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Geometry and Physics

Shing-Tung YauHarvard University

Master Lectures–the Legacy of Emmy Noetherand Gottingen mathematics

December 19, 2016

Mathematics, rightly viewed, possesses not only truth, butsupreme beauty cold and austere, like that of sculpture,without appeal to any part of our weaker nature, without thegorgeous trappings of painting or music, yet sublimely pure,and capable of a stern perfection such as only the greatest artcan show. The true spirit of delight, the exaltation, the senseof being more than Man, which is the touchstone of thehighest excellence, is to be found in mathematics as surely asin poetry.

Bertrand Russell

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In February 2016, there was an announcement by NSF,Caltech and MIT that they have found gravitational wave aspredicted by Einstein 100 years ago.

This is a triumph of the theory of Einstein who proposed thatgravity should be looked at as effects of curvature ofspace-time. My title of Geometry and Physics is very muchrelated to this proposal of Einstein.

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Before this theory of Einstein, scientists followed the viewpoint of Newton: the space is static and gravity allows actionat a distance simultaneously.

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When special relativity was discovered at 1905 by Einstein,with the helps by several people including Lorentz andPoincare, it was found that physical information should nottravel faster than light, and the principle of action at adistance that was used by Newtonian gravity is not compatibleto the newly found special relativity.

Lorentz Poincare

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In a vague sense, Einstein and other physicists knew that spaceand time cannot be distinguished under the rules of relativity.

It was not until 1908 that Minkowski, teacher of Einstein,proposed the concept of Minkowski four dimensionalspace-time where a metric of Riemann type is used to describeall phenomena that appear in special relativity. The group ofmotion of this space-time is given by the Lorentzian group.

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After this important discovery of Minkowski, Einstein realizedthat the description of gravity should be given by a fourdimensional object. By thought experiments, he realized thatthe quantity to describe gravity has to depend on directions ateach point. When an observer is moving in some direction, thedistance he or she measures will change depending on thedirection he is measuring. (When it is measured in thedirection perpendicular to his movement, nothing changes butis different when it is parallel to his movement.)

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After he consulted his college friend Grossmann, Einsteinunderstood that gravity should be measured by a tensor: aquantity that was invented by geometers (Christoffel, inspiredby Riemann’s works) in late nineteenth century. In fact, thetensor he needed was first invented by Riemann in 1854, withdifferent signature. This was a great breakthrough asNewtonian gravity was measured by a scalar function, not by atensor.

Grossmann Riemann Christoffel

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When Einstein wanted to generalize the equation of Newtonon gravity, he needed some quantity that was obtained bydifferentiating the above metric tensor two times. (InNewtonian gravity, it was determined by the Laplacian of thescalar gravitational potential.) But he wanted to make sureeverything obeys the equivalence principle: every law ofphysics is the same independent of frame of observer. Hencethe result of this second derivative of the metric tensor mustbe a tensor again.

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Well, it is known that all tensors that are obtained bydifferentiating the metric tensor twice must be a combinationof the curvature tensor of the metric and the curvature tensoris the only tensor that depends linearly on the secondderivatives of the metric. If there are other physical fieldscoming from different matter (other than gravity) there is amatter stress tensor. Then a simple generalization of Newton’sequation is try to equate the above curvature tensor with thismatter tensor.

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There is only a couple of curvature tensors that can do thejob. One is called the Ricci tensor which was found in thelibrary by Grossmann for Einstein. It was invented by Ricci inthe end of nineteenth century.

Ricci

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Einstein and Grossmann wrote two papers in 1912 and 1913,where they wrote down the equation for gravity in tensorialform. But Einstein was not able to use the equations toexplain perihelion of Mercury. He was tempted to give up theequivalence principle by choosing a suitable coordinate system.The Einstein-Grossmann equation did not satisfy conservationlaw and has to be modified!

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Einstein struggled on this question until he met Hilbert in1915. By November, Hilbert and Einstein arrived at thederivation of the Einstein equation around the same time.Hilbert also found the Hilbert action whose variation will givethe Einstein equation. It should be noted that Hilbert gave alot of credit to Emmy Noether on helping him to achieve theworks. The great accomplishment of Einstein also relied on hisunderstanding of the physical meaning and the application ofthe equation to explain astronomical events.

Noether Hilbert

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The geometers Euler, Gauss, Riemann, Ricci, Christoffel,Bianchi, Minkowski, Hilbert, Levi-Civita and others had greatimpact on the creation of the subject of general relativity. Butthe creation of general relativity has tremendous input on thedevelopment of Riemannian geometry in the twentieth centuryup to present days.

We shall discuss about such development now.

Euler Gauss Levi-Civita

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The idea of using symmetry to dictate geometry andphysical phenomena:

Some physicists claimed that Einstein was the first one to usesymmetry to derive an equation of interest. Actually his workwas inspired by his teacher Minkowski who used Lorentziangroup as the group of symmetry to derive the Minkowskispace-time.

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In fact, the idea of using symmetry to understand geometrywent back to the nineteenth century where the works ofSophus Lie and Klein helped us to create invariants ofgeometry by continuous symmetries. Klein’s famous ErlangenProgram in 1872 was to classify geometry according to theirgroup of symmetry.

Klein Lie

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The classification of the structure of the Lie groups is one ofthe most glorious chapter in mathematics. It starts fromSophus Lie, Killing, Klein and continued into the twentiethcentury by E. Cartan, and H. Weyl. The power ofrepresentation theory of finite and compact groups hasrepeatedly appeared in geometry and physics.

Cartan Killing

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Weyl was the key person to pioneer the theory for compact Liegroups. Hermann Weyl, Eugene Wigner and others appliedsuch theory to quantum mechanics and brought fruitfulresults. It has been one of the most important method instudying particle physics, where groups U(1), SU(2), SU(3)and SU(5) are related to electric magnetic field, weakinteraction, strong interaction and grand unified fields.

Weyl Wigner

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The work of Emmy Noether on the action principle had directinfluence on modern physics and geometry. In fact, she was inGottingen in 1915 when Hilbert was working on the actionprinciple of general relativity. Hilbert acknowledged theinfluence of her ideas on his work.

The Noether theorem, which was published in 1918, inspiredthe modern treatment of mechanics and modern symplecticgeometry and the concept of moment map.

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The Noether theorem:

If a system has a continuous symme-try property, then there corresponds aquantity that is conserved in time.

Noether E. Invariante Variationsprobleme.

Nachr. Konig. Gesellsch. Wiss. Zu Gottin-

gen, Math-phys. Klasse. (1918) 235–257.

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Development of gauge theory:

Immediately after the success of Einstein on general relativity,there was great desire to unify all the known forces by usingideas similar to general relativity. At that time, the mostimportant field is electricity and magnetism as is dictated byMaxwell equations.

Maxwell

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There were two approaches: one is the gauge theory ofHermann Weyl and the other one is the Kaluza-Klein model ofGeneral relativity in five dimensions.

Both of these two developments lay the foundations of moderngeometry and modern physics.

Kaluza Klein

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In fact, in the theory of electromagnetism, Riemann-Silbersteinvector which combines electric and magnetic fields, is alsoattributed to Riemann. This is a complex vector of the formF = E + c

√−1B where c is the speed of light.

This vector is an origin of what physicists later generally called‘dualities’ in the framework of string theory and quantum fieldtheory. For example, the interpretation of GeometricLanglands program in the work of Kapustin and Wittenoriginates from a generalization of this ‘electro-magnetic’duality to nonabelian contexts.

At the level of particles, this is the duality between electronand magnetic monopole, which gave birth to theSeiberg-Witten theory in physics.

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Hermann Weyl was the first one who introduced the conceptof Gauge theory. (he was the one who coined thisterminology.) While gravity can be considered as a gaugetheory with gauge group given by the group ofdiffeomorphisms, Weyl succeeded to show that Maxwellequations is also a gauge theory with gauge group given byU(1). The development went through a nontrivial process.

The group that Weyl proposed at the beginning was noncompact and cannot preserve length. This was criticized byEinstein. But after a few years, Weyl learned from the worksof London et al in quantum mechanics that the group shouldbe U(1). Once the group is chosen right, length is preservedunder parallel transportation and Maxwell equation becomes agauge theory.

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While Weyl accomplished the remarkable interpretation ofMaxwell equations in terms of gauge theory around 1928, thetheory of connection was developed by several geometers. In1917 Levi-Civita studied parallel transport of vectors inRiemannian geometry. In 1918 Weyl in his book “Raum, Zeit,Materie” introduced affine connections. Cartan in 1926studied holonomy group for general connections.

Levi Civita and E. Cartan were interested in another approachto extend Einstein theory of general relativity by looking intoconnections with nontrivial torsion.( Einstein was usingLevi-Civita connections which has no torsion.) The connectionstill preserves metric. This is in fact a form of gauge theory onthe tangent bundle. But Weyl’s point of view was differentand did not restrict himself to tangent bundles.

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In 1944, Chern studied Hermitian connections on complexbundles and, using the curvature of the Hermitian connections,introduced the Chern classes of the bundles. They give rise tothe de Rham classes of the space which turns out to beintegral classes.

Chern de Rham

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Upon seeing the definitions, Weil interpreted Chern’s theory interms of invariant theory. This is called the Chern-Weil theory.It is remarkable that Weil said that at that time Chern classesmay be used to quantize physical theory.

The modern formulation of connections on arbitrary bundleswas introduced by C. Ehresmann in 1950.

Weil Ehresmann

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Modern development of high energy physics and theory ofcondensed matter shows that the prediction of Weil isaccurate. In fact, not only Chern classes play an importantrole in modern quantum field theory, the Chern-Simonsinvariant, which is derived from curvature representation ofChern classes, also play an important role in condensed matterphysics and string theory, which in turn influence the study ofknot theory in geometry, as was shown by Witten that it canbe used to explain the Jones polynomial of the knots.

From this point of view, an important insight was gained tocalculate the volume of a complete hyperbolic metric on theknot complement. This is called Volume conjecture due toKashaev, Murakami and Murakami.

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The volume conjecture states that in a certain limit when thenumber of colorings N approach infinity in the N-coloredJones polynomial for a knot, the value of the colored knotJones polynomial evaluated at the N-th root of unity is theexponential of N-times the simplicial volume of the knotcomplement divided by 2π. The knot complement can beuniquely decomposed into hyperbolic pieces and Seifert fiberedpieces. The simplicial volume is then the sum of the hyperbolicvolumes of the hyperbolic pieces of the decomposition.

The volume conjecture dictates the convergence of thequantum Chern-Simons path integral with non-compact gaugegroup and has nontrivial connections with three dimensionalquantum gravity. This connection was anticipated by Wittenand later studied also by Gukov and Vafa.

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In 1954, there were two independent developments related togauge theory. One was the work of Yang and Mills who lookedat an action on the space of connections on higher rankbundles over a manifold. The group of parallel transportationwill preserve a higher dimensional Lie group whose dimensionis, in general, greater than the circle group as was used byWeyl. This group is in general not commutative. Hence thisgeneralization of gauge theory of Weyl is called nonabeliangauge theory. The action that Yang and Mills used was the L2

norm of the curvature of the connection.

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The theory of Yang-Mills was finally quantized by ’t Hooft inearly seventies, based on preliminary works of Faddeev-Popov.This was a difficult work as there is a problem of choice ofgauge. The works of ’t Hooft was continued by Veltman,Gross et al. It laid the foundation for the theory of standardmodel of modern particle physics.

Yang ’t Hooft

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The equation of motion defined by the Yang-Mills action givesrise to a nice elliptic system (in a suitable choice of gauge)which was not studied by geometers. At late sixties and earlyseventies, during the process of quantization of gauge theory,’t Hooft and Polyakov became interested in the concept ofmonopoles and instantons defined on four dimensionalEuclidean space. They give important special solutions ofYang-Mills equation, by minimizing Yang-Mills energy in termsof topological data. Because of the last property, they playedimportant roles in topological quantum field theory.

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There were extensive efforts by physicists and mathematiciansin the seventies to find these instantons. The equations ofinstantons were rewritten in 1977 by C.N. Yang in terms ofCauchy-Riemann equations. In fact, what Yang did was thatthe instanton bundle should be extended to be a holomorphicbundle over complex projective space and, after extension, theinstanton connection satisfies certain equation on thisholomorphic bundle which we later calledHermitian-Yang-Mills connection.

The concept of Hermitian-Yang-Mills connection can begeneralized to higher dimensional Kahler manifolds. They are“super symmetric” and played important roles in thedevelopment of string theory and algebraic geometry.

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The most important work in this direction was due toDonaldson for algebraic surfaces and due to Uhlenbeck-Yau forarbitrary Kahler manifolds.

In the meanwhile, Donaldson observed that the moduli spaceof instantons can be used to define topological invariants forfour dimensional manifolds. Hence he made the first majorachievement in the theory of topology of smooth fourmanifolds. While Donaldson invariants have been fundamental,it is not so easy to compute. Ten years later, Seiberg andWitten found a simpler invariant for four manifolds whichenjoy similar properties as Donaldson invariants. Taubes madefundamental contributions to the subject of symplecticgeometry by constructing pseudoholomorphic curves based onnonvanishing of Seiberg-Witten invariants. Many major resultswere solved by the work of Taubes.

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The other important development was due to E. Calabi. Hewas also interested in the Yang-Mills action on the space ofmetrics. Within the space of Kahler metrics with the sameKahler class, Calabi showed that the critical point of theYang-Mills functional gives rise to the Kahler-Einstein metric ifthe Kahler class is proportional to the first Chern class.

The existence of such critical points was not known at thetime. It is called the Calabi conjecture when the first Chernclass is zero.

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The third major development, after the great discovery in1915, was the work of Kaluza, followed by Oscar Klein. Theyproposed a remarkable approach to create the Maxwellequations from vacuum Einstein equations. They consideredthe vacuum Einstein equation on a four dimensional manifoldproduct with a circle and demanded all the fields to beinvariant under the rotation of the circle group.

In this way, they found a (Lorentzian) metric tensor, a vectorfield and a scalar on the four manifold. The vector fieldsatisfies the Maxwell equations which couple with the metrictensor. This is a beautiful theory, except that the extra scalarfield cannot be found in nature. Nonetheless , this is abeautiful theory and Einstein likes it.

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This theory is the forerunner of the compactification theory inmodern string theory. The circle is replaced by a sixdimensional manifold satisfying certain constraints. Thoseconstraints give rise to the Calabi-Yau manifolds which areKahler manifolds with a non-vanishing holomorphic volumeform.

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The existence of a Kahler metric with zero Ricci curvature wasproved by me in 1976. Its use in string theory was proposed byCandelas-Horowitz-Strominger-Witten in 1984. The proposalwas that, with the right choice of such Calabi-Yau manifolds,we can calculate the basic physical quantities in nature,including number of generations of fermions, grand unificationscale and Yukawa couplings. Many important algebraic andenumerative properties was proved for Calabi-Yau manifoldsbased on intuition coming from string theory.

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A very major property that arises in physics is duality betweenCalabi-Yau manifolds which gives a very effective tool tocalculate interesting geometry objects that are of greatinterest to geometers. Different branches of mathematics werebrought in to study such dualities.

It is rather exciting to watch these branches merge in a naturalmanner within string theory.

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The basic tools to study above theories came from someclassical theory in mathematics and in physics. Hilbert, in hissystematical way to organize the theory of integral equations,introduced the concept of Hilbert space. The study ofself-adjoint and non-self-adjoint operators on Hilbert spacesplay fundamental roles in quantum mechanics.

It was a coincidence that abstractly defined spectrum of anoperator coincide with the spectrum found in nature. Weylalso found the fundamental Weyl law for the asymptoticbehavior of the spectrum of linear elliptic operators based onthe question on black body radiation, which was a questionraised by Lorentz.

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Study of index of elliptic operators relating index of theoperator to the topology of the manifold has made muchcontribution to modern geometry and particle physics. Thisgave rise to the famous Atiyah-Singer index formula.

Atiyah and Singer

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One of many physics applications of the index formula is tothe study of anomaly in Quantum Field Theory. Fujikawaamong others derived the chiral anomaly using index theorem.Alvarez-Gaume and Witten applied similar methods to thestudy of anomaly in quantization of gravity. Green andSchwarz used it to derive the correct grand unification groupfor the Heterotic string, setting off the 1st string theoryrevolution.

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The very important tool towards such development waspioneered by Hodge in 1941. This actually went back to theworks of nineteenth century where periods of integral werestudied extensively by Riemann, Abel, Lagrange, Jacobi andothers.

Hodge Abel

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A significant part of the later interpretation of Riemann’s workowes to Hermann Weyl. Weyl’s book “Die Idee derRiemannschen Flache” formulated Riemann’s results in modernterms concerning the existence of polarized Hodge structures.

Weyl’s book was published in 1913 as the fifth volume in theseries of Gottingen Lectures on mathematics, the previous fourvolumes were by Klein, Minkowski, Voigt, and Poincare.

Weyl developed the theory later called Hodge theory in hisbook based on a motivation from fluid dynamics . Hisphilosophy was influenced by the book of F. Klein, “OnRiemann’s Theory of Algebraic Functions and their Integrals”.

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Period is the integral of a closed form over a cycle. Accordingto the theory of de Rham, this gives a pairing betweentopological cycles and space of closed forms modulo thosewhich are exact. Hodge proposed forms that are closed andcoclosed to be harmonic forms. He proved that periods can berealized by harmonic forms.

Historically, the question of period was studied for twodimensional surfaces at the beginning. Some part of thistheory was developed by the theory of two dimensional fluiddynamics in nineteenth century. Klein wrote about it in hisbook. In the 1913 book on Riemann surface, Weyl establishedthe theory of harmonic forms on Riemann surfaces based onthe Dirichlet principle of Riemann.

In 1930s, Hodge generalized the theory to higher dimensionalmanifold based on the theory of parametrix of Hadamard.Hodge’s work appeared in 1942.

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Hodge stated that the main purpose of his book “The theoryand applications of Harmonic integrals”, published in 1941,was to prove, using differential forms and the then recent deRham theorem, Lefschetz’s results on the topology ofalgebraic varieties.

He derived several theorems of de Rham using his mainexistence theorem for harmonic integrals. This led to theHodge decomposition theorem.

It is interesting to note that Weyl published a paper called“Method of Orthogonal Projection in Potential Theory”,which he demonstrated how to apply the theory to studyHodge theory in Riemann surfaces. This method was laterused extensively to study Hodge theory.

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Right after Hodge published his work, a gap was found in theproof of existence of harmonic forms with prescribed periodwhich was fixed by Weyl and Kodaria via different approaches.

Weyl’s proof appeared in the paper “On Hodge’s theory ofHarmonic integrals” in 1943. Kodaira gave another proofindependent of Weyl’s for the prescribed period problem in1942. Kodaira later generalized it to the existence of harmonicforms with prescribed singularities (and periods).

In early fifties, Milgram and Rosenblum introduced the heatequation method to give a proof of the Hodge theory. Themethod of heat flow has tremendous influence in laterdevelopment in geometry.

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In the above discussion on Weyl estimate on asymptoticbehavior of eigenvalues, we should mention that estimate ofeigenvalues of Laplacian went back to early works of appliedmathematicians and physicists such as lord Kelvin, lordRayleigh, Rellich, Hilbert in nineteenth century and also Polya,Szego, Courant, Carleman in the early twentieth century.

Some approaches are based on studying the heat kernel andthe Tauberian theorems (in the last fifty years, wave equationmethod was brought in by Hormander, developed by Ivrii,Victor Guillemen, Sternberg, Duistermaat, Boutet de Monvel,Sjostrand, Taylor, Zelditich, Melrose, Ralston and others),which was the one used by Weyl to deduce his asymptoticestimates. Another approach was based on variationalprinciple, or min-max principle characterization of eigenvalues.

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The later method brought in close relation of spectrum ofLaplacian with geometry, especially the isoperimetric inequalityof various kind. Polya and Szego gave many importantdiscussions on how those quantities give rise to effectiveestimates of eigenvalues of Laplacian. The idea appeared laterin the paper of Cheeger where he discussed the estimate offirst eigenvalue on a compact manifold. This is now calledCheeger inequality and is being studied extensively in thetheory of graph.

While the idea of using wave kernel has been able to linkeigenvalues of Laplacian with the length of closed geodesics ofthe manifold, the method of heat kernel gives rise to manydelicate estimates in geometry, including the local indexformula of Atiyah-Singer. This later work has played importantroles in modern quantum field theory.

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Several important events occurred in the nineteen seventiesthat changed geometers’ view towards physics. Besides theimportant discussion of instanton solutions to Yang-Millsequations which led to the revolution of topology of fourmanifolds, we have the discussion of positive mass theorem ingeneral relativity.

While this conjecture was settled by Schoen any myself in1978, it has far reaching consequence in understandinggeometry of manifolds with positive scalar curvature. Thelater work of Witten in proving the positive mass conjectureusing Dirac spinors gives a different and powerful venue tounderstand classical general relativity.

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In the past thirty years, both the methods of Schoen-Yau andWitten have developed to be important powerful tools inclassical general relativity and the theory of manifolds withpositive scalar curvature.

Perhaps one should mention that the method of harmonicspinor dated back to the important work of Lichnerowicz on hisfamous vanishing theorem, which, coupled with Atiyah-Singerindex theorem, gave the first instinct of topologicalobstruction for metrics with positive scalar curvature.

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A very major turning point for implication of ideas ongeometry was the famous paper of Witten on analytictreatment of Morse theory appeared in 1984 in Journal ofDifferential Geometry. This paper has deep influence in laterdevelopment of supersymmetric quantum field theory anddifferential geometry. Immediately afterwards, Floer extendedthe ideas to build the Floer theory in symplectic geometrywhere he was able to prove the Arnold conjecture in case themanifold has trivial higher homotopy groups.

The idea of Witten was motivated by quantum field theoryand also became the foundation of later development oftopological field theory.

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The idea of keeping track of the change of a theory whensome parameter is moving, is a very important one. In thecase of the heat equation proof of the Hodge theory or theindex formula, when the temperature is very high, we see theharmonic forms or solutions of some linear elliptic system. Butwhen the temperature is low, the classical effect comes fromthe metric or from the coefficients of the operators. Hence ifsome object (such as the index of the elliptic operator) isinvariant when the temperature varies, we can relate it on theone hand to quantum mechanical property and on the otherhand to classical properties.

In Witten’s interpretation of the Morse theory, space ofharmonic forms is related to critical point of a function definedon the manifold. In 1994, Seiberg and Witten, based onsimilar philosophy, also connected the Donaldson invariants onfour manifolds to some topological invariant which are easierto compute.

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A very important contribution of the Seiberg-Witten typeinvariants is the fundamental result of Cliff Taubes who wasable to make use of the non-vanishing of the (topological)Seiberg-Witten invariant to construct pseudo-holomorphiccurves for four dimensional symplectic manifold with an almostcomplex structure.

Many important open problems in four dimensional symplecticgeometry were solved by this theorem of Taubes. In particular,Taubes solved an old problem that the symplectic structure onthe complex projective plane is unique.

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The subject of symplectic geometry probably already startedafter Newtonian mechanics was invented. But the moderndevelopment mainly started from the work of Emmy Noetherwhere she published the important foundational work in 1918.Many modern ideas such as moment map can be traced to herworks.

The theory of geometric quantization and moment map playedan important role in geometry and physics, influencing thetheory of representation and differential geometry.

Atiyah and Bott initiated the idea of interpreting Yang-Millsaction of bundles over Riemann surfaces. This point of viewhas deep influence on Donaldson and other mathematicians inOxford, who started to look at similar situations on complexalgebraic surfaces.

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However, the theory becomes much more complicated astheory of nonlinear elliptic equation associated to the subjectin this number of dimensions is not solid enough. In mostcases, it was used as a motivation rather than a proof.

On the other hand, symplectic geometry has become muchmore developed in the last thirty years after it was realizedthat the theory of mirror symmetry, motivated by physicalconsideration, called for a symmetric treatment of symplecticgeometry with complex geometry. Roughly speaking , thesymplectic geometry of one Calabi-Yau manifold is supposedto be isomorphic to the complex geometry of the mirrorCalabi-Yau manifold. The actual situation is much morecomplicated as we need to find the so called quantumcorrections to the symplectic theory. But the quantumcorrections contain many interesting geometric objects that welike to learn, for example pseudo-holomorphic curves.

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Pseudo-holomorphic curves are also called worldsheetinstantons by physicists. In fact, when Candelas et. al.computed the instanton correction for an importantCalabi-Yau manifold, he found a closed formula for thecounting of rational curves within the manifold, revealing itsdeep geometric properties.

It solved a long standing question in enumerative geometry.Nobody in algebraic geometry expected that it can be done insuch an elegant way. Up to now , there is no other way to findthe formula of Candelas et. al. based on algebraic geometryalone although the formula was proved rigorously to be true byGivental and Lian-Liu-Yau independently, a few years afterCandelas and collaborators announced their result.

This new chapter of enumerative geometry is called the theoryof Gromov-Witten invariants. Major contributions on thesubject can be viewed as joint efforts of mathematicians andphysicists.

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In the past ten years, we have witnessed that topologicalquantum field theory is starting to play important roles in alsocondensed matter theory, for example, as seen in the works ofCharles Kane, S. C. Zhang, on topological insulators andKitaev and X. G. Wen on topological phases. SophisticatedChern-Simons theory calculations and higher order tensorcategory theories were used in some of these studies, especiallyregarding questions on classification of topological orders.

Recently we proposed, in collaboration with J. Wang and X.G. Wen, the quantum statistics of anyon excitations incondensed matter systems satisfy consistency relations arisingfrom surgery on three and four manifolds. This is deeplyrelated to the work of Witten on TQFTs.

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A very important goal in fundamental physics and geometry inthe twenty first century is to build a solid foundation for atheory that is capable to incorporate quantum theory in thesmall scale of the spacetime. Insights from physics andgeometry have to play a fundamental role.

String theory, theory of quantum entanglement and concept ofnoncommutative geometry were brought in. Theunderstanding of quantum entanglement, may offer a deeperlook into the nature of spacetime, and important geometricconcepts related to gravitation such as quasi-local massstudied by me and collaborators.

There is no doubt that great activities in the interactionsbetween geometry and physics will go on today and in thefuture. And both subjects will greatly benefit from it. Weexpect to see much further interactions between geometry andphysics in the next fifty years.

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Thank you!

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