Answer by just-someone for New grand projects in contemporary math
Closely related to the theory of hyperbolic conservation laws already mentioned by András Bátkai is the study of dispersionless (i.e., rewritable as first-order quasilinear homogeneous) systems...
View ArticleAnswer by Gil Kalai for New grand projects in contemporary math
The concentration of measure phenomenon and (related to it) the modern developments related to isoperimetric relations can be regarded as a large mathematical programme which involves analysis,...
View ArticleAnswer by Nick Gill for New grand projects in contemporary math
Within the realm of finite permutation group theory there are a series of projects that could be collectively entitled The classification of finite combinatorial objects subject to transitivity...
View ArticleAnswer by András Bátkai for New grand projects in contemporary math
The theory of nonlinear dispersive equations, hyperbolic conservation laws, etc., seeTerece Tao's book on the subject,Jean Bourgains bookorHelge Holdens co-authored monographs.
View ArticleAnswer by John McKay for New grand projects in contemporary math
The CFSG (classification of finite simple groups) yields L: The finite groups of Lie type,and S: the non-Lie groups = 26 sporadic simples. We do not know how natural this taxonomy is.One approach is...
View ArticleAnswer by Alexander Chervov for New grand projects in contemporary math
In information theory (error-correcting codes) the grand achievements in 90-ies are turbo-codes and LDPC codes. Recent 2009 discovery which became hottest topic is polar codes. It istempting to say...
View ArticleAnswer by Adrien Hardy for New grand projects in contemporary math
Universality phenomena for determinantal point processes and relatives. After the deep results obtained by many great researchers concerning independent random variables, lot of attention has been...
View ArticleAnswer by Timothy Chow for New grand projects in contemporary math
Graph minor theory. The first success of this theory was the proof of Wagner's conjecture that in every infinite set of finite graphs, one is a minor of the other. However, the theory developed over...
View ArticleAnswer by Abdelmalek Abdesselam for New grand projects in contemporary math
Work on the mathematical foundations of quantum field theory.See for instance the recent review by Michael Douglas: "Foundations of quantum field theory"inProc. Symp. Pure Math. vol 85, 2012. See also...
View ArticleAnswer by Dirk for New grand projects in contemporary math
In numerical mathematics there is a recent new grand theme called randomized numerical linear algebra (RandNLA). One example (probably even a paradigm) is the "randomized range finder" from the paper...
View ArticleAnswer by Jonny Evans for New grand projects in contemporary math
One grand project which has generated much work is the quest for mathematical understanding of mirror symmetry (via homological mirror symmetry, or the Strominger-Yau-Zaslow/Gross-Siebert picture)....
View ArticleAnswer by Gil Kalai for New grand projects in contemporary math
A successful grand project is the classification of finite simple groups. It was completed but several exciting follow-up projects are ongoing.
View ArticleAnswer by Kevin R. Vixie for New grand projects in contemporary math
I have three answers. The first two involve mathematics with an applied flavor, with strong connections to mathematics of a purer flavor. The last one is purer in origin, but full of potential for...
View ArticleAnswer by Johannas for New grand projects in contemporary math
I might say that algebraic stacks and their development might be a "grand project" (although, admittedly, this is something that I do not know a whole lot about). In particular, there is the "Stacks...
View ArticleAnswer by JSE for New grand projects in contemporary math
I think it is certainly appropriate to denote as a "grand project" the remarkable new progress in the area sometimes called additive combinatorics or additive number theory, though the subject has...
View ArticleAnswer by Joseph O'Rourke for New grand projects in contemporary math
This is a question rather than an answer.Shmuel Weinberger wrote a (to me, amazing) book about a decade ago, entitled"Computers, rigidity, and moduli. The large-scale fractal geometry of Riemannian...
View ArticleAnswer by Aleksandar Bahat for New grand projects in contemporary math
The field with one element $\mathbb{F}_1$ (a.k.a. F-un).Having a precise notion of such a field would allow us to further exploit the analogy between number fields and function fields (much like...
View ArticleAnswer by Gil Kalai for New grand projects in contemporary math
A grand programme in mathematics with some enthusiastic supporters is that of computerized proofs for mathematical theorem.Another related (but different) programme is that of automatic verification of...
View ArticleAnswer by Gil Kalai for New grand projects in contemporary math
Theory of computing and more specifically computational complexity and the NP=!P problem represet an important new paradigm in mathematics. This offers new look on classical issues like the study of...
View ArticleAnswer by Sam Hopkins for New grand projects in contemporary math
Is quantum computing too far away from pure math to qualify? Perhaps it has not shaken up mathematics so much, but it has brought together theoretical physics, computer science, and mathematics in a...
View ArticleAnswer by alvarezpaiva for New grand projects in contemporary math
The simultaneous study of a space $X$ and its observables $F(X)$ (real, complex, or operator-valued functions on $X$) is an old topic, but with quantum groups and non-commutative geometry it has been...
View ArticleAnswer by Ricky for New grand projects in contemporary math
The Langlands program. It goes back to the sixties, but in the last years, with the proof of the fundamental lemma by Ngô Bảo Châu and with several results in the local case, it became one of the most...
View ArticleAnswer by Benoît Kloeckner for New grand projects in contemporary math
Optimal transport. Both its study (generalizations, Monge problem, regularity issues, and geometric properties to cite the part I work in) and its applications (to geometry notably with the Work of...
View ArticleAnswer by Benoît Kloeckner for New grand projects in contemporary math
Ricci flow. It did solve Poincaré's conjecture and the $1/4$-pinching conjecture, but has also become an object of study. More generally, it has launched a large amount of work on geometric flows (mean...
View ArticleAnswer by user30304 for New grand projects in contemporary math
Hyperfields"Krasner, Marshall, Connes and Consani and the author came tohyperfields for different reasons, motivated by different mathematicalproblems, but we came to the same conclusion: the...
View ArticleAnswer by AAK for New grand projects in contemporary math
There is the derived algebraic geometry program of Jacob Lurie, starting with his thesis in 2007 and building on the work of ..., Simpson, Toën-Vezzosi, etc. In the words of Lurie, this is basically...
View ArticleAnswer by Yuichiro Fujiwara for New grand projects in contemporary math
Does compressed sensing count as math? If it does, here is a blog post from the horse's mouth.Edit: For those who would like a popular article, here is a good one in Wired (by JSE if I'm not mistaken)....
View ArticleAnswer by David Corwin for New grand projects in contemporary math
Manjul Bhargava's new field of arithmetic invariant theory is a perfect example of a new grand project. It began with Manjul's doctoral thesis, in which he presented a completely new view of Gauss's...
View ArticleAnswer by user30304 for New grand projects in contemporary math
Tropical mathematicsThe algebraic geometry, analysis and other mathematics over the tropical semiring instead of the real numbers.
View ArticleAnswer by user30304 for New grand projects in contemporary math
Large Networks and Graph Limits - new AMS book by László Lovász"The theory has rich connections with other approaches to the study of large networks, such as ``property testing'' in computer science...
View ArticleNew grand projects in contemporary math
When I was a graduate student in math (mid-late eighties and early nineties) the arena was dominated by a few grand projects: for instance, Misha Gromov's hyperbolic groups, which spread into many...
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