Career - What's a mathematician to do? It's not mathematics that you need to contribute to. It's deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics? Such a question is not possible to answer in a purely intellectual way, because the effects of our actions go far beyond our understanding. We are deeply social and deeply instinctual animals, so much that our well-being depends on many things we do that are hard to explain in an intellectual way. That is why you do well to follow your heart and your passion. Bare reason is likely to lead you astray. The product of mathematics is clarity and understanding. The world does not suffer from an oversupply of clarity and understanding (to put it mildly).
I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Unitary group. The Fundamental Group — A Primer. Being part of the subject of algebraic topology, this post assumes the reader has read our previous primers on both topology and group theory. As a warning to the reader, it is more advanced than most of the math presented on this blog, and it is woefully incomplete. Nevertheless, the aim is to provide a high level picture of the field with a peek at the details. An Intuitive and Totally Uncomputable Topological Invariant Our eventual goal is to get comfortable with the notion of the “homology group” of a topological space. That is, to each topological space we will associate a group (really, a family of groups) in such a way that whenever two topological spaces are homeomorphic their associated groups will be isomorphic.
In other words, we will be able to distinguish between two spaces by computing their associated groups (if even one of the groups is different). The simplest example of this is a circle has a two-dimensional “hole” in it (the hollow interior). Paths and Homotopy to be a map. Morse theory. "Morse function" redirects here. In another context, a "Morse function" can also mean an anharmonic oscillator: see Morse potential The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory. Basic concepts[edit] Contour lines around a saddle point To each of these three types of critical points – basins, passes, and peaks (also called minima, saddles, and maxima) – one associates a number called the index. Define Ma as f−1(−∞, a]. These figures are homotopy equivalent.
One therefore appears to have the following rule: the topology of Mα does not change except when α passes the height of a critical point, and when α passes the height of a critical point of index γ, a γ-cell is attached to Mα. This rule, however, is false as stated. Formal development[edit] For the functions from R to R, f has a critical point at the origin if b=0, which is non-degenerate if c≠0 (i.e. f is of the form a+cx2+...) and degenerate if c=0 (i.e. f is of the form a+dx3+...). Morse lemma[edit] . Maryam Mirzakhani Is First Woman Fields Medalist. The Banach-Tarski Paradox. The Axiom of Choice is Wrong | The Everything Seminar. When discussing the validity of the Axiom of Choice, the most common argument for not taking it as gospel is the Banach-Tarski paradox. Yet, this never particularly bothered me. The argument against the Axiom of Choice which really hit a chord I first heard at the Olivetti Club, our graduate colloquium.
It’s an extension of a basic logic puzzle, so let’s review that one first. 100 prisoners are placed in a line, facing forward so they can see everyone in front of them in line. The warden will place either a black or white hat on each prisoner’s head, and then starting from the back of the line, he will ask each prisoner what the color of his own hat is (ie, he first asks the person who can see all other prisoners). Any prisoner who is correct may go free. The answer to this in a moment; but first, the relevant generalization. I should give credit where it is due. First, lets review the solution the the basic problem.
Hmm, he could say the color of the hat on the guy in front of him. . Jeremy Kun. The Graph of Math | Gödel's Lost Letter and P=NP. Kurt Gödel is said to have been a latecomer to appreciating the power of Model Theory. He was of course the greatest architect of Proof Theory, which stands in contrast to Model Theory. Model Theory concerns itself with what could be true, while Proof Theory deals with what can be proved. The latter sounds more definite, but they are supplementary: a statement is capable of being true somewhere precisely when its negation cannot be proved. The question is, where is that somewhere? And when? Two years ago by our reckoning, Dick and I put that question to Gödel, at the close of our interview whose first and second parts we posted at this time the past two years. As before, we took advantage of time-reversed communication that was made possible by Gödel’s own solution to Albert Einstein’s equations of general relativity.
We must confess we are unable to recover the beginning of this part of the interview. Well Gödel didn’t like this question. Interview, Part III GLL: To build? GLL: Take your. The Existential Risk of Mathematical Error. Welcome — Statistics Done Wrong. HoTT/book. Wikipedia-size maths proof too big for humans to check - physics-math - 17 February 2014. If no human can check a proof of a theorem, does it really count as mathematics? That's the intriguing question raised by the latest computer-assisted proof. It is as large as the entire content of Wikipedia, making it unlikely that will ever be checked by a human being. "It might be that somehow we have hit statements which are essentially non-human mathematics," says Alexei Lisitsa of the University of Liverpool, UK, who came up with the proof together with colleague Boris Konev.
The proof is a significant step towards solving a long-standing puzzle known as the Erdős discrepancy problem. It was proposed in the 1930s by the Hungarian mathematician Paul Erdős, who offered $500 for its solution. Imagine a random, infinite sequence of numbers containing nothing but +1s and -1s. Adding up the numbers in a sub-sequence gives a figure called the discrepancy, which acts as a measure of the structure of the sub-sequence and in turn the infinite sequence, as compared with a uniform ideal.
Universal Properties. Previously in this series we’ve seen the definition of a category and a bunch of examples, basic properties of morphisms, and a first look at how to represent categories as types in ML. In this post we’ll expand these ideas and introduce the notion of a universal property. We’ll see examples from mathematics and write some programs which simultaneously prove certain objects have universal properties and construct the morphisms involved. A Grand Simple Thing One might go so far as to call universal properties the most important concept in category theory.
This should initially strike the reader as odd, because at first glance universal properties are so succinctly described that they don’t seem to be very interesting. In fact, there are only two universal properties and they are that of being initial and final. Definition: An object in a category is called initial if for every object there is a unique morphism . Is called final if for every object. . , the Hom set consists of a single element).