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My research is in analysis, but it moved to the area that requires algebraic topology. I have some working knowledge in that area, but I always feel that I am on a shaky ground and I need to go back and study algebraic topology again. However, that would also require refreshing my knowledge in algebra.Read more

I have a list of equations. Some of them are algebraic equations like 2x == y / (y+1), others contains derivatives in them(e.x. df/dx == 0). Note they are not differential equations. As they can be simplified from other algebraic equations before taking the (partial) derivative. After taking the partial derivative, they will all beRead more

I am on second semester so I have to choose between geometric transformation and algebraic structures?? I don’t know what to pick so I need your help to guide me, what are these subject all about ( first semester-I didn’t study algebra, I study discrete math and mathematics 1. Whats do u think is bestRead more

For $ A = (A_1,\cdots,A_d)\in\mathcal{L}(E)^d$ , the algebraic spectral radius of $ A$ was given by $ $ r_a(A)=\lim_{n\to+\infty}\left\|\sum_{f\in F(n,d)} A_f^* A_f\right\|^{\frac{1}{2n}} , $ $ where $ F(n,d):=\{f:\,\{1,\cdots,n\}\longrightarrow \{1,\cdots,d\}\}$ and $ A_f:=A_{f(1)}\cdots A_{f(n)}$ , for $ f\in F(n,d)$ . Further the geometric spectral radius of $ A$ was given by $ $ r_g(A)=\max\{\|\lambda\|_2,\;\lambda=(\lambda_1,\cdots,\lambda_d) \in \sigma(A)\}.$Read more

I’m relatively new to universal algebra, but it seems that lots of theorems from specific algebraic theories (groups, rings) can be stated in the context of universal algebra, perhaps I’m wrong. Examples being Jordan-Holder, Lasker-Noether, Hilbert Nullstellensatz (maybe not?) and many more, where one can talk about sobobjects, and congruences instead of “subgroups” and “ideals”.Read more

Let $ X$ be a smooth quasi-projective variety over a field $ k$ , $ f : Y\to X$ an étale surjective map, with $ Y$ connected. $ f$ induces a pullback map $ f^* : Z^r(X)\to Z^r(Y)$ on codimension $ r$ cycles. Is $ f^*$ an isomorphism? Note that for a surjective smooth $Read more

Let $ X$ be a smooth surface of general type with $ q=p_g=0$ , define a set $ A=\{C\subset X|$ C is smooth rational and $ C^2<-1\}$ , let $ k=|A|$ which is cardinality of $ A$ . What is the minimal value of $ k$ can be? Can you give me some example ofRead more

Is there an example of a smooth complex algebraic variety whose underlying complex manifold is not Kähler?Read more

Since today is the Chow-variety day, I’m going to ask my question here. Suppose I have a smooth projective variety $ X$ over a field of characteristic zero, and a smooth hyperplane $ p: H\hookrightarrow X$ . Is the pullback $ p^* : C^m(X)\to C^m(H)$ between Chow varieties of $ m$ -codimensional cycles, a morphismRead more

$ t^n=a$ , we get one solution to the equation: $ $ t=e^{\frac{1}{n}\int^a_1 \frac{1}{x}}$ $ , generalizing this result by replacing the exponential with an elliptic modular function and the integral with hyperelliptic integrals, we can get every solution to an algebraic equation $ x$ in $ a_0+a_1x+a_2x^2+\cdots+a_nx^n=0$ with degree above 5 by formulation ofRead more