Imperial College London

Professor Richard Thomas FRS

Faculty of Natural SciencesDepartment of Mathematics

Professor of Pure Mathematics
 
 
 
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Contact

 

richard.thomas Website

 
 
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Location

 

659Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
to

56 results found

Kool M, Thomas RP, 2018, Stable pairs with descendents on local surfaces I: the vertical component, Pure and Applied Mathematics Quarterly, Vol: 13, ISSN: 1558-8599

We study the full stable pair theory --- with descendents --- of theCalabi-Yau 3-fold $X=K_S$, where $S$ is a surface with a smooth canonicaldivisor $C$. By both $\mathbb C^*$-localisation and cosection localisation we reduce tostable pairs supported on thickenings of $C$ indexed by partitions. We showthat only strict partitions contribute, and give a complete calculation forlength-1 partitions. The result is a surprisingly simple closed product formulafor these "vertical" thickenings. This gives all contributions for the curve classes $[C]$ and $2[C]$ (andthose which are not an integer multiple of the canonical class). Here theresult verifies, via the descendent-MNOP correspondence, a conjecture ofMaulik-Pandharipande, as well as various results about the Gromov-Witten theoryof $S$ and spin Hurwitz numbers.

JOURNAL ARTICLE

Tanaka Y, Thomas RP, Vafa-Witten invariants for projective surfaces I: stable case, Journal of Algebraic Geometry, Pages: 562-562, ISSN: 1534-7486

On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a ℂ∗ action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations.When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover the first terms of modular forms predicted by Vafa and Witten.

JOURNAL ARTICLE

Tanaka Y, Thomas RP, 2018, Vafa-Witten invariants for projective surfaces II: semistable case, Pure and Applied Mathematics Quarterly, Vol: 13, Pages: 517-562, ISSN: 1558-8599

We propose a definition of Vafa–Witten invariants counting semistable Higgs pairs on a polarised surface. We use virtual localisation applied to Mochizuki/Joyce–Song pairs.For KS≤0we expect our definition coincides with an alternative definition using weighted Euler characteristics. We prove this for degKS<0 here, and it is proved for Sa K3 surface in “Sheaf counting on local K3 surfaces” [D. Maulik and R. P. Thomas, arXiv:1806.02657].For K3 surfaces we calculate the invariants in terms of modular forms which generalise and prove conjectures of Vafa and Witten.

JOURNAL ARTICLE

Segal E, Thomas R, 2018, Quintic threefolds and Fano elevenfolds, JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, Vol: 743, Pages: 245-259, ISSN: 0075-4102

JOURNAL ARTICLE

Thomas RP, 2018, Notes on homological projective duality, American-Mathematical-Society Summer Research Institute on Algebraic Geometry, Publisher: American Mathematical Society, Pages: 585-609, ISSN: 2324-707X

CONFERENCE PAPER

Jiang Y, Thomas RP, 2017, VIRTUAL SIGNED EULER CHARACTERISTICS, JOURNAL OF ALGEBRAIC GEOMETRY, Vol: 26, Pages: 379-397, ISSN: 1056-3911

JOURNAL ARTICLE

Pandharipande R, Thomas RP, 2016, THE KATZ-KLEMM-VAFA CONJECTURE FOR K3 SURFACES, FORUM OF MATHEMATICS PI, Vol: 4, ISSN: 2050-5086

JOURNAL ARTICLE

Calabrese JR, Thomas RP, 2016, Derived equivalent Calabi-Yau threefolds from cubic fourfolds, MATHEMATISCHE ANNALEN, Vol: 365, Pages: 155-172, ISSN: 0025-5831

JOURNAL ARTICLE

Pandharipande R, Thomas RP, 2016, Notes on the proof of the KKV conjecture, ADVANCES IN GEOMETRY AND MATHEMATICAL PHYSICS, Vol: 21, Pages: 289-311, ISSN: 1052-9233

JOURNAL ARTICLE

Gholampour A, Sheshmani A, Thomas R, 2014, Counting curves on surfaces in Calabi-Yau 3-folds, MATHEMATISCHE ANNALEN, Vol: 360, Pages: 67-78, ISSN: 0025-5831

JOURNAL ARTICLE

Addington N, Thomas R, 2014, HODGE THEORY AND DERIVED CATEGORIES OF CUBIC FOURFOLDS, DUKE MATHEMATICAL JOURNAL, Vol: 163, Pages: 1885-1927, ISSN: 0012-7094

JOURNAL ARTICLE

Katz S, Klemm A, Pandharipande R, 2014, On the motivic stable pairs invariants of K3 surfaces

For a K3 surface S, we study motivic invariants of stable pairs moduli spacesassociated to 3-fold thickenings of S. We conjecture suitable deformation anddivisibility invariances for the Betti realization. Our conjectures, togetherwith earlier calculations of Kawai-Yoshioka, imply a full determination of thetheory in terms of the Hodge numbers of the Hilbert schemes of points of S. Thework may be viewed as the third in a sequence of formulas starting withYau-Zaslow and Katz-Klemm-Vafa (each recovering the former). Numerical datasuggest the motivic invariants are linked to the Mathieu M_24 moonshinephenomena. The KKV formula and the Pairs/Noether-Lefschetz correspondence togetherdetermine the BPS counts of K3-fibered Calabi-Yau 3-folds in fiber classes interms of modular forms. We propose a framework for a refined P/NLcorrespondence for the motivic invariants of K3-fibered CY 3-folds. For the STUmodel, a complete conjecture is presented.

JOURNAL ARTICLE

Kool M, Thomas R, 2011, Reduced classes and curve counting on surfaces I: theory, Algebraic Geometry, Vol: 1, Pages: 334-383

JOURNAL ARTICLE

Kool M, Thomas R, 2011, Reduced classes and curve counting on surfaces II: calculations, Algebraic Geometry, Vol: 1, Pages: 384-399

JOURNAL ARTICLE

Pandharipande R, Thomas RP, 2014, 13/2 ways of counting curves, Moduli Spaces, Publisher: Cambridge University Press, Pages: 282-333, ISBN: 9781107279544

BOOK CHAPTER

Huybrechts D, Thomas RP, 2014, Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes (vol 346, pg 545, 2010), MATHEMATISCHE ANNALEN, Vol: 358, Pages: 561-563, ISSN: 0025-5831

JOURNAL ARTICLE

Pandharipande R, Thomas RP, 2014, ALMOST CLOSED 1-FORMS, GLASGOW MATHEMATICAL JOURNAL, Vol: 56, Pages: 169-182, ISSN: 0017-0895

JOURNAL ARTICLE

Odaka Y, Thomas RP, 2013, Separatedness of moduli of K-stable varieties

Given a one parameter flat family of polarized algebraic varieties, we showthat any K-stable limit is unique. In particular, moduli spaces of K-stablepolarized varieties are automatically Hausdorff when they exist. We also give a characterization of K-stable limits in terms of the CM linebundle, and some applications to moduli. Our methods work for arbitraryprojective schemes in any characteristic.

JOURNAL ARTICLE

Ross J, Thomas R, 2011, WEIGHTED BERGMAN KERNELS ON ORBIFOLDS, JOURNAL OF DIFFERENTIAL GEOMETRY, Vol: 88, Pages: 87-107, ISSN: 0022-040X

JOURNAL ARTICLE

Ross J, Thomas R, 2011, WEIGHTED PROJECTIVE EMBEDDINGS, STABILITY OF ORBIFOLDS, AND CONSTANT SCALAR CURVATURE KAHLER METRICS, JOURNAL OF DIFFERENTIAL GEOMETRY, Vol: 88, Pages: 109-159, ISSN: 0022-040X

JOURNAL ARTICLE

Kool M, Shende V, Thomas RP, 2011, A short proof of the Gottsche conjecture, GEOMETRY & TOPOLOGY, Vol: 15, Pages: 397-406, ISSN: 1364-0380

JOURNAL ARTICLE

Stoppa J, Thomas RP, 2011, HILBERT SCHEMES AND STABLE PAIRS: GIT AND DERIVED CATEGORY WALL CROSSINGS, BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE, Vol: 139, Pages: 297-339, ISSN: 0037-9484

JOURNAL ARTICLE

Huybrechts D, Thomas RP, 2010, Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes, MATHEMATISCHE ANNALEN, Vol: 346, Pages: 545-569, ISSN: 0025-5831

JOURNAL ARTICLE

Pandharipande R, Thomas RP, 2010, STABLE PAIRS AND BPS INVARIANTS, JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 23, Pages: 267-297, ISSN: 0894-0347

JOURNAL ARTICLE

Maulik D, Pandharipande R, Thomas RP, 2010, Curves on K3 surfaces and modular forms, JOURNAL OF TOPOLOGY, Vol: 3, Pages: 937-996, ISSN: 1753-8416

JOURNAL ARTICLE

Thomas RP, 2010, An Exercise in Mirror Symmetry, International Congress of Mathematicians (ICM 2010), Publisher: HINDUSTAN BOOK AGENCY-INDIA, Pages: 624-651

CONFERENCE PAPER

Pandharipande R, Thomas RP, 2009, Curve counting via stable pairs in the derived category, INVENTIONES MATHEMATICAE, Vol: 178, Pages: 407-447, ISSN: 0020-9910

JOURNAL ARTICLE

Rollenske S, Thomas R, 2009, Smoothing nodal Calabi-Yau n-folds, JOURNAL OF TOPOLOGY, Vol: 2, Pages: 405-421, ISSN: 1753-8416

JOURNAL ARTICLE

Pandharipande R, Thomas RP, 2009, The 3-fold vertex via stable pairs, GEOMETRY & TOPOLOGY, Vol: 13, Pages: 1835-1876, ISSN: 1465-3060

JOURNAL ARTICLE

Ross J, Thomas R, 2007, A study of the Hilbert-Mumford criterion for the stability of projective varieties, JOURNAL OF ALGEBRAIC GEOMETRY, Vol: 16, Pages: 201-255, ISSN: 1056-3911

JOURNAL ARTICLE

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