

History 4:
Atiyah, Green, Schwarz, Witten, Ramond,
Weinberg, Salam, Greene, CalabiYau 



Atiyah
Green
Green
It is a most beautiful and aweinspiring 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.
From the above brief review, we find there are three versions of geometrization of nongravitational gauge interactions: 1. Fibrebundle 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 spacetime, or just a mathematical structure? 2. KaluzaKlein version, in which extra space dimensions which compactify in lowenergy 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 SYZneeds to be explored much
further, both in terms of construction of special Lagrangian cycles and
the perturbation of semiflat 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 (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 R_{4} and an internal structure, or set of states, labeled by elements g of G. Atiyah
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 spacetime. Also, the geometry of spacetime is
determined by the distribution of energy and momentum. The basic
equation of motion is
In this equation G_{mn} describes the spacetime geometry, G is Newton's constant characterizing the strength of gravitation, and T_{mn} 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
[All] chemical binding is electromagnetic in origin, and so are all phenomena of nerve impulses. 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 visuallybased
symmetry groups and their associated differential invariants have, in
recent years, assumed great significance in practical image processing
and object recognition. §
Calabi et al. Not so simple CalabiYau space









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