ajb

Okay, so correct me! Looking at the syllabus of a few universities it does not seem that a typical student leaving with an undergrad education will have been exposed to any high brow mathematics - the same is true of typical physics educations. After that things will depend a lot more on the tastes and interests of the individuals - chemistry research covers a lot of things including things boarding with theoretical physics.

Can you point to some results to back up what you are supporting?

I guess the only thing that can be done is to learn the mathematics and mathematical language needed. One may only need a working knowledge rather than a deep understanding.

Can you expand on that with some nice examples that really are beyond arithmetic, basic calculus, linear algebra and integral transforms? One thing that has been touched upon is 'mathematical chemistry', which does use some graph theory, combinatorics and topology. There is also quantum chemistry, but I think as dealing with more than a few interacting particles explicitly is impossible one resorts to numerical methods - I am not sure that many people in this field really need deep results from topological algebra, operator theory and spaces. Chemical engineering - as suggested in this thread - uses numerical modelling of various phenomena. There will also be the need for some basic statistics and data analysis - it depends on who you ask if this is really mathematics. I know that some group theory can also be useful, but I doubt many chemists are well-versed in groups. For sure group theory is needed in spectroscopy - but how much beyond basic group theory does one need here? I would be very happy to be corrected and shown how advanced mathematics is part of a typical chemists tool kit.

Chemistry is far from my interests - however I am sure you can find MSc programmes on mathematics and chemistry. You have have to leave Kosovo though.

So you are thinking of an MSc or PhD in `mathematical chemistry'? I accept that.

At what level? Generally, I would say that there is almost no mathematics in chemistry apart from basic numeracy. There are exceptions like quantum chemistry, molecular dynamics and similar. And even then, the level of mathematics will depend on personal preferences to a large extent.

Great suggestion - you really can picture what is going on.

Some of these questions you should be able to find reasonable answers to via wikipedia... But as to the point of calculus, well there are two at first seemingly different topics in calculus i) Differentiation ii) Integration The first as you say deals with the instantaneous rates of change of functions (and similar objects). For example, a straight line can be described by y(x) = mx +c. Taking the derivative gives dy/dx = m. This is like the `velocity'. Taking the derivative (w.r.t. x) gives zero - the rate of change of the gradient or slope of a straight line is zero, thus it does not change. There is then an inverse operation of differentiation - given up to an additive constant - this we call the antiderivative. Following the simple example, if I have a constant function m, then the antiderivative is y(x) = mx+c, but the c is arbitrary as there is no way to fix this without some further information. On to integration, this is usually introduced as a calculational tool for working out the areas under a curve y(x) between x0 and x1 (say). This then can be generalised to find volumes and so on. The loose idea is to cut up the area under the curve into thin strips and then add up all these strips. For a finite number of strips you get an approximation to this area. If you consider an infinite number of strips then you get the area - but to make sense of this you need limits. This gives us the definite integral as it is between two points. There is also the indefinite integral, where no bounding points are given. Now, the amazing thing is the fundamental theorem of calculus tells us that the antiderivative and the indefinite integral are the same thing. Thus differentiation and integration are tightly related.

I think that could have been a possibility. For sure special relativity via the Poincare transformations is `written into' Maxwell's equations, as are other important things in modern physics like conformal invariance, gauge invariance and electromagnetic duality (in vacuum). Maxwell's work really was the starting place of a lot of modern physics - so like Einstein and Lorentz's work on time dilation the philosophy of physics was changed by the understanding of the mathematics.

What time dilation tells us is that the old notion of time ticking away in the 'background' is not really the right way to view the Universe. A global time for everyone works okay for Newtonian physics - well this is actually written into the maths - but this is only an approximation and relativity gives us a deeper view of time. Still, lots of things we don't really understand about time and its direction...

I think that you understand this well, but Tim88 I am not so sure... Just for future reference, the Poincare group (Lorentz + translations) is a Lie group - that is both a smooth manifold and a group. Minkowski space-time can be considered as a homogeneous space of the Poincare group. In fact this approach is more used with the supersymmetric extensions or spaces like ADS and DS, but it is worth knowing.

Just to clarify something, one does not think of the manifold of space-time as a field - this is just nonsense - but the metric is a field. Classical fields are sections of various fibre bundles over a manifold. In slightly less technical language, fields are well defined mathematical objects that you attach to space-time.