How can I show that $ax^2 + 2hxy + by^2 +2gx + 2fy+c = 0$ represents a point when $\Delta = 0$ and $h^2 - ab < 0$. I know I can do it by completing the square. But it turns out to be too tedious. Can anyone help me ?

I am about to finish working through Williams's Probability With Martingales. I have studied analysis up to the first five chapters of Folland's text but have not studied any combinatorics yet. It seems like 'combinatorial' probability topics like percolation, probability on graphs and networks,...

I'm trying to learn about the Kushner-Stratonovich-Pardoux equations in filtering theory. I'm familiar with Itô calculus at the level of Øksendal's book (but struggle with much of Karatzas and Shreve, for example). My PDE theory is pretty weak. I know about the Fokker-Planck equations, and that...

I haven’t found any posts on this, so I figured I would ask. Beyond learning basic algebra (rings, groups, fields) and complex analysis, what must one study if they want to start learning a good amount of iwasawa theory? In what sequence should they study this material?

I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and Homological Algebra), self study of Geometry (Manifolds, Differential Geometry, Riemannian Metrics, Curvat...

Next semester I may study a course where the ultimate goal is to get to the Borel - Weil - Bott (BWB) Theorem, if not at least try to understand it in the case that we have $G = \text{SL}_n$. I have studied some representation theory of Lie groups from Brian Hall's book Lie groups, Lie algebras ...

Let me start off with a description of my background. I am an undergraduate student, with some background in real analysis (Rudin, Principles of Mathematical Analysis), measure theory (Royden, Real Analysis: Parts I and III) and a little functional analysis (first five chapters of Rudin's 'Real a...

I am now considering about studying algebraic topology. There are a lot of books about it, and I want to choose the most comprehensive book among them. I have a solid background in Abstract Algebra, and also have knowledge on Homological Algebra(in fact I am now study Tor and Ext functors). But m...

I think a useful combination of resources for universal algebra would ideally, when taken together: Provide ample motivation behind the various developments in the field. Either provide powerful intuitions, or avoid them altogether, in an effort to not cripple the learner's imagination. Provide...