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16

One sparkling gem at the intersection of number theory and geometry is Aubry's reflective generation of primitive Pythagorean triples, i.e. coprime naturals $\,(x,y,z)\,$with $\,x^2 + y^2 = z^2.\,$ Dividing by $z^2$ yields $\,(x/z)^2\!+(y/z)^2 = 1,\,$ so each triple corresponds to a rational point $(x/z,\,y/z)$ on the unit circle. Aubry showed that we can ...


9

How about Furstenberg's proof of the infinitude of prime numbers?


9

The Nielsen–Schreier theorem (a subgroup of a free group is itself free) can be proven using methods of elementary algebraic topology. My personal favourite is the pretty deep result that every finite dimensional divison algebra over $\mathbf{R}$ has dimension $1,2,4$, or $8$. This result seems to be due Kervaire and Milnor; the proof uses methods of ...


9

We would like to conjecture that two important mechanisms are involved with a mathematical description of the material world: Abstraction, giving rise to Science Idealization, giving rise to Mathematics Schematically: Abstraction Etymology. Perfect passive participle of abstraho ("draw away from"). Certain properties of the whole thing are preserved in ...


8

I can't comment on other countries, but in the USA the purpose of a master's thesis is to get you a master's degree. It need not be original, but of course there are quality standards. Anything that you have discovered by yourself that you have not seen anywhere else could be considered original research. What academics generally mean when they say ...


7

What follows is an attempt to motivate this beautiful and difficult (in my opinion) subject. It is just an attempt, I cannot promise it will be useful. Suppose you have a family of curves over $\mathbb A^1=\textrm{Spec }\mathbb C[t]$, like for instance the family $$\pi:\textrm{Spec }\mathbb C[x,y,t]/(xy-t)\to \mathbb A^1$$ given by $t\mapsto t$. As it is ...


6

Francis Su described in 1999 ("Rental Harmony: Sperner's Lemma in Fair Division", Amer. Math. Monthly, 106, 1999, 930-42) how to apply Sperner's Lemma---which says that every so-called Sperner coloring of a triangulation of an $n$-simplex contains a cell colored with a complete set of colors---to produce a list of variously sized rents for rooms in a shared ...


5

The proof of the Cayley–Hamilton theorem in the case of different eigenvalues is very easy. The extension to general case in any field is possible using the Zariski topology.


5

There is an area called arithmetic geoemetry that exploits links between arithmetic and algebro-geometric questions. For example, Fermat's famous equations $X^n + Y^n = Z^n$ can be thought of as a curve in projective space, called Fermat curves, and one can use geometric tools to study it. The affine part, so $X^n + Y^n = 1$ is somewhere between a circle ...


5

I try to write down a sentence explaining what exactly I'm counting on both sides of the equality, to convince myself that they are the same. For example, for the identity ${n \choose k} + {n \choose k+1} = {n+1 \choose k+1}$ I would write down the the left hand side first counts all subsets of $[n+1]$ of size $k+1$ that include the element $n+1$, plus ...


5

First, manifolds "exist in their own". However, every closed oriented 3-manifold does bound a compact 4-manifold (the latter is far from being unique). My suggestion is to pick up a copy of Munkres' "Topology" book and read first few chapters. This will help to clear many issues that you currently have. Edit: Of course, Munkres does not discuss ...


5

Let $R$ be a ring, and $\{r_\alpha\}_{\alpha\in I}\subset R$. The ideal generated by $\{r_\alpha\}$ consists of all the finite sums$$\sum_{i=1}^na_ir_{\alpha_i},$$where the coefficients $a_i$ are just elements of $R$. Take for example $\mathbb{Z}[x]$, the ring of polynomials in one variable with integer coefficients, and consider the ideal generated by ...


5

I have a PhD and have served as direct advisor for a few PhD students and on committees for several, though in physics and electrical engineering and computer science, not mathematics. A few thoughts: First, look deep into yourself and review your career so far and understand your strengths and weaknesses, and choose an advisor accordingly. Some of my ...


4

The Brouwer fixed-point theorem. The Wikipedia lists the following methods: Homology Stokes's theorem Combinatorial (Sperner's lemma) Reducing to the smooth case and using Sard's theorem Reducing to the smooth case and using the COV theorem Lefschetz fixed-point theorem Using Hex


4

My best suggestion for enumeration problems is to set it up as a sequence of explicitly defined steps being multiplied by multiplication principle (or adding a disjoint set by addition principle). Then, check your steps by mock answering them. See then also if there is a different sequence of choices that arrives at the same answer. If there is, then you ...


4

The Necklace Splitting problem in combinatorics has been beautifully solved by Alon and West using the Borsuk-Ulam theorem. http://en.wikipedia.org/wiki/Necklace_splitting_problem


3

The fundamental theorem of algebra Every nonconstant polynomial with coefficients in $\mathbb{C}$ has a root in $\mathbb{C}$ can be shown using algebraic topology. See for example Hatcher (http://www.math.cornell.edu/~hatcher/AT/AT.pdf) Page 31. Theorem 1.8. Sketch of the proof: Take a polynomial: $p(z) = z^n +a_1 z^{n−1} +\cdots+a_n$ . Then ...


3

One of the most important things is that you understand exactly which structures you want to count (making sure when two of them are the same for example). Once you know this you can test if your formula works for small cases which you can work out by hand.


3

"I always thought that it was beyond me and that there's no way for me to do well in it." This is your first problem, its called a limiting belief, you need to change that to the belief that you can master analysis. "I spend so many hours and put in so much effort into studying" This is the second problem, more time studying does not produce better ...


3

There is no right or wrong answer on your decision to not pursue something. Will you look back later in life and regret that decision? I don't know so maybe you will or maybe you wont. If you know analysis is not what you want to do in math, that is perfectly fine since mathematicians around the world specialize in many different areas of mathematics, and ...


3

Let me concisely but properly address your questions. 1) Learn about tensor products and multilinear algebra in general. (Since you mention that you read Lang's linear algebra book, there is a more advanced algebra book by Lang, called - guess what - Algebra; it is a standard reference.) These things are totally crucial in e.g. commutative algebra, ...


2

You should definitely check this out: The Most Common Errors in Undergraduate Mathematics. The first section, "Errors in Communication," will be of interest to you. It addresses things such as teacher hostility or arrogance, student shyness, etc. Reading that would be an excellent place to start.


2

Stop asking her and start asking your questions here. This might work!


2

The sequence 2, 2, 2, 3, 4 is not increasing, but it's non-decreasing. The sequence 2, 3, 4, 5, 6 is both increasing and non-decreasing. Compare the difference between "less than" and "not greater than." (I have never seen the words ascending/descending used in this context but I imagine it's the same).


2

There are several problems here: There is not "the standard model of set theory". There are notions of "standard models" (note the plural), but there is no "the standard model". With respect to the real numbers there are several possible scenarios: It might be the case that there is a standard model containing all the reals. This model, if so, has to be ...


2

I would recommend "Complex Analysis (Universitext)" by Freitag & Bufsam. It contains all your requirements and has complete solutions in the back. There is also a second volume if you want to go deeper into the subject.


2

I recommend Marsden's Basic Complex Analysis and Bak's Complex Analysis. It might help if you have this Schaum's outline on the side to accompany one (or both) of these texts.


2

Hindman's Finite Sums Theorem: If you partition $\mathbb N$ into finitely many classes, there is an infinite sequence $a_1\lt a_2\lt a_3\lt\cdots$ in $\mathbb N$ such that all of the finite sums $a_{i_1}+a_{i_2}+\cdots+a_{i_k}$, where $i_1\lt i_2\lt\cdots\lt i_k$ and $k\in\mathbb N$, belong to the same partition class. The proof uses some kind of topological ...


2

The Borsuk-Ulam Theorem states that for any continuous map $f : \mathbb{S}^n \to \mathbb{R}^n$, there are two antipodal (opposite) points of $\mathbb{S}^n$ that $f$ maps to the same value. (This follows from the fact that every antipodes-preserving map from the sphere to itself has odd degree.) For example, in the case $n = 2$, we can conclude that there ...


2

How could you do better than the problems and solutions themselves for past Putnam exams? 1938-1964 1965-1984 1985-2000 For a Putnam archive of past exam questions and solutions, check out this website maintained by Kiran Kedlaya. There are, of course, many other resources, but these are by far the best I know to date.



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