history4

In classical, or non-quantum-mechanical, general relativity a particle moves along the world line that minimizes the so-called action of the particle: its energy as it moves through time. The action is proportional to the length of the world line, and so a path of least action is a geodesic, or the shortest distance between two points in spacetime. The motion of a string is treated in an analogous way. In a non-quantum mechanical approximation the string also moves in a way that minimizes its action. The action is proportional to the area swept out by the string, and so the world sheet must be a surface of minimum area. If time is regarded as a spatial dimension, the world sheet swept out by a closed string can be thought of as a kind of soap film that joins the string at its starting point and at the end of its path in spacetime. There is an enormous symmetry embodied in the condition that the motion of the string is determined by minimizing the area of its world sheet.

It is a most beautiful and awe-inspiring fact that all the fundamental laws of classical physics can be understood in terms of one mathematical construct called the action. It yields the classical equations of motion, and analysis of its invariances leads to quantities conserved in the course of the classical motion. In addition, as Dirac and Feynman have shown, the action acquires its full importance in quantum physics.

It is increasingly clear that the symmetry group of nature is the deepest thing that we understand about nature today.

[All] chemical binding is electromagnetic in origin, and so are all phenomena of nerve impulses.

From the above brief review, we find there are three versions of geometrization of non- gravitational gauge interactions:

1. Fibre-bundle version, in which the gauge interactions are correlated with the geometrical structures of internal space. [...] the essence of the internal space is still a vexing problem: Is it a physical reality as real as space-time, or just a mathematical structure?

2. Kaluza-Klein version, in which extra space dimensions which compactify in low-energy experiments are introduced and the gauge symmetries by which the forms of gauge interactions are fixed are just the manifestation of the geometrical symmetries of the compactified space. [...] The assumption of the reality of the compactified space is substantial and is in principle testable [...]

3. Superstring version, in which the introduction of extra compactified space dimensions is due to different considerations from just reproducing the gauge symmetry.

String theory has provided a very rich background to study geometry of Ricci flat metrics. Duality concepts have provided very powerful tools. The construction of SYZ
needs to be explored much further, both in terms of construction of special Lagrangian cycles and the perturbation of semi-flat Ricci flat metrics to Ricci flat metrics in terms of holomorphic disks. The fundamental questions in complex geometry are

(1) To find a topological condition so that an almost complex manifold admits an integrable complex structure.
(2) To find a way to determine which integrable complex structure admits Kahler metrics, or weaker form of Kahler metrics, e.g., balanced metrics. There are Hermitian metrics ! so that d(!n􀀀1) = 0:
(3) To find a way to deform a Kahler manifold to a projective manifold.
(4) To characterize those projective manifolds in terms of algebraic geometric data that can be defined over Q
(5) Study algebraic cycles and algebraic vector bundles (or more generally, derived category of algebraic manifolds).
(6) To understand moduli space of algebraic structures and the above algebraic objects.

Yau

	History 4: Atiyah, Green, Schwarz, Witten, Ramond, Weinberg, Salam, Greene, Calabi-Yau
Atiyah	Atiyah In classical, or non-quantum-mechanical, general relativity a particle moves along the world line that minimizes the so-called action of the particle: its energy as it moves through time. The action is proportional to the length of the world line, and so a path of least action is a geodesic, or the shortest distance between two points in spacetime. The motion of a string is treated in an analogous way. In a non-quantum mechanical approximation the string also moves in a way that minimizes its action. The action is proportional to the area swept out by the string, and so the world sheet must be a surface of minimum area. If time is regarded as a spatial dimension, the world sheet swept out by a closed string can be thought of as a kind of soap film that joins the string at its starting point and at the end of its path in spacetime. There is an enormous symmetry embodied in the condition that the motion of the string is determined by minimizing the area of its world sheet. Green Schwarz It is a most beautiful and awe-inspiring fact that all the fundamental laws of classical physics can be understood in terms of one mathematical construct called the action. It yields the classical equations of motion, and analysis of its invariances leads to quantities conserved in the course of the classical motion. In addition, as Dirac and Feynman have shown, the action acquires its full importance in quantum physics. Ramond Weinberg It is increasingly clear that the symmetry group of nature is the deepest thing that we understand about nature today. [All] chemical binding is electromagnetic in origin, and so are all phenomena of nerve impulses. From the above brief review, we find there are three versions of geometrization of non- gravitational gauge interactions: 1. Fibre-bundle version, in which the gauge interactions are correlated with the geometrical structures of internal space. [...] the essence of the internal space is still a vexing problem: Is it a physical reality as real as space-time, or just a mathematical structure? 2. Kaluza-Klein version, in which extra space dimensions which compactify in low-energy experiments are introduced and the gauge symmetries by which the forms of gauge interactions are fixed are just the manifestation of the geometrical symmetries of the compactified space. [...] The assumption of the reality of the compactified space is substantial and is in principle testable [...] 3. Superstring version, in which the introduction of extra compactified space dimensions is due to different considerations from just reproducing the gauge symmetry. Cao Yau String theory has provided a very rich background to study geometry of Ricci flat metrics. Duality concepts have provided very powerful tools. The construction of SYZ needs to be explored much further, both in terms of construction of special Lagrangian cycles and the perturbation of semi-flat Ricci flat metrics to Ricci flat metrics in terms of holomorphic disks. The fundamental questions in complex geometry are (1) To find a topological condition so that an almost complex manifold admits an integrable complex structure. (2) To find a way to determine which integrable complex structure admits Kahler metrics, or weaker form of Kahler metrics, e.g., balanced metrics. There are Hermitian metrics ! so that d(!n􀀀1) = 0: (3) To find a way to deform a Kahler manifold to a projective manifold. (4) To characterize those projective manifolds in terms of algebraic geometric data that can be defined over Q (5) Study algebraic cycles and algebraic vector bundles (or more generally, derived category of algebraic manifolds). (6) To understand moduli space of algebraic structures and the above algebraic objects. Yau	We shall now recall the data of a classical theory as understood by physicists and then reinterpret them in geometrical form. Geometrically or mechanically we can interpret this data as follows. Imagine a structured particle, that is a particle which has a location at a point x of R4 and an internal structure, or set of states, labeled by elements g of G. Atiyah Green The thing that impressed other physicists most about the general theory of relativity is that it is based on very general physical principles—the equivalence principle and general coordinate invariance—and very beautiful mathematical concepts. The relevant mathematics is called different- ial geometry (specifically, Riemannian geometry). The idea is that gravity is a manifestation of the curvature of space-time. Also, the geometry of space-time is determined by the distribution of energy and momentum. The basic equation of motion is In this equation G_mn describes the space-time geometry, G is Newton's constant characterizing the strength of gravitation, and Tmn describes the distribution of energy and momentum. Ramond Furthermore, and now this is the point, this is the punch line, the symmetries determine the action. This action, this form of the dynamics, is the only one consistent with these symmetries ... This, I think, is the first time that this has happened in a dynamical theory: that the symmetries of the theory have completely determined the structure of the dynamics, i.e., have completely determined the quantity that produces the rate of change of the state vector with time. Weinberg Salam For most of us, or perhaps all of us, it's impossible to imagine a world consisting of more than three spatial dimensions. Are we correct when we intuit that such a world couldn't exist? Or is it that our brains are simply incapable of imagining additional dimensions— dimensions that may turn out to be as real as other things we can't detect? Groleau Epistemologically it is not without interest that in addition to ordinary space there exists quite another domain of intuitively given entities, namely the colors, which forms a continuum capable of geometric treatment. Weyl Calabi Lie group methods have proven to play a vital role in modern research in computer vision and engineering. Indeed, certain visually-based symmetry groups and their associated differential invariants have, in recent years, assumed great significance in practical image processing and object recognition. [...] Our approach to differential invariants in computer vision is governed by the following philosophy. We begin with a finite- dimensional transformation group G acting on a space E, representing the image space, whose subsets are the objects of interest. In visual applications, the group G is typically either the Euclidean, affine, similarity, or projective group. We are particularly interested in how the geometry, in the sense of Klein, induced by the transformation group G applies to (smooth) submanifolds contained in the space E. A differential invariant I of G is a real-valued function, depending on the submanifold and its derivatives at a point, which is unaffected by the action of G. In general, a transformation group admits a finite number of fundamental differential invariants, I1; : : : ; IN, and a system of invariant differential operators D1; : : : ;Dn, equal in number to the dimension of the submanifold, and such that every other differential invariant is a function of the fundamental differential invariants and their successive derivatives with respect to the invariant differential operators. This result dates back to the originalwork of Lie (1884)... Calabi et al. Simple Calabi-Yau space	fiber
Green
Schwarz			relativity
Ramond			action
Weinberg Salam Greene Calabi Kaluza Klein Yau			state vector symmetry
			EM

			Calabi-Yau compactify