In 1994 or 1995, some colleagues and i made a list of problem areas in mathematics that would be enormously helpful in several computing applications - among the topics were: (0) the mathematics of self-reference (we learned of a book, non-well-founded sets, by peter aczel, that was a good start for that work), (1) the mathematics of big (not infinite, but too large to treat individually and too small or diverse to treat collectively), (2) the mathematics of abstraction and mapping among different formal spaces (as in different formal models of real-world phenomena), (3) the mathematics of logics of importance and relevance (analogies and significance instead of likelihood), (4) the mathematics of integration among different uncertainly measures (both probability and other kinds), and a few others i don't quite remember now

we saw much discussion and some progress, but much more is needed

Solutions of non linear differential equations.

Thanks Claude. Could you please expand your answer a bit more to clarify your suggestion? As far as I know, one can write down fairly large classes of solutions for integrable nonlinear PDEs (http://en.wikipedia.org/wiki/Integrable_system#Solitons_and_inverse_spectral_methods), and quite often a number of solutions can be found using symmetries (for starters see e.g. my answer to this question: https://www.researchgate.net/post/I_started_my_research_in_Controlling_Chaos_There_are_lots_of_methods_to_solve_such_non_linear_PDEs_how_far_is_theHomotopy_Analysis_method_efficient ). However, expecting that the mathematicians will some day be able to provide general solutions for *generic* nonlinear PDEs sounds a bit too optimistic to me...

Non linear PDE and integrability constitute indeed a very new field of research in mathematics. It is not possible to precisely say what is needed since we don't know what will be found. But it can be forcast that it will have an increasing role in fundamental physics, and will inspire entirely new concepts. The twistors that are central in integrability theory were first thought out by physicists for physics. The inverse spectral method itself makes use of wave functions and Hamiltonians, the very basis of quantum mechanics. Let's mention fiber bundles too. So it is less about "ready-to-wear" solutions than methods and concepts.

The Mori-Zwanzig equation is an exact consequence of classical and statistical mechanics. It ,\takes an statistico-mechanical time evolution look sort of like the Langevin equation with memory. However, the memory kernel of the Mori equation has a time evolution determined by a projected time evolution operator, and little is known about the effect of projection on time evolution. I expect that something related to Lie algebras (not Lie groups) might be useful here.

Mathematics an important tool for every researcher . i think COMPUTATIONAL MATHEMATICS is importance branch of Mathematics in view of its application to other branch of sciencs.

For this thread, it would be helpful if Mathematica included the results of mappings between the tangent spaces for differential manifolds.

'C with classes' once was the precursor of C++. Sometimes I think that 'Mathematica with classes' would be a great thing. Thinking deeper I become aware that Mathematica is already 'with classes' but in a cryptic sense which does not allow using classes as easy as in the typical OO languages (among which C++ is my favorite). In 'Mathematica with classes' differentiable manifolds could be preformed as a class and James' desire concerning mappings between tangent spaces could be satisfied in a natural manner.

In molecular modeling is a significant problem. Not that it was not solved at all, but it was not solved effectively. The problem is old, but the decision was never found.

When we calculate the trajectories of particles we do it in small steps. The step size is limited to rapid movements in the model. In practice the steps are 0.5-1 fs. A natural solution is to partition the system into parts with different characteristic velocities and calculate with different steps. Such algorithms are available and 2 fs steps are allowed. However, at the 5 fs starts instability which does not allow to make great steps, even if we have some very slow motions in system.

There are some algorithms that could theoretically improve the situation somewhat, but all have problems impeding their practical application.

If a suitable algorithm will be developed, it would have significantly improved our simulation capabilities.

@Artur, one of these is Lyapunov and Krasovskii theory for stability study!

@Ljubomir: thanks. Could you please expand a bit as to which open problems in the stability theory you would like to see solved and for which sort of scientific applications are these problems important?

There are several coarse-graining approaches that may sometimes be helpful. The Mori-Zwanzig treatment is an example.

@Artur, I do a lot of research in stability, both theory and practice. Recently, I have found nice paper on that issue by my colleague Antonio Loria.

He says: " Firstly, we recall an open problem on output feedback control of Euler-Lagrange systems and show that the stumbling block to prove global asymptotic stability is the lack of proof of global boundedness of the trajectories. Secondly, we show how the latter may be formulated in terms of Input-to-State Stability. More precisely, it is desired to construct a Lyapunov function for systems satisfying "La

Salle's" conditions in the absence of inputs"

There seems to be a very simple and very robust (stable) method of integration, which I tried to promote in a RG question:

"Who knows of an integrator for the general first order ODE which is second order and needs only one evaluation of the equation's rhs per time step?"

This question created an intensive and instructive discussion. One of the participants (H.E. Lehtihet) proposed example systems which made evident a certain kind of weakness of the method which I finally was able to overcome in an elegant and simple manner. The final method is now available from arxiv:

http://www.arxiv.org/abs/1311.6602

I call it ADALF for averaged densified asynchronous leap-frog (method).

https://www.researchgate.net/post/Who_knows_of_an_integrator_for_the_general_first_order_ODE_which_is_second_order_and_needs_only_one_evaluation_of_the_equations_rhs_per_time_step

In 1994 or 1995, some colleagues and i made a list of problem areas in mathematics that would be enormously helpful in several computing applications - among the topics were: (0) the mathematics of self-reference (we learned of a book, non-well-founded sets, by peter aczel, that was a good start for that work), (1) the mathematics of big (not infinite, but too large to treat individually and too small or diverse to treat collectively), (2) the mathematics of abstraction and mapping among different formal spaces (as in different formal models of real-world phenomena), (3) the mathematics of logics of importance and relevance (analogies and significance instead of likelihood), (4) the mathematics of integration among different uncertainly measures (both probability and other kinds), and a few others i don't quite remember now

we saw much discussion and some progress, but much more is needed

I enjoyed learning about the list conveyed by Christopher L. I particularly support the point (0), i.e. how to understand (and compute) the problem of self-references in order to tackle problems in biology and AI. To me it is not a fruitful idea to avoid self-references via logical extension (e.g. non-well-founded sets) of various sorts.

Why not embrace self-references like Cantor did in connection with his theory of transfinite numbers. There is no problem to treat self-references in naive set theory and linear algebra if you allow for general complex symmetric representations with Segrè characteristics > 1.

I would also like the mathematicians to develop analytic function theories in more than one variable.

My dear friends, again downvotes! What are the reason for? It is interesting to point out that same people downvote many threads and answers, without taking part in a discussion! Do You recognize him my dear @Artur?

since many biological systems are subject to periodic external influences, both environmental and human-induced (e.g. therapies) , a theoretical "quantum leap" in the field of nonautonomous dynamical systems (both deterministic and stovhastic) would be of extreme relevance.