Mathematics has been always one of the most active field for researchers but the most attentions has gone to one or few subjects in one time for several years or decades. I'd like to know what are the most active research areas in mathematics today?
Yes, mathematics has been always not only one of the most active field of the researchers, it was for a long time along with philosophy one of the first sciences. But, it’s hard to say what are the most active research areas in mathematics today or what are the most important scientifically explored in mathematics. Less and less support is provided for purely fundamental mathematics is nowadays, and more and more is required to solve specific problems by "someone else" i.e. mathematics turns into a servant of other sciences
Can I ask you which area of mathematics you are interested in? Something which touch many fields is "big data" (engineering, statistics, Earth sciences, ... ).
you may be interested in the Swiss CSEM photovoltaic center and their latest research outputs - https://www.csem.ch/pv-center
Jean-Philippe Montillet my main interest is geometry, specially analytic hyperbolic geometry and its connection with other fields, but also I'd like to know about the most active subjects of math
I am not an expert in this field.However, there are applications in engineering (complex networks - see https://arxiv.org/pdf/1006.5169.pdf) Mathematical Physics (https://scholarworks.lib.csusb.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=2099&context=etd-project) and some new developments in Riemann geometry. You may want to look to the discussions in https://www.researchgate.net/topic/Hyperbolic-Geometry
- Best -
I take it from responses you would most interested in applicable math.
The most usfull areas, in my taste, here would be
a) Extensions of vector analysis
The language developed by Elie Cartan, making use of the external or wedge product (grassman algebra) supplementd by ideas due to Hodge has permitted for example electromagnetic theory expressed in different language ie, in
Roger Penrose , road to Reality. The subject, geometrical algebra in David Hestenes is also along these lines, but with many more results and for versions of vector analysis in larger dimensions.
b) Quantum theory has resulted in interest for eigenalue problems, whose more general expression is contained within Sturm Liouville theory which pre dated the Quantum. Many systems of Special functions result..many other problems are tackled, such as aeronautical problems. Schrodinger is a special case of Sturm Liouville with weight function equal to 1, and another function equal to 1
Some versions of the Quantum (not very accepted at least yet) alllow this generalization; but no fundamental physical principle is actually violated.
c)If you want to try a field not so applicable, but fascinating; you can ask what sorts of number theories encompass the Complex numbers.
Most say along the lines of Quaternions and normed division algebras. However
this is partly false, because quaternions do not commutate with product, and complex numbers do.
Then one can seek commutating numbers in the area of rings instead of fields.
(field is a special case of ring)..and here you find numbers with any dimension.
But this is not very developed yet. You find anologs of Cauchy Riemann and other properties of the Complex, do not know of applications yet.
Division si not allowed for each nonzero numbers, but can be over most of them.
There is a determinant instead of a norm.
For reasons I do not understand, some characterize this as geometrical algebra.
In my opinion I would say it's homotopy and manifold. In these areas researches are being carried out on daily basis.
Look at arxiv.org under Mathematics. Today's papers are by definition what mathematicians are working on right now.
Dear Mahfouz Rostamzadeh,
`most active´ is a difficult question. One has to know all. But only could give his personal vote.
Near to the most important research I think may be that which I found.
1. Proof of multiplication (2500 year problem)
The binoms of the Babyloniens seem to be the proof for the result of factorization including signs (prefixes).
To us three were teached: (+a + b)2; (+a – b)2 and (+a + b) (+a – b)
But with full variation of the prefixes there must be 16.
Who defined those (two of all with new result), which produces
1.1 –a2 – 2ab – b2 ? [(+a + b) (–a – b) or (–a – b) (+a + b)]
1.2 – a2 + 2ab – b2 ? [(+a – b) (–a + b) or (–a + b) (+a – b)]
2. New definition of the square-root (250 year problem)
To take the square-root makes leaving information of the factors which produced the radicand. Thereby confusion of imaginary numbers come.
Complete on my homepage.
3. Field
Two, in hierarchie, different operations in one group gets disproved.
Inverse operation can´t be excluded from the group; it comes intrinsic by negative prefix vice versa.
4. Set theory
Disproved and new formulated.
[Addition and subtraction in the set; multiplication as well as the inverse operation `division´ generates a new set. Naturals in a set are equal one to another. So three and (plus) seven results to `two natural numbers´; not to ten! If you look for if three or seven, you sort and don´t look at the property `natural´ only.]
5. Euler-Equivalence
Disproved.
6. Cantor (150 year problem)
Second diagonal-argument disproved.
And more. So good luck with this new research and hello again at my homepage.
Cheers, Peter
Yes, mathematics has been always not only one of the most active field of the researchers, it was for a long time along with philosophy one of the first sciences. But, it’s hard to say what are the most active research areas in mathematics today or what are the most important scientifically explored in mathematics. Less and less support is provided for purely fundamental mathematics is nowadays, and more and more is required to solve specific problems by "someone else" i.e. mathematics turns into a servant of other sciences
Back to applied would not be a bad thing. Today everybody talking about something different in math. You have to be in a specilized nitch; more unity needed.Too many definitions. Thats why I mostly stick to physics.
Vector analysis is what Ive used more than anything else, I think. Fourier ,special functions and differential equations a close second. We dont get around to proving everything, just to see if results are reasonable.
thanks dear @Juan weisz
one of my main interests is mathematical physics, thise area has a lot to say
@ Mahfouz
If your interest is on mathematical physics, I am interested in your opinion to the (only?) unit in mathematics, the imaginary unit.
Does math have the power to nominate a unit or is it for physics only?
Why should there be a unit including only one expression?
Physical units must have three: number (of amount), measure, unit. 1[m] length i.e. .
Are you interested in my solution to that problem?
To Juan Weisiz
Excuse me, but I do not know at what level you are dealing with physics. Everything you mentioned about mathematics is classics - the level of study, not "high" science in physics. I taught students of physics linear algebra, group theory, and Theory of representations of groups. For serious scientific work in physics, a lot of mathematics is needed, and sometimes new fields of mathematics are created for solving problems in physics.
Mathematics needs more or less almost all sciences, but mathematics is without applications science itself -- "the queen of science."
How would you treat the philosophy, which, along with mathematics, is the oldest science, which has no application and does not benefit ?
@ Mirjana Vukovic
Physics will beat mathematics — look at my reform!
By that mathematics become a natural science. Without the reform math hasn´t to do anything with reality (because there is no bijectivity (isometry) to it). The only unit math deals with is the imaginary one. But that´s an unsolved problem which get´s solved by my reform.
Mathematics is a science that creates, models, describes, explains, applies, and of course its areas of research are always new. Ask about a branch in recent development and of interest to the scientific community is to prepare to find countless answers. Of course, the investigator's self-interest will guide him to appropriate topics.
I have read some answers to this question, published in this medium and I am surprised. They do not do science a favor with them.
What is the reason for writing " Physics will beat mathematics — look at my reform! " . What reform? Somebody knows about that reform?
Physics is a great science, and from its observations mathematics has developed very serious theories (Fourier - Heat, Gauss - sound). They were times of illustration. And conversely, Physics has found that without the development of mathematics there are many observations that could not be modeled.
But, returning to the initial question, I think you should look for an answer, and in that I agree with @ Mirjana Vukovic , if your interest is in pure or applied mathematics.
Greetings from Venezuela
Dear Jorge,
it would seem very impertinent to give ever and ever one´s own adress for homepage.
But for the reform you may look here:
www.mathe-neu.de
You may look on RG what I have written (and answered). For that the button `follow´stays. And you may ask me personal.
Many know my reform, look at the counter.
Mirjana
As a physicist, my reaction is that more empirisicim and less speculation is needed now. Physics has not advanced very fast recently, the last thing is
perhaps the gravitation wave detection. Some say the same as 70 years ago.
People try to fill this void with all sorts of stuff, often involving higher math.
However physics is different. Before applying the math you have to know what is really going on. That is why what we often use is consolidated math from 100 years ago. We do not have to chase the latest math wave. Physics is about measurable stuff, with units.
Algebra and groups that you mention is usefull.
Perhaps philosophy might help someone to live better.
It is difficult to say this or that is the most active research area. mathematics has expanded in various directions;exploded like a bomb. And if you go across any direction you can find new/active areas of research.
The original question focused on current trends in mathematics. Notice that a number of responses to this question focus on applied mathematics.
Pure Mathematics. For a bird's eye view of what is happening among young researchers in pure mathematics, see the current issues of
Notices of the American Mathematical Society
https://www.ams.org/notices
Applied Mathematics.
@ Juan Weisz : As a physicist, my reaction is that more empirisicim and less speculation is needed now.
For a good sampler in applied mathematics, see the bi-monthly
Mathematics of Computation
http://www.ams.org/publications/journals/journalsframework/mcom
One of the most active applications of mathematics is theoretical physics, in particular, quantum field theory and statistical physics. Many of the ideas that emerged in quantum field theory and statistical physics gave rise to important theorems, the proof of which is to be expected by future mathematicians. Many mathematicians are engaged in problems of theoretical physics. At mathematical congresses, mathematics quite often discusses the problems of theoretical physics.
Dear Mr. Rostamzadeh,
More and more people should get interested in solving the Millenium Prize problems especially for specific applied scientific fields like we have done for the new Econophysical Network Field of AGGNNNetworks which are the fundamental Dbranes string theoretic field for cybernetic Econophysics String Matching Field Theory. You can take a look at our publication on www.researchgate.net/profile/Soumitra_Mallick.You may be able to solve for Nuclear information-energy technologies for civilian use by using our developed field and method. We think our European and American and Indian Professors, particularly will like it if you do so.
Soumitra K. Mallick, Corresponding Author,
for S.K.Mallick, S.Raychaudhury, S.Mallick
of The RHMHM School,
USA, Japan and India.
It is not an answer to the original question, as currently only myself does research in this field. But it is one of the most promising new areas of research.
I have discovered a completely new way to describe general topology and this looks very promising. The following Web page contains all my research freely available:
http://www.mathematics21.org/algebraic-general-topology.html
@ Victor Lvovich Porton
Sorry, but your `discovered way´ uses theses which got disproved by my papers (Cantor, Proof, Cantor 3).
C (choice) of ZFC uses elements of sets more times (actual).
Infinite sets couldn´t be graduated (in potency, cardinality). There are no properties for infinite sets among themselves like `small´ or `less´.
You may have a view at my work: www.mathe-neu.de
WARNING: Peter Kepp states that all infinite sets have the same cardinality.
ALL-CLEAR
A proof exists!
IN as well as IQ are classified as to be countable (Cantor `slash 1´).
The power of a set is equal to the power of another set if there exists a bijection among themselves (definition).
Excluding the (illogical) empty set, the development of finit power-sets goes:
{1} => P{1} = {1}
{2} => P{2} = {{1}; {2}; {1,2}}
{3} => P{3} = {{1}; {2}; {3}; {1,2}; {1,3}; {2,3}; {1,2,3}}
And so on (if one wants, including {0}).
One never could depict all elements of an infinite set; so a symbol is used for that.
For the naturals: IN. For the rationals: IQ. For the reals: IR.
For whole the power-set of IN: P{IN}.
Cantor couldn’t depict all rationals. He introduced his first slash-argument by representing the development of them. This also could be done for the power-set of an infinite set, as I did above.
Now (the development of) the relations:
1 {1};
2 {1}; 3 {2}; 4 {1,2};
5 {1}; 6 {2}; 7 {3}; 8 {1,2}; 9 {1,3}; 10 {2,3}; 11 {1,2,3};
And so on.
But the relations get surjective. Is IN by that more powerful than P(IN)?
It´s easy to recognize that each higher developed line includes its predecessor.
Excluding all double relations makes the relation bijective. By that you are ever at the last line, the development actually (IN P{IN})!
All of you are invited to disprove or accept this.
Good luck further with the conclusions (except `slash 1´) of Cantor. Don’t get the blame.
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