Atlas Of Mathematics

[Demeter Exekutor]

In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.

A transition map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.)

One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of differentiation of functions on a manifold, then it is necessary to construct an atlas whose transition functions are differentiable. Such a manifold is called differentiable. Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.

If each transition function is a smooth map, then the atlas is called a smooth atlas, and the manifold itself is called smooth. Alternatively, one could require that the transition maps have only k continuous derivatives in which case the atlas is said to be C k \displaystyle C^k .

Very generally, if each transition function belongs to a pseudogroup G \displaystyle \mathcal G of homeomorphisms of Euclidean space, then the atlas is called a G \displaystyle \mathcal G -atlas. If the transition maps between charts of an atlas preserve a local trivialization, then the atlas defines the structure of a fibre bundle.

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The "L-functions and Modular Forms Database," or LMFDB, is a detailed atlas of mathematical objects that maps out the connections between them. The LMFDB exposes deep relationships and provides a guide to previously uncharted territory that underlies current research in several branches of physics, computer science, and mathematics. This coordinated effort is part of a massive collaboration of researchers around the globe.

The scale the computational effort that went into creating the LMFDB is staggering: Multiple teams of researchers spent a total of nearly 1,000 years of computer time on calculations. One recent computation by Principal Research Scientist Andrew Sutherland at MIT used more than 72,000 cores of Google's Compute Engine to complete in one weekend a tabulation that would have taken more than a century on a single computer. As noted by Sutherland, "Computations in number theory are often amenable to massive parallelization, and this allows us to scale them to the cloud." Some of these calculations are so intricate that only a handful of experts know how to do them, and some are so big that it makes sense to run them only once, and then share the verified results.

Prime numbers have fascinated mathematicians throughout the ages. The distribution of primes is believed to be random, but proving this remains beyond the grasp of mathematicians to date. Under the Riemann hypothesis, the distribution of primes is intimately related to the Riemann zeta function, which is the simplest example of an L-function. The LMFDB contains more than 20 million L-functions, each of which has an analogous Riemann hypothesis that is believed to govern the distribution of a wide range of more exotic mathematical objects. Patterns found in the study of these L-functions also arise in complex quantum systems, and there is a conjectured connection to quantum physics.

The connections exploited by Wiles are just a small part of the Langlands program, a vast web of conjectures proposed by Robert Langlands in the late 1960s that has been called the Rosetta Stone of mathematics. The Langlands program is enormous in scope but vague in some of its details; it serves as a framework for the millions of connections now cataloged by the LMFDB. The exact nature of these connections is the subject of a great deal of current research that will be accelerated by the LMFDB.

Several simultaneous events on May 10 in North America and Europe will celebrate the launch of the LMFDB, including public presentations and lectures at Dartmouth College in Hanover, New Hampshire; the American Institute for Mathematics in San Jose, California; and the University of Bristol in the United Kingdom.

The LMFDB project is funded by the U.S. National Science Foundation, the U.K. Engineering and Physical Sciences Research Council, the American Institute of Mathematics, the EU 2020 Horizon Open DreamKit Project, and the Institute for Computational and Experimental Research in Mathematics, and involves researchers from Arizona State University, Dartmouth College, Duquesne University, Oregon State University, the University of California at San Diego, the University of Bristol, the University of Warwick, the University of Washington, the University of Waterloo, and other institutions.

An atlas is a collection of consistent coordinate charts on a manifold, where "consistent" most commonly means that the transition functions of the charts are smooth. As the name suggests, an atlas corresponds to a collection of maps, each of which shows a piece of a manifold and looks like flat Euclidean space. To use an atlas, one needs to know how the maps overlap. To be useful, the maps must not be too different on these overlapping areas.

The overlapping maps from one chart to another are called transition functions. They represent the transition from one chart's point of view to that of another. Let the open unit ball in be denoted . Then if and are two coordinate charts, the composition is a function defined on . That is, it is a function from an open subset of to , and given such a function from to , there are conditions for it to be smooth or have smooth derivatives (i.e., it is a C-k function). Furthermore, when is isomorphic to (in the even dimensional case), a function can be holomorphic.

A smooth atlas has transition functions that are C-infty smooth (i.e., infinitely differentiable). The consequence is that a smooth function on one chart is smooth in any other chart (by the chain rule for higher derivatives). Similarly, one could have an atlas in class , where the transition functions are in class C-k.

In the even-dimensional case, one may ask whether the transition functions are holomorphic. In this case, one has a holomorphic atlas, and by the chain rule, it makes sense to ask if a function on the manifold is holomorphic.

It is possible for two atlases to be compatible, meaning the union is also an atlas. By Zorn's lemma, there always exists a maximal atlas, where a maximal atlas is an atlas not contained in any other atlas. However, in typical applications, it is not necessary to use a maximal atlas and any sufficiently refined atlas will do.

Scientists have unveiled the most compete atlas of the human brain ever created. The work, part of the National Institutes of Health BRAIN Initiative, is the culmination of five years of research, and includes the location and function of more than 3,000 cell types.

This story is from The Checkup, our weekly newsletter giving you the inside track on all things medicine and biotech. Sign up to receive it in your inbox every Thursday.

To Gbor Domokos, a professor at the Budapest University of Technology and Economics, an ordinary-looking rocky outcrop in the hills over Budapest is not just a respite from the busy city, but a wellspring of mathematical questions.

Classification of Finite Simple Groups, one of the most monumental accomplishments of modern mathematics, was announced in 1983 with the proof completed in 2004. Since then, it has opened up a new and powerful strategy to approach and resolve many previously inaccessible problems in group theory, number theory, combinatorics, coding theory, algebraic geometry, and other areas of mathematics. This strategy crucially utilizes various information about finite simple groups, part of which is catalogued in the Atlas of Finite Groups (John H. Conway et al.), and in An Atlas of Brauer Characters (Christoph Jansen et al.). It is impossible to overestimate the roles of the Atlases and the related computer algebra systems in the everyday life of researchers in many areas of contemporary mathematics.

The main objective of the conference was to discuss numerous applications of the Atlases and to explore recent developments and future directions of research, with focus on the interaction between computation and theory and applications to number theory and algebraic geometry. The papers in this volume are based on talks given at the conference. They present a comprehensive survey on current research in all of these fields.

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