Let $G$ be a topological group and $\mathcal{B}$ its Baire $\sigma$-algebra (i.e. the smallest $\sigma$-algebra for which all continuous functions $G\rightarrow\mathbb{R}$ are measurable). Consider the product map $\cdot :G\times G\rightarrow G$ given by $(g,h)\mapsto g \cdot h$. Question: Is the...

Let $X$ be a singular, projective (complex) variety of dimension $n$ with at worst isolated singularities. We know that taking cup-product with the cohomology class of $X$ induces a morphism from the cohomology group to the Borel-Moore homology: $$ \cap [X]: H^{2p}(X) \to H_{2n-2p}^{\mathrm{BM}}(...

In his PhD thesis, Gaitsgory (in his "remark 6") makes the following claim: Consider two complexes of holonomic $D$-modules with regular singularities on a variety $X.$ Suppose that at each point, the complexes have the same Euler characteristic; then in fact the Euler characteristics of the hype...

Let $\mathrm{G}$ be a complex reductive group and let $\mathrm{O}_g$ be the adjoint orbit of $g\in \mathrm{G}$. I wonder is the centralizer $\mathrm{C}_{\mathrm{G}}(\mathrm{O}_g)$ still a reductive group in general? If not, how about the identity component of $\mathrm{C}_{\mathrm{G}}(\mathrm{O}_g...

Is there any example of a locally conformally flat manifold that is neither a space form nor a product of space forms?

Suppose we have a ring $R$ with involution and a positive natural number $n$ such that the free module $R^{n}$ of rank $n$ is isomorphic to free module $R^{n+1}$ as $R$-module. Consider $L_{\ast}(R)$ the algebraic L-theory groups for quadratic forms. Can we say something about these groups ? Are ...

I came across multiple research papers where they have mentioned that two 1-generic reals $x,y$ can be constructed such that $x \oplus y \equiv_{T} \textbf{0'}$ where $\textbf{0'}$ is the degree containing the halting set. I was wondering how to prove this fact. I know some basic computability th...

The probabilistic method uses arguments from probability to prove deterministic statements. This has been applied to diverse fields such as combinatorics, topology and number theory. In this method, you constructs a random variable based on the theorem that you want to prove. Using probabilistic ...