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Dror Bar-Natan:
Talks:
# A Homological Construction of the Exponential Function

## Graduate Student Seminar, University of Toronto

### BA6183, 12PM, March 8, 2007

**Abstract. ** Often in mathematics one needs to solve certain
non-linear functional equations and sometimes one can be helped by
certain techniques from homological algebra. After quickly mentioning a
few typical examples for that need I will concentrate on the simplest
example I'm aware of - finding a solution *e* of the non-linear
functional equation *e(x+y)=e(x)e(y)* within the algebra
**Q**[[x]] of power series in the variable *x* with
rational coefficients. The point of course is the technique, not the
actual solution, which I'm sure you all know well.

It is worth noting that in some a priori sense the existence a solution
of *e(x+y)=e(x)e(y)* is unexpected. For *e* must be
an element of the relatively small space **Q**[[x]] of
power series in one variable, but the equation it is required to
satisfy lives in the much bigger space **Q**[[x,y]] of
power series in two variables. Thus in some sense we have more
equations than unknowns and a solution is unlikely. How fortunate we
are that exponentials do exist, after all!

Also see my paperlet The
Existence of the Exponential Function (with contributions by Omar
Antolin Camarena).