Let G be a finite group and consider k[G] where k is a field. In the scenario where char(k) divides |G|, how can one show that the dimension of Z(k[G]/rad k[G]) is strictly less than dimension of Z(k[G])? I'm trying to use this to show that in the case where char(k) divides |G|, the number of si...

Let G be a finite group and consider k[G] where k is a field. In the scenario where char(k) divides |G|, how can one show that the dimension of Z(k[G]/rad k[G]) is strictly less than dimension of Z(k[G])? I'm trying to use this to show that in the case where char(k) divides |G|, the number of si...

Recently , I read one paper titled Modular equations and approximations to π by Ramanujan, in which there are some formulas for $q=\pi i \tau$( where $\tau=x+yi, y>0$, hence $|q|<1)$ : $$\prod_{n=1}^\infty\left(1+q^{2n-1}\right)=2^{\frac{1}{6}} q^{\frac{1}{24}}(kk')^{-\frac{1}{12}} ~~~ (1)$$ ...

I searched "Tower of Hanoi" on arXiv: The Group of Symmetries of the Tower of Hanoi Graph The Tower of Hanoi and Finite Automata Shortest paths in the Tower of Hanoi graph and finite automata Asymptotic aspects of Schreier graphs and Hanoi Towers groups A search on Wikipedia returned the Towe...

I think you can get it from the Jacobi triple product identity \[ \prod_{m=1}^\infty (1-x^{2m})(1-x^{2m-1}y^2)(1+x^{2m-1}y^{-2}) = \sum_{n=-\infty}^\infty x^{n^2}y^{2n} \] We can set $x = q^{1/2}, y = q^{1/2}$. \[ \prod_{m=1}^\infty (1-q^{m})(1-q^{m+\frac{1}{2}})(1+q^{m-\frac{3}{2}}) = \sum_{n=...

I'd say it's closer to an oriented manifold with corners (corners happening where the divisor is singular), or even that times a coefficient. In these papers Khesin, Rosly, and later Thomas build a homology theory based on this analogy.

I understand the OP is frustrated with tight job market and those are valid concerns, but his/her description of the geometric analysis as a shallow subject is ridiculous. Parts of geometric analysis certainly attract top people and have seen remarkable recent progress. How a new Ph.D. could fai...

In Dubuc and de la Vega's very nice write-up of Grothendieck's Galois theory they cover the pro-group case when the fibre functor is representable (see section 5.5): Consider a category $C$ and an object $A \in C$. Axioms on $C$: R1) $C$ has a terminal object and pullbacks (thus all fin...

For details I recommend looking at the papers of Yagasaki on arxiv especially this paper. Let me answer 1. Consider be the diffeomorphism group of $\mathbb R^2$ that fixes $0$ and acts as the identity at the tangent space to $0$. The group acts on your space of embeddings by postcomposition. B...

Don't need stable. It's the stack of all G bundles on a curve. The right hand side is the derived category of local systems, which are vector bundles with flat connection (G-bundles, for the Langlands dual of G in this equation). As for background, that depends on how deep an understanding you...

One way to tell how active a field is is by looking at what's appearing on the arXiv in that area. I think that will show you that operator algebra is a robust subject with a lot of activity. In the comments, MaoWao points out that UC Berkeley and UCLA have very strong operator algebra groups, a...

I'm currently studying with Melvyn Nathanson,who is really considered one of the experts on additive number theory.His texts,ADDITIVE NUMBER THEORY:THE CLASSICAL BASES and ADDITIVE NUMBER THEORY:INVERSE PROBLEMS are really the standard introductions to the subject.They are both published by Sprin...

Let $G$ be a finite group and consider $k[G]$ where $k$ is a field. In the scenario where $\mathrm{char}(k)$ divides $|G|$, how can one show that the dimension of $Z(k[G]/\operatorname{rad}k[G])$ is strictly less than dimension of $Z(k[G])$? I'm trying to use this to show that in the case where ...