Geometry, from the ancient Greek geo (earth) and metron (measurement), is often considered a universal quality in human thinking. In fact, this idea of an innate ability to “know” geometry dates back to Plato. In the dialogue Meno, written about 380 BC by Plato, the philosopher Socrates draws out an accurate answer to a geometric puzzle from a young slave who had never studied the subject. With geometry in mind, be it intrinsic or learned, SCGP News invited mathematicians and physicists to share their thoughts on What is Geometry? The only criteria was to provide a candid, perhaps personal perspective, from any angle at any length.
Dennis Sullivan: Geometry Derived From Space and Number
The Old Babylonian scribes from around 2000 BC taught their students how to solve problems utilizing lengths, areas, and volumes.
Euclid discovered a path to knowledge by stating explicit self-evident laws to be used in demonstrations of statements about the geometry of space. This is the beginning of modern geometry. Euclid employed this logical method with his algorithm of division to demonstrate the infinity of prime numbers. Thus also began arithmetic, which relates to geometry.
Descartes invented an explicit coordinate plane and thus introduced real numbers−and hence analysis− into geometry.
Riemann used small right triangles and the infinitesimals of Newton and Leibniz to formulate curvature in models of space in any number of dimensions.
Klein formulated a larger conception of homogeneous geometry in terms of congruences between objects. Congruences can be combined leading to group theory and to Lie theory.
Einstein realized the force of gravity could be interpreted using geometry via Riemann’s curvature of four-dimensional spacetime.
Poincaré simplified geometry, breaking space into benign pieces, whose contiguity properties contained useful and calculable information, beginning the field of topology.
Elie Cartan combined all of the above ideas using parallel transport defined infinitesimally, which added local or gauge symmetry to geometry. This connected to topology via integrands and multivalued functions.
Physicists use observed and conceived geometric symmetries in nature to create models. Gauge symmetry in geometry revealed a richness in the theory of four-dimensional manifolds, Donaldson theory.
Thurston used topology and bounded angle distortion geometry to obtain a complete geometric vision of three-dimensional manifolds, verified by Perelman using a dynamical process on the space of Riemannian geometries.
Gromov created a geometry in the large for groups that gives much insight into the theory of those discrete groups that possess only very thin large triangles. He also created, via an infinite dimensional continuous group, symplectic geometry containing the topological string theory of physics.
Grothendieck, building on Descartes, formulated geometry purely in terms of rings of number-like objects, which can be added and multiplied without regard to order. This geometry is based critically on Euclid’s infinitude of primes and there is a perfect duality between geometry and algebra.
Tate founded a rigid geometry for each prime p motivated by questions that led to the final solution of the Fermat Theorem. Recently, Scholze perfected this setting of Tate, his contribution being like Newton’s addition of limits to classical Greek geometry. This event seems to be the beginning of a new epoch in rigid geometry.
Finally, the discrete or quantum nature of charge plus the reality of space in nature renders plausible the appearance of all of the geometric and arithmetic aspects of mathematics that occur simultaneously in today’s physics models.
What first attracted me to geometry was learning about differential forms, the d operator, and Stokes theorem. The de Rham theorem came next, and this was soon followed by learning about Lie groups and Lie algebras. I was hooked!
A large part of the appeal of geometry, for me, comes from the interaction with what one might call symbolic methods. We have some algebraic formula, say, or a differential equation, which we think about by manipulating symbols on the page. Then we draw a picture and “see” what is going on. The delight is in the interaction between the two ways of thought.
The sight is the richest sensation humanity owns. When we try to explain something, we often draw a figure to visualize the contents for an explanation. The role of geometry in mathematics is not so much to give an answer to a problem, since providing an answer to a mathematical question by calculation etc., is usually the role of algebra. The role of the geometer is to say what the answer (often a number) means. Geometry in that sense is not so practical since the answer is given anyway. However, humankind is unable to understand a complicated and chaotic calculation. We need something–a notion, a concept, or a vision–to understand why we calculate that way. That is, I think, the role of geometry in mathematics. Figures appearing in modern mathematics are usually beyond the ability of one’s eyesight. Nevertheless, we visualize important concepts too complicated to understand without geometric meaning. It is amazing how much we (people studying modern mathematics) came far away from our physical eyesight, free from restriction of the human body, as does modern science and technology. The frontier of science and technology requires reaching beyond the border of our imagination, which, I believe, modern geometry can provide.
Virtually every mathematical notion involves some form of spatial image in our mind: the arrow of a map f: A -> B or an element x of a set A. Replace these by the words ‘transformation’ or ‘point in a space’ and the abstract becomes more geometrical. Geometrical objects in mathematics carry with them an aura of mental concepts—visual, tactile analogues of the physical world around us—which help us to open up avenues of exploration using whatever mathematical tools are at hand.
Einstein has taught us that the force of gravity is due to the geometry of space and time. A geometry that is constrained by Einstein’s equations. A fascinating fact is that the geometry of the universe at large scales is almost flat, but not exactly. It has small deformations that seem to originate from small quantum fluctuations produced at the beginning of the universe. The cosmological shape of our universe is like that of a giant microscope which amplifies small quantum effects to the scale of the whole universe. This fascinating connection between the very small and the very big lies at a very active frontier of current exploration.
I know of no satisfactory all-encompassing definition of what geometry is, but only examples of what it is. Sufficiently broadly defined it can include all we do in physics and mathematics these days. But perhaps for a narrower definition we can use the age old view of what geometry is: the study of all things that can be measured by a ruler!
Editors Note: The following contribution by Jean-Pierre Bourguignon has been edited from an interview:
Concerning the question ‘what is geometry?’—for me, the key characteristic of geometry has been, after a long period of uniqueness, its remarkable capacity to broaden itself several times. One thing I find marvelous and that, actually, lies at the heart of a place like the Simons Center for Geometry and Physics is the perpetual struggle scientists face to understand what is space. This question is critical for physics of course. And the whole point of geometry is to study structures that enrich spaces. So, for me, geometry is the theory of spaces, endowed with all kinds of extraordinary structures. •