Interesting thesis topic on statistical inference that is sufficiently mathematical

Hello , I am a student who's gonna start honours in mathematics . Currently , I am at the stage of finding a suitable honours thesis topic . I've chosen my supervisor , who's research interest is on statistical inference and probability theory , but more on inference , I suppose.

My undergraduate coursework consists mostly of pure mathematics , and I have not really much knowledge in statistical science , but I am really interested in the theory of statistics , that's why I chose a supervisor doing stat.

But since my honours is in mathematics , my thesis should be sufficiently pure mathmatical. So my supervisor suggest that I could do something in shrinkage estimation , but he is even struggling on finding the right paper on it . And I guess I'd better conduct some research myself .

I have no suffcient background in statistics , as I mentioned above , but I've done a lot of courses on mathematics( group ,rings ,field extensions , galois theory , representation theory , analysis,probability theory, general topology , algebraic topology ,moduli space etc. ) and small project also . Now I really wanna combine theses things with theory of statists, but dont know where to start with.

I've heard that it is possible to put geometries into statistical theory , and this looks very attractive :D . I am particularly interested in geometry and topology .

I am also thinking doing something on statistical inference and stochastic analysis.

However , as my supervisor suggest , shrinkage estimation is a very good choice . So maybe it would be better to follow his suggestion ?

Anyway , I am really excited , but also struggling on choosing the right topic. Could anyone give me some advice on it ? Both reference to good papers on shrinkage estimation or perhaps suggestions on other related good topics would be both great . Thanks in advance !

Relatedly, Ackerman, et al. (2011) demonstrate a computable random variable $(X,Y)$ with non-computable conditional distribution $P(Y \mid X)$.