As a student I have grown a curiousity on Research Topics in Mathematics. Anyways I would like to know what are the current Research Topics in Mathematics for Phd students? I just want an idea of what they are researching on, to give me a focus in a particular area of study of the basics needed to understand such research. As well, another question I have. Can the Mathematics of General Relativity be considered a branch in Pure Mathematics? As to describe Geometrical Spaces leaving out the physics, or somewhat incorporating physics. Im interested particularly in Einstein Field Equations as a Pure Mathematics subject rather then a Theoretical Physics subject. Mainly I think my main problem in physics, is that my ideologies in Mathematics conflicts with Physics. I've always been somewhat of a Platonist believing in a Transcendental Mathematics of the Universe. Whereas I've tried rediculously to form understandings in Physics from Pure Mathematics with no applications of Physical Phenomenon which has led to my ultimate failure in understanding Physics. However I do understand that Einsteins General Theory of Relativity has a huge application of Riemann Geometry, is general relativity without the physics an understanding of Riemann Geometry? As for that, please list if you know of, the current research topics in Pure Mathematics for Phd students. I would be thankful for such.
http://www.math.caltech.edu/general/phddesc.html That's the page detailing the PhD program in Mathematics at Cal Tech. You can see the different classes offered and areas of research speciality of their faculty. The three areas of focus for the program and Real and Complex Analysis, Abstract Algebra, and Topology/Geometry. I'm sure you can find a similar page at the site for most universities.
here's the curriculum that's testable on the PhD candidacy exams at my school (in algebra, analyis & topology), including recommended textbooks: http://www.math.uvic.ca/grad/phd/algebra.html http://www.math.uvic.ca/grad/phd/realanalysis.html http://www.math.uvic.ca/grad/phd/topology.html (I assume going through all that would enable someone to understand pretty much any paper on those subjects, and thus be able to do independent research in those general areas) I would say no, General Relativity would be considered part of applied math. Riemannian geometry would be pure math, but when used in GR it would applied.
Actually an enormous number of PURELY mathematical questions have come from studying physical problems and even applications of exisiting mathematics to physical problems. In the case you mention general relativity does actually lead to an interesting set of problems of a purely mathematical nature. One such group has been the topic of a very interesting area of research in the last decade. The subject of differential geometry which is one of the main branches of mathematics involved in general relativity has some imporatnt unanswered questions. Many have nothing to do with GR per se but the proof of a very famous long standing open problem has recentely been accomplished using tools which originally were developed to deal with some issues in GR. The famous Poincare conjecture and the related Thurston's conjecture make general statements about certain kinds of 3 dimensional geometric spaces (actually they basically classify them). A big part of the classification scheme presented by these two have to do with so called "einstein metrics". The question of whether you can always assign a certain geometric object to a space in the way presceribed by einstein's equations is an important one in low dimesional geometry. But though many geometers know GR quite well and many have contributed to issues realting to the physics of GR, they certainly study issues which pertain to math proper. Of course there are many many more examples of "pure math" research which stem from physics issues. For along time it could be said that much of the math realtes to physics was "clean up", in the sense that physicist used mathematics that was essentailly correct and known but they they did so in an informal way and consequently the work of "mathematical physics" was simply to rigorize the statements of physicists. But twice in the last century significant exceptions to this idea came about. The devlopment of quantum mechanics and later quantum feild thoery forced a large amount of activity in the area of "functional analysis". though much was known before QM questions that arose because of physical theories turned out to require study in the purely mathematical sense. Then with the birth of geometric and topological quantum field theories in the 70-90's many interesting question in the fields of pure topology and geometry were either re opened or addressed ina new way. Some surprising connections to seemingly unrelated areas of mathematics also developed from physics itseld in this era/area. So yes while GR and diff geo are far from the main stage in math research these days (you should have been ready in the 60's) there is still some important stuff to do related to this "physics".
I recently have found a deep interest in Differential Geometry through Einsteins GR. Particularly the solutions of GR which are quite interesting such as the Black Hole solution and the not far away, worm hole solution to the equations. The geometry of black hole and worm hole are what I find interesting. I also came across Nash's Embedding Theorems which also caused a huge spark in my interest in Differential Geometry and topology. I want to specialize particularly in Differential Geometry and hopefully to contribute to the progression of the subject. As well, thank you fourier and lose for the links. I also find the books listed in fourier's link a bit more tempting and will definatly come to purchasing my own copies where ever I can find available.
well in that case look up some of the following things: "spectrum of the laplacian" "einstein metrics" "ricci flow" "comparison geometry" How much mathematical background do you have? You will have trouble learning diff geom basics unless you have a solid grip on algebra. In particular you will need to know about bi linear forms (and multilinear forms and tensors) direct sums and products, inner products, exterior products and basic group theory. It pays to learn a little point set topology too and the definition of a topological manifold. Perhaps the easiest place to start is a good grounding in vector calculus and what is sometimes called 'analysis on manifiolds or elementary differential topology.
here are the rest (graph theory, stats, etc), if anyone cares: http://www.math.uvic.ca/grad/phd/phd-index.html
And hovering somewhere between physics and mathematics is the class of so-called "toy theories." I wish I could remember the website, but there is one that considers what Einstein's field equation might look like in a spacetime with one dimension of time but less than three dimensions of space. As I recall, one of the constant coefficients in the equation had to be different from Einstein's in order for the thing to make sense.
Here is a list of the current areas of faculty research at University of Maryland, College Park. PhD candidates will be doing related research. UMCP Math Faculty Research Interests
Actually i'd really like to know as well. Though the sites that these people listed don't really work. Could anybody tell me?
Hi In the 1940s Murray and von Neumann developed the mathematical framework underlying Quantum Mechanics by means of Hilbert space operator theory and the analysis of Rings of Operators. In contemporary times operator theory plays an important role in diverse areas of mathematics and its applications, including, for example, the emerging mathematical methodology for Quantum computers. In their seminal papers Murray and von Neumann identified the need to classify the simple algebras (those with trivial centre) and the topic of classification remains a central one in the modern theory. Research at Lancaster has made diverse contributions to the classification and structure of various antisymmetric operator algebras, in particular, to limit algebras and Lie semigroup operator algebras Thersey cleaning tulsa
To describe research topics, except for the really big problems, in mathematics requires some fairly specialized knowledge. The topics are usually intelligible only to specialists in the specific area, and not even the average PhD mathematician would understand the cutting edge research problems, except for the major ones, outside of his speciality. For an idea of the really major open problems in mathematics you might consider two sources. The most easily accessible are the Millenium Problems of the Clay Mathematics Institute. Each is a major problem and each comes with a description written by a very senior mathematician. http://www.claymath.org/millennium/ The other set of significant problems, many of which have been solved are the Hilbert Problems of 1900. The AMS has a two-volume set that discusses them, or you mght start with this Wiki article. http://en.wikipedia.org/wiki/Hilbert's_problems As far as the mathematics of general relativity goes, that is indeed an area of mathematical researchy. It is called Riemannian (or sometimes pseudo-Riemannian or semi-Riemannian) geometry, and is a branch of differential geometry. Major contributions have been made by the eminent geometer Shing-Tung Yau whose name is also associated with the Calabi-Yau manifolds that are of great interest in string theory.