The framing of your question is a bit ambiguous and perhaps there are two different questions
here depending on the context and interpretation. One could approach your question from the point of view of intrinsic scientific content and ask: why QFT seems to be intrinsically more related to topology than analysis? (Question A). But one can also
approach the question from the angle of how this is reflected in human activity (mathematicians doing mathematics)
in these subjects.
Namely, you could ask the question: why, among mathematicians interested in QFT, there are more topologists than
analysts? (Question B).

Here is a stab at answering these two very different questions.

**Question A:**

This is moot since it is based on a false premise. QFT is not only related to topology but also to analysis and, I even venture to say, to almost all of mathematics. The reason for this is that QFT or the problem of rigorously defining functional integrals is the logical and natural continuation of the development of calculus
as I explained in this MO answer. After the usual calculus sequence (I, II, III) concerning the finite-dimensional situation, it is natural to explore differentiation (Calc IV) and integration (Calc V) in infinite dimension. Although Calc IV can be traced back to the early work on the calculus of variations by Maupertuis, Euler and Lagrange, I think its mathematical development started in earnest with the work of Volterra. As for Calc V, Wiener's construction of Brownian motion would come to mind as
an important early milestone. The intrinsic programmatic content of Calc V
is exactly what the analysis of QFT is about.
As Robert mentioned, this area of mathematics already exists and is called *constructive quantum field theory* (CQFT), although
nowadays it is also called *rigorous renormalization group theory*.

To get an idea of what is going on in the field, have a look at the reports for the two recent Oberwolfach meetings:

Another recent meeting around the new developments by Hairer and others in the strongly related field of stochastic quantization
is:

The problem of defining a QFT functional integral is a well posed mathematical problem
(see this MO answer for details).
In a nutshell, one starts by putting UV and IR cutoffs as is familiar in the theory of
Schwartz distributions and one lets bare couplings vary with these cutoffs. The problem is to find the set of all weak
limits for the corresponding probability measures on Schwartz distributions. The main difficulty is to construct such weak limit points that are not Gaussian or free measures. One would also like to parametrize this collection of weak limits by a *finite number* of parameters called
renormalized couplings. The main tool to do this is the renormalization group (RG).
In this MO answer
I briefly explained what the RG is, but I did not give details about how the RG provides a strategy for solving
the above problem about weak limit points. For more explanations about this strategy see my article
"QFT, RG, and all that, for mathematicians, in eleven pages"
and my answer to the physics.stackexchange question Wilsonian definition of renormalizability.

What Robert said "I think there is a feeling that the "easy" questions have been answered, and much of what remains may be impossibly hard"
is not quite correct. There are plenty of doable problems to work on at present in CQFT other than $YM_4$.

For example one has analogous conjectures for the 2d Gross-Neveu model and the 2d $\sigma$-model.
These are not impossibly hard like $YM_4$ and they do not really require extraterrestrial "new ideas".
As in the millenium problem, what one has to do is a construction of the model without UV cutoffs and in infinite volume
together with a proof of mass gap.

Another interesting problem (the one I focused on in the above references) is to make contact with conformal field theory.
The good class of examples to study in this regard are three-dimensional
$N$ component phi-four models with fractional Laplacian
$(-\Delta)^{\alpha}$ in the kinetic term.

The cases $N=1$, $\alpha=\frac{3}{4}+\epsilon$ as well as $N$ large, $\alpha=1$ should be not be impossibly hard.

Other problems of current interest are: proving the operator product expansion and conformal invariance using the RG.
**As for what I personally think is most important problem in constructive quantum field theory today, it is to develop a rigorous Wilsonian RG formalism for handling space-dependent couplings**.

**Question B:**

This one is not moot.
It is a fact that there are more mathematicians working on the topological aspects of QFT rather than its analytical aspects. I think this state of affairs is simply due to the status quo, i.e., it's just the way things are.
With regards to the North American situation in particular, I think the main explanation is that if a graduate
student would like to work on the analysis of QFT, chances are there would simply be nobody in their department to teach them the subject
to the point of being research-ready.
I think there is nothing more to it, but this could change in the future.

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