As the name suggests, this is the study of geometry using algebra. In this example, we get a circle! Some mathematicians spend their whole lives on this majestic subject. If we can figure out how to reduce topology to algebra, it might help us formulate a theory of quantum gravity. As a kid, I liked physics better than math.
My parents lived in Washington, D. This was fascinating. When I was 8, he gave me a copy of the college physics textbook he wrote. I found long division insufferably boring and refused to do my math homework, with its endless repetitive drills. But later, when I realized that by fiddling around with equations I could learn about the universe, I was hooked. The mysterious symbols seemed like magic spells.
And in a way, they are. Science is the magic that actually works. I already knew that there was too much math to learn it all, so, in grad school, I tried to focus on what mattered to me. As a graduate student, this seemed like a terrible limitation. Why does algebraic geometry restrict itself to polynomials? By restricting the breadth of their investigations, algebraic geometers can dig deeper into these mysteries. But for its true devotees, it is an end in itself. I once met a grad student at Harvard, and I asked him what he was studying. To penetrate even the first chapter of Hartshorne, you need quite a bit of background.
To read Hartshorne is to try to catch up with centuries of geniuses running as fast as they could. From about to , Grothendieck revolutionized algebraic geometry as part of an epic quest to prove the Weil Conjectures, which relate varieties to solutions of certain problems in number theory. Grothendieck guessed that the Weil Conjectures could be settled by strengthening and deepening the link between geometry and algebra.
He had a concrete idea for how this should turn out. But making this idea precise required a huge amount of work. To carry it out, he started a seminar.
He gave talks almost every day, and enlisted the help of some of the best mathematicians in Paris. Working non-stop for a decade, they produced tens of thousands of pages of new mathematics, packed with mind-blowing concepts. In the end, using these ideas, Grothendieck succeeded in proving all the Weil Conjectures except the final, most challenging one. As a grad student I steered clear of them, since I was busy struggling to learn physics: There, too, centuries of geniuses have been working full-speed, and anyone wanting to reach the cutting edge has a lot of catching up to do.
Had I taken a different path, I might have come to grips with his work through string theory. String theorists postulate that besides the visible dimensions of space and time—three of space and one of time—there are extra dimensions of space curled up too small to see.
- String theory and noncommutative geometry.
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In some of their theories these extra dimensions form a variety. So, string theorists easily get pulled into sophisticated questions about algebraic geometry. And this, in turn, pulls them toward Grothendieck.
Symposium on Noncommutative Geometry
For example, string theory is shockingly good at counting how many curves of different kinds you can draw on certain varieties. Thus, we now see string theorists talking with algebraic geometers, each able to surprise the other with their insights. In particular, his page unpublished manuscript Pursuing Stacks , written in , made a big impression on me. At first this idea may sound just like algebraic geometry, where we use algebra to describe geometrical shapes, like curves or higher-dimensional varieties.
Thanks to my work on physics, his proposal was tremendously exciting when I came across it. It seems that until we do this, our understanding of the fundamental laws of physics is doomed to be incomplete. One reason is that quantum physics is based on algebra, while general relativity involves a lot of topology.
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But that suggests an avenue of attack: If we can figure out how to reduce topology to algebra, it might help us formulate a theory of quantum gravity. My physics colleagues will let out a howl here, and complain that I am oversimplifying. View cookies policy. Algebraic Geometry Zoom Zoom. Availability Usually despatched within 2 weeks. With Free Saver Delivery.
Geometry and physics
Facebook Twitter Pinterest Share. Description Reviews More Details. Description This is a monograph about non-commutative algebraic geometry, and its application to physics. The main mathematical inputs are the non-commutative deformation theory, moduli theory of representations of associative algebras, a new non-commutative theory of phase spaces, and its canonical Dirac derivation. The book starts with a new definition of time, relative to which the set of mathematical velocities form a compact set, implying special and general relativity. With this model in mind, a general Quantum Theory is developed and shown to fit with the classical theory.
Related Geometry of Time-Spaces : Non-Commutative Algebraic Geometry, Applied to Quantum Theory
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