Kefeng Liu

Kefeng Liu (Chinese: 刘克峰; born December 1965), is a Chinese mathematician who is known for his contributions to geometric analysis, particularly the geometry, topology and analysis of moduli spaces of Riemann surfaces and Calabi-Yau manifolds. He is a professor of mathematics at University of California, Los Angeles, as well as the Executive Director of the Center of Mathematical Sciences at Zhejiang University.

Biography

Liu was born in Kaifeng, Henan Province, China. In 1985, Liu received his B.A. in mathematics from the Department of Mathematics of Peking University in Beijing. In 1988, Liu obtained his M.A. from the Institute of Mathematics of the Chinese Academy of Sciences (CAS) in Beijing.^[1] Liu then went to study in the United States, obtaining a Ph.D. from Harvard University in 1993 under Shing-Tung Yau.^[2]

From 1993 to 1996, Liu was C. L. E. Moore Instructor at MIT. From 1996 to 2000, Liu was an assistant professor at Stanford University. Liu joined the UCLA faculty in 2000, where he was promoted to full professor in 2002.^[3] In September 2003, Liu was appointed as the head of Zhejiang University's mathematics department.^[4] Liu is currently the Executive Director of the Center of Mathematical Sciences at Zhejiang University.

Contributions to mathematics

Elliptic genus

By using the results of Kac-Peterson-Wakimoto about the modular invariance of the characters of affine Lie algebras, under a very natural assumption on the first equivariant Pontryagin class, Liu proved the rigidity of the Dirac operator on loop space twisted by positive energy loop group representations of any level, and generalized the rigidity theorems to the so-called non-zero anomaly cases.

The Â-vanishing theorem for loop spaces with spin structures is one of the corollaries. This is a loop space analogue of the Atiyah-Hirzebruch Â-vanishing theorem for group actions and the loop space Â-genus, or the Witten genus. An analogue of the Lawson-Yau's vanishing theorem for non-abelian group action is also derived. The proof involves index theory and certain subtle properties of the Jacobi theta functions.

Using the modular invariance of the characters of Kac-Moody algebras in a substantial way, Liu proved general vanishing theorems associated to loop groups. These theorems provide new obstructions for group actions on manifolds.

Liu's general rigidity theorem associated to loop groups generalizes Witten rigidity conjectures proved by Taubes, Bott-Taubes, Hirzebruch, Krichever, Landweber-Stong and Ochanine, which is the first time that Kac-Weyl character formula entered the fields of geometry and topology.

Liu proved a generalization of the 12-dimensional miraculous cancellation formula by Alveraz-Gaume and Witten to arbitrary dimensions and general vector bundles. Together with with Weiping Zhang, he found relations between elliptic genus and other geometric invariants, such as holonomy, the APS eta-invariants and the Rokhlin invariants.

Liu described an approach to the geometric construction of elliptic cohomology by using the K-group of infinite dimensional vector bundles, and proved a Riemann-Roch type theorem for such K-group. With X. Ma and W. Zhang, Liu proved several rigidity and vanishing theorems for the family indices of elliptic operators.

With X. Ma and W. Zhang, Liu proved certain general rigidity and vanishing theorems of elliptic genus for foliated manifolds by using classical equivariant index theory. The main techniques are the Jacobi theta functions and the construction of a new class of elliptic operators associated to foliations.

With C. Dong and X. Ma, Liu proved a rigidity theorem for vertex operator algebra bundles, which strongly indicates that the geometric construction of elliptic cohomology is related to vertex operator algebras.

Heat kernel and moduli spaces

Liu derived general formulas for the intersection numbers on the moduli spaces of flat connections over a Riemann surface using explicit formulas of heat kernels on Lie groups, the Reidemeister torsion and symplectic geometry, hence proved several conjectural formulas derived by Witten by using path-integral method. This new method brings new vanishing formulas for the intersection numbers. These formulas contain all the information needed for the Verlinde formula. Liu's results are for general compact semi-simple Lie groups, and generalizes to the cases when the moduli spaces are singular, as well as when the Riemann surface has several boundary components. It inspired Bismut and Labourie's work on the general Verlinde formula.

This method gives several very general new vanishing theorems about the characteristic numbers of the moduli spaces, which actually follows from the delta function property of heat kernels. Some generalized previous results by Atiyah-Bott and Witten.

Mirror principle and rational curves

With B. Lian and S.-T. Yau, Liu introduced the general notion of Euler data. These are sequences of equivariant cohomology classes in the linear sigma models, the simplest compactifications of the moduli spaces of holomorphic maps from curves into certain projective manifolds with symmetry. They gave a complete proof of the mirror conjecture which relates the counting series of rational curves in a Calabi-Yau quintic manifold to the hypergeometric series of its mirror, as proposed by Candelas and his collaborators. An independent proof was given by Givental.

The method works for computing general characteristic classes on the moduli spaces of stable maps into projective balloon manifolds, which include toric manifolds and homogeneous manifolds, and many of their submanifolds. The mirror principle lays a foundation for almost all of the formulas as conjectured by string theorists for counting rational curves.

Mariño-Vafa conjecture and string duality

Inspired by the large N duality between Chern-Simons and string theory, Mariño and Vafa made a remarkable conjecture about the generating series of certain triple Hodge integrals on moduli spaces of curves. In joint works with C.-C. Liu and J. Zhou, Liu proved this conjecture by localization on moduli space of relative stable maps. Together they proved a two partition analogue of Mariño-Vafa formula.

Topological vertex theory was developed by Vafa and his collaborators since 1998. It gives the most effective way to compute both open and closed Gromov-Witten invariants on toric Calabi-Yau manifolds, through a gluing rule of the vertex data. In a joint work with J. Li, C.-C. Liu and J. Zhou, Liu developed a rigorous mathematical theory of the topological vertex.

With J. Li and J. Zhou, Liu proved that the generating series of Gromov-Witten invariants can be identified with the generating series of the equivariant indices of elliptic operators on the ADHM moduli. This result not only proves some cases of the Gopakumar-Vafa conjecture, but also proves the duality of gauge theory and string theory.

Canonical metrics on the moduli spaces of Riemann surfaces

With X. Sun and S.-T. Yau, Liu introduced new complete Kähler metrics on the moduli and Teichmüller spaces of Riemann surfaces about the two new complete Kähler metrics, the Ricci metric and the perturbed Ricci metric. They proved that the new metrics have bounded geometry, and gave the precise asymptotic behaviors of the metric and its curvatures. Also proved that the perturbed Ricci metric has bounded negative Ricci and holomorphic sectional curvature. This is the first known such metric on the moduli spaces. They also proved that the Kähler-Einstein metric has strongly bounded geometry, the log cotangent bundle of the moduli spaces of Riemann surfaces of genus larger than 1 are Mumford stable, and the Weil-Petersson metric and the new metrics are good in the sense of Mumford.

Tautological ring of the moduli space of curves

In 1990s, Faber proposed a series of conjectures about the structure of the tautological ring of moduli spaces of curves. In particular, the Faber intersection number conjecture is a remarkable identity giving top intersections of tautological classes. With H. Xu, Liu gave the most concise proof of the Faber intersection number conjecture by applying the Witten-Kontsevich theorem. There are other proofs due to Getzler-Pandharipande, Goulden-Jackson-Vakil (for up to 3 points) and Buryak-Shadrin.

Liu and Xu found new effective recursion formulas for computing top intersections of tautological classes. Their work was relevant in Faber's discovery that exactly one relation is missing in codimension 12 forbidding the tautological ring in genus 24 to be Gorenstein. Liu-Xu's formula was programmed to calculate how many and which additional relations are required besides the Faber-Zagier relations, which contain all currently known tautological relations, in order to guarantee the Gorensteiness of tautological rings of moduli spaces of curves (marked or unmarked) when the genus is large.

Mean curvature flows of higher codimension

With H.W. Xu, F. Ye and E. Zhao, Liu studied the convergence of the mean curvature flow of arbitrary codimension in Riemannian manifolds with bounded geometry; proved that if the initial submanifold satisfies a pinching condition, then along the mean curvature flow the submanifold contracts smoothly to a round point in finite time; proved convergences of mean curvature flows of higher codimension in various space forms under certain pinching conditions; generalized several fundamental results of mean curvature flows to higher codimension cases.

With Y. Li, Liu introduced a new geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature; proved the global existence of the solution; discussed the global convergence of its solution and possible applications to the Hopf conjecture, as well as its relation to the Navier-Stokes equations on manifolds.

Asymptotic expansion of the Bergman kernel

With X. Dai and X. Ma, Liu established the asymptotic expansion of the Bergman kernel on a polarized compact Kähler manifold through heat kernel method and local index theory. The proof works for orbifolds and symplectic manifolds as well. They proved a conjecture of Donaldson that certain Bergman-type operators is asymptotically related to the Laplace operator. These work has important application in complex geometry.

Hyperbolic geometric flows

With D. Kong, Liu introduced the hyperbolic geometric flow. This kind of flow is very natural to understand certain wave phenomena in the nature as well as the geometry of manifolds. It possesses many interesting properties from both mathematics and physics.

With W. Dai, C. He and D. Kong, Liu introduced dissipative hyperbolic geometric flow and Hyperbolic mean curvature flow, established the short-time existence and uniqueness theorem for them, and discuss the relation between the equations for hyperbolic mean curvature flow and the equations for extremal surfaces in the Minkowski spacetime.

Modular forms, anomaly cancellation and factorization formulas

With F. Han, Y. Wang, W. Zhang, Liu proved that all of the fundamental anomaly cancellation and factorization formulas in string theory, including the Alvarez-Gaume-Witten miraculous anomaly cancellation formula and the Green-Schwarz anomaly factorization formula for certain gauge groups, and the Horava-Witten anomaly factorization formula can be uniformly derived by using modular forms. The method gives more general and new anomaly cancellation and factorization formulas which are expected to have applications in string theory.

Duality and link invariants

Based on large N Chern-Simons and topological string duality, Labastida, Mariño, Ooguri and Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. By exploring the structure of the generating series with cut-and-join equations and other refined algebraic structures, Liu and Peng gave a proof of this conjecture. Their proof sheds new light on the volume conjecture in knot theory and the structure of general open Gromov-Witten invariants for Calabi-Yau conifold. With Q. Chen P. Peng and S. Zhu, Liu derived an infinite product formula for Chern-Simons partition functions, the generating function of quantum invariants for all colors, and various new higher order congruent skein relations for colored HOMFLY-PT polynomials.

Hodge theory and deformations of Kähler and Calabi-Yau manifolds

With S. Rao and X. Yang, Liu obtained a method to construct globally convergent power series of integrable Beltrami differentials on Calabi-Yau manifolds, and used a new iteration method to construct global canonical families of holomorphic (n,0)-forms on the deformation spaces of Kähler manifolds. These lead to a global convergence result of the Beltrami differentials in the deformation theory of Kodaira-Spencer-Kuranishi of Calabi-Yau manifolds, and a global expansion of the holomorphic (n,0) forms on deformation spaces of Kähler manifolds.

With F. Guan, A. Todorov, Liu proved the existence of holomorphic affine structure on the Teichmüller space of Calabi-Yau type manifolds, which naturally extends to give an affine flat structure on the Hodge metric completion. By using these holomorphic affine structures we can prove that the period map is affine map into the unipotent orbit of the period domain, and proves a global Torelli theorem for Calabi-Yau type manifolds which asserts that the period maps from the Teichmüller spaces to the period domains are injective.

Geometry of moduli spaces and vanishing theorem

With X. Sun, X. Yang and S.-T. Yau, Liu obtained new vanishing theorems about ample vector bundles; derived strong positivity and negativity about the curvatures of Weil-Petersson metrics. They used analytic method instead of the Leray-Borel-Le Potier spectral sequence.

With Y. Zhang, Liu proved the existence of a Kähler metric on any quasi-projective manifold which effectively parametrizes families of a large class of projective manifolds. This Kähler metric has nonpositive bisectional curvature, negative holomorphic sectional curvature, and Ricci curvature, both with negative upper bound.

Awards and Honors

• Frederick E. Terman Fellow (1997-2001)
• Sloan Fellowship (1998-2001)
• Guggenheim Fellow (2002)
• Silver Morningside Medal (1998)
• Morningside Gold Medal in Mathematics (2004)
• invited speaker, 2002 International Congress of Mathematicians

Editorial Work

• Communications in Analysis and Geometry, Editor-in-Chief
• Pure and Applied Mathematical Quarterly, Co-Editor-in-Chief.
• Asian Journal of Mathematics, Editor.
• Pacific Journal of Mathematics, Editor.
• Notices of the International Congress of Chinese Mathematicians, Co-Editor-in-Chief.
• Advanced Lectures in Mathematics, Executive Editor.
• Science China Mathematics, Editor.
• Mathematics and Humanities, Co-Editor-in-Chief.

References

External links

• Homepage of Kefeng Liu at UCLA
• Homepage of Kefeng Liu at ZJU (Chinese)
• The Mathematics Genealogy Project – Kefeng Liu
• Knowledge, Technique, and Imagination – Kefeng Liu Hangzhou, April 2004