User marc - MathOverflow

Answer by marc for Is there any geometry where the triangle inquality fails?

2010-04-30T19:56:29Z

In Information Geometry, the Kullback-Liebler divergence is commonly used in the manner of a metric, but it does not satisfy the triangle inequality. It localizes to the Fisher Information Metric on the probability simplex, the interior of which is diffeomorphic to the geometry of the sphere with the inherited metric from Euclidean space under the map $x \mapsto 2\sqrt{x}.$ This geometry is used in evolutionary game theory to study natural selection.

There are many other information divergences that are not symmetric and do not satisfy the triangle inequality. You can form a divergence that does satisfy the triangle inequality.

Answer by marc for Homological Algebra for Commutative Monoids?

2009-11-17T18:21:59Z

Presummably you could use the forgetful functor to sets and its adjoint to construct a homology theory (using the walking adjunction, simplicial sets, etc) analogously to how group homology arises from the forgetful-free adjunction to set.

The details are probably rather dense and tedious though.

Answer by marc for Examples of mathematics motivated by technological considerations

2009-11-23T03:16:03Z

Contemporary work in biology has lots of examples. With the wealth of sequencing data coming out of the machines at ever lower costs there is a huge need for new methods and models to analyze and understand the data.

Google is another good example. Better (or even acceptable) internet search was desired and it took a mathematical advance to move the technology forward.

Answer by marc for Programming Languages based on Category Theory

2009-11-17T21:51:16Z

Also see here.

Answer by marc for Relating Category Theory to Programming Language Theory

2009-11-17T21:44:29Z

If you want to get your hands dirty, look into the programming language Haskell, which has functors and natural transformations.

Functors implement structure. For instance, consider the category of types T and a functor L that sends a type t to the type L(t), lists of t. This is somewhat similar to the functor on Set that sends a set X to the set of all finite lists on the set X (i.e. the underlying set of the free monoid on X). You could also consider a functor B that sends a type t to the type B(t) of a binary tree on t.

Now you can define a natural transformation from B(t) to L(t) such as flatten, which takes a binary tree and flattens it to a list. So as functors implement structure, natural transformations alter structure.

It gets interesting when you bring in monads. Using the list functor above, think about L(L(t)), lists of lists. You can concatenate a list of lists to a single list, which corresponds to the monad map L(L(t)) to L(t).

Check out this link for more.

Answer by marc for How do I check if a functor has a (left/right) adjoint?

2009-11-17T06:28:21Z

There's the Freyd Adjoint Functor Theorem.

A right adjoint functor is continuous (commutes with limits) and a left adjoint functor is cocontinuous (commutes with colimits). So, if a functor has a left adjoint then it is continuous because it is a right adjoint. The adjoint functor theorem is a partial converse to this fact in the case that the domain category is complete (has all small limits) and the functor satisfies a "smallness condition".

Answer by marc for How do you keep your research notes organized?

2009-11-09T08:10:39Z

I'll echo what Charles said with the addition of using git. You can literally branch your notes just like you would source code and it plays nicely with latex. You can roll back, merge, and share with others very easily. You can tag commits with brief explanations and search those later or use a git visualizer to quickly find something.

Plus all your work is still in text files so you can still use grep and all the usual console tools.

Comment by marc

2010-05-03T21:41:52Z

In evolutionary game theory the Fisher information metric is called the Shahshahani metric. See "Evolutionary Game Theory" by Hofbauer and Sigmund. The information divergence can be used to give a Lyapunov function for the replicator equation. I believe this goes back to Akin; others use an exponentiated form (e.g. Hofbauer and Sigmund).

Comment by marc

2009-11-23T19:34:24Z

Just to emphasize on part of this response, note that an adjoint pair provides a global solution (category-wide) as opposed to a local solution provided by universal maps. So universality is more general, in that if the universal map exists for every object in the category, the family of universal maps can be used to define an adjoint.

Comment by marc

2009-11-23T15:06:19Z

Well it depends on what you mean by mathematical advance. If your metric is the creation of an entire new field, such as graph theory itself in this case, then I guess not.

Comment by marc

2009-11-17T18:16:41Z

It's not all that hard to verify the definition directly in a lot of cases. Do you mean "not tedious"-to-check?