pdf|archive-date=2018-10-02|title=Academic Promotions 2005|publisher=Imperial College London}}

Thomas has made contributions to algebraic geometry, differential geometry, and symplectic geometry. His doctoral thesis, which introduced the invariants that later became known as Donaldson–Thomas invariants, was published in the Journal of Differential Geometry as `A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations'.{{cite journal | url=https://projecteuclid.org/euclid.jdg/1214341649 | doi=10.4310/jdg/1214341649 | title=A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations | year=2000 | last1=Thomas | first1=R. P. | journal=Journal of Differential Geometry | volume=54 | issue=2 | s2cid=119163745 | doi-access=free | arxiv=math/9806111 }} Motivated by homological mirror symmetry, he produced braid group actions on derived categories of coherent sheaves in joint work with Paul Seidel.{{cite journal | url=https://projecteuclid.org/euclid.dmj/1091737124 | doi=10.1215/S0012-7094-01-10812-0 | title=Braid group actions on derived categories of coherent sheaves | year=2001 | last1=Seidel | first1=Paul | last2=Thomas | first2=Richard | journal=Duke Mathematical Journal | volume=108 | hdl=10044/1/220 | s2cid=458756 | hdl-access=free }} With Shing-Tung Yau he formulated a conjecture (now known as the Thomas–Yau conjecture) concerning the existence of a special Lagrangian in the Hamiltonian deformation class of a fixed Lagrangian submanifold of a Calabi–Yau manifold.{{cite journal

| last1 = Thomas | first1 = R. P.

| last2 = Yau | first2 = S.-T. | author2-link = Shing-Tung Yau

| doi = 10.4310/CAG.2002.v10.n5.a8

| issue = 5

| journal = Communications in Analysis and Geometry

| mr = 1957663

| pages = 1075–1113

| title = Special Lagrangians, stable bundles and mean curvature flow

| volume = 10

| year = 2002| arxiv = math/0104197

}} Together with Rahul Pandharipande he formulated a refinement of the Donaldson–Thomas invariants for the special case of curve counting, the Pandharipande–Thomas (PT) stable pair invariants.{{Cite journal |doi = 10.1007/s00222-009-0203-9|title = Curve counting via stable pairs in the derived category|journal = Inventiones Mathematicae|volume = 178|issue = 2|pages = 407–447|year = 2009|last1 = Pandharipande|first1 = R|last2 = Thomas|first2 = R. P|arxiv = 0707.2348|bibcode = 2009InMat.178..407P|s2cid = 14113803}} With Martijn Kool and Vivek Shende, he used the PT invariants to prove the Göttsche conjecture—a classical algebro-geometric problem going back more than a century.[https://arxiv.org/abs/1010.3211 A short proof of the Göttsche conjecture] With Davesh Maulik and Pandharipande he proved the Katz–Klemm–Vafa (KKV) conjecture,{{Cite journal |doi = 10.1112/jtopol/jtq030|title = Curves on K 3 surfaces and modular forms|journal = Journal of Topology|volume = 3|issue = 4|pages = 937–996|year = 2010|last1 = Maulik|first1 = D|last2 = Pandharipande|first2 = R|last3 = Thomas|first3 = R. P|citeseerx = 10.1.1.756.7349|s2cid = 54186798}} establishing links between the Gromov–Witten theory of K3 surfaces and modular forms. His collaboration with Daniel Huybrechts led to contributions to the deformation theory of complexes.{{Cite journal |doi = 10.1007/s00208-009-0397-6|title = Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes|journal = Mathematische Annalen|volume = 346|issue = 3|pages = 545–569|year = 2009|last1 = Huybrechts|first1 = Daniel|last2 = Thomas|first2 = Richard P|arxiv = 0805.3527|s2cid = 115164648}} With Nick Addington he established a compatibility result for two rationality conjectures on cubic fourfolds.{{cite journal | url=https://projecteuclid.org/euclid.dmj/1404824304 | doi=10.1215/00127094-2738639 | title=Hodge theory and derived categories of cubic fourfolds | year=2014 | last1=Addington | first1=Nicolas | last2=Thomas | first2=Richard | journal=Duke Mathematical Journal | volume=163 | issue=10 | arxiv=1211.3758 | s2cid=119147791 }}

He coauthored a book on mirror symmetry.{{cite book

| last1 = Hori | first1 = Kentaro

| last2 = Katz | first2 = Sheldon

| last3 = Klemm | first3 = Albrecht

| last4 = Pandharipande | first4 = Rahul

| last5 = Thomas | first5 = Richard

| last6 = Vafa | first6 = Cumrun

| last7 = Vakil | first7 = Ravi

| last8 = Zaslow | first8 = Eric

| isbn = 0-8218-2955-6

| mr = 2003030

| publisher = Clay Mathematics Institute | location = Cambridge, MA

| series = Clay Mathematics Monographs

| title = Mirror symmetry

| url = https://www.claymath.org/library/monographs/cmim-1.pdf

| volume = 1

| year = 2003}} Thomas also wrote expository notes on derived categories,{{Cite journal |arxiv = math/0001045|last1 = Thomas|first1 = R. P|title = Derived categories for the working mathematician|journal = Proceedings of the Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Cycles, Harvard, January|volume = 1999|issue = 2001|year = 2000|bibcode = 2000math......1045T}} curve counting,{{Cite book |arxiv = 1111.1552|last1 = Thomas|first1 = R. P|title = Moduli Spaces|pages = 282–333|year = 2011|doi = 10.1017/CBO9781107279544.007|isbn = 9781107279544|s2cid = 117183792}} and homological projective duality.{{Cite arXiv |eprint = 1512.08985|last1 = Thomas|first1 = R. P|title = Notes on HPD|class = math.AG|year = 2015}} He appeared in the documentary film 'Thinking space' by Heidi Morstang.{{cite web | url=https://www.lms.ac.uk/library/frames-of-mind | title=Frames of Mind | London Mathematical Society }} Thomas has played an important part in promoting geometry in the UK, encouraging younger mathematicians,{{cite journal | doi=10.1112/S0024609305004431 | title=Prizewinners 2004 | journal=Bulletin of the London Mathematical Society | year=2005 | volume=37 | issue=3 | pages=468–476 | doi-access=free }} and in bringing more geometry to Imperial college: "[...] There was little geometry in Imperial then, but now, thanks largely to the drive of my colleague Richard Thomas, we have one of the main centres for research in this area." - Simon Donaldson{{Cite web |url=http://www.shawprize.org/en/shaw.php?tmp=3&twoid=12&threeid=41&fourid=22&fiveid=11 |title=The Shaw Prize - Top prizes for astronomy, life science and mathematics |access-date=2018-04-08 |archive-date=2018-03-17 |archive-url=https://web.archive.org/web/20180317232004/http://www.shawprize.org/en/shaw.php?tmp=3&twoid=12&threeid=41&fourid=22&fiveid=11 |url-status=dead }}

In 2004, Thomas was awarded the London Mathematical Society's Whitehead Prize and the Philip Leverhulme Prize, in 2010 the Royal Society Wolfson Research Merit Award. From the Whitehead prize citation:

"Thomas has made seminal contributions across an unusually broad range of topics.

Much of his work is related to mirror symmetry and Calabi–Yau geometry, and thus

has an important bearing on exciting contemporary interactions with mathematical

physics. [...] This involved the combination of deep, original insights and

sophisticated technical proofs that is characteristic of Thomas’s work."{{cite journal | doi=10.1112/S0024609305004431 | title=Prizewinners 2004 | journal=Bulletin of the London Mathematical Society | year=2005 | volume=37 | issue=3 | pages=468–476 | doi-access=free }}

In 2010 he also was invited speaker for the algebraic geometry section at the International Congress of Mathematicians in Hyderabad, where he delivered a lecture on mirror symmetry.{{cite web | url=https://www.mathunion.org/icm-plenary-and-invited-speakers?combine=&page=14&order=field_speakers_congress_year&sort=desc | title=ICM Plenary and Invited Speakers | International Mathematical Union (IMU) }} Thomas was elected a Fellow of the Royal Society (FRS) in 2015.[https://royalsociety.org/people/richard-thomas-12414/ FRS announcement][http://www3.imperial.ac.uk/newsandeventspggrp/imperialcollege/newssummary/news_5-5-2015-15-46-48 Professors in Mathematics and Chemistry honoured with Royal Society Fellowships]

His contributions to algebraic geometry led to his election to the 2018 class of fellows of the American Mathematical Society.{{citation|url=http://ams.org/profession/ams-fellows/new-fellows|title=2018 Class of the Fellows of the AMS|publisher=American Mathematical Society|accessdate=2017-11-03}} For 2025 he was awarded the Oswald Veblen Prize in Geometry, shared with Soheyla Feyzbakhsh.[http://www.ams.org/news?news_id=7386 Oswald Veblen Prize in Geometry 2025]

{{reflist|35em}}

• {{Google Scholar id|Xi8nyIsAAAAJ}}

{{FRS 2015}}

{{Authority control}}

{{DEFAULTSORT:Thomas, Richard}}

Category:Living people

Category:20th-century British mathematicians

Category:21st-century British mathematicians

Category:Fellows of the Royal Society

Category:Fellows of the American Mathematical Society

Category:Alumni of the University of Oxford

Category:Academics of Imperial College London

Category:Algebraic geometers

Category:British string theorists

Category:Royal Society Wolfson Research Merit Award holders

Category:Year of birth missing (living people)