Spiral Challenge — Continuous Everywhere but Differentiable Nowhere

Megan Schmidt is obsessed with spirals. Her obsession got me hooked — for hours — on a math problem. I thought it would take maybe an hour or two, but I’m still at it and I’ve probably been working four or five hours. I’ve been having so much fun with it. Here’s the problem. Look at the […]

via Spiral Challenge — Continuous Everywhere but Differentiable Nowhere

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Sara Zahedi won one of the 10 EMS prizes

SZ.png

Sara Zahedi

* 1981 in Teheran (Iran)
Assistant Professor
Royal Institute of Technology (KTH), Sweden

EMS Prize 2016
„For her outstanding research regarding the development and analysis of numerical algorithms for partial differential equations with a focus on applications to problems with dynamically changing geometry.“

Research Interests
Sara Zahedis research interests lie in the development and analysis of computational methods, in particular finite element methods, for solving partial differential equations on dynamic geometries. The main application she has in mind is multiphase flows. She is also interested in numerical methods for representing and evolving interfaces separating immiscible fluids.

Curriculum Vitae: 

  • 2014: Assistant Professor in Numerical Analysis, KTH, Stockholm (Swe)
  • 2011: Postdoctoral Position, Uppsala University (Swe)
  • 2011: PhD in Numerical Analysis, KTH, Stockholm (Swe)
  • 2006: Master of Science with a major in Mathematics, KTH (Swe)
  • 2006: Teaching Assistant, Dept. of Numerical Analysis, KTH (Swe)
  • 2003: Teaching Assistant, Stockholm University (Swe)

More Information: 1 , 2 , 3 and 4 

Baum–Connes conjecture

In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the C*-algebra of a group and the K-homology of the corresponding classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, with the K-homology being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the reduced {displaystyle C^{*}}C^*-algebra is a purely analytical object.

The conjecture, if true, would have some older famous conjectures as consequences. For instance, the surjectivity part implies the Kadison–Kaplansky conjecture for a discrete torsion-free group, and the injectivity is closely related to the Novikov conjecture.

The conjecture is also closely related to index theory, as the assembly map {displaystyle mu }mu is a sort of index, and it plays a major role in Alain Connesnoncommutative geometry program.

The origins of the conjecture go back to Fredholm theory, the Atiyah–Singer index theorem and the interplay of geometry with operator K-theory as expressed in the works of Brown, Douglas and Fillmore, among many other motivating subjects.

From Wikipedia, the free encyclopedia

More:

  1.  Introduction to the Baum-Connes Conjecture Alain VALETTE (PDF)
  2. A Survey on the Baum-Connes Conjecture (PDF)
  3. https://en.wikipedia.org/wiki/Baum–Connes_conjecture

 

Feynman Diagrams – Visual Depictions of Physics — Delightful & Distinctive COLRS

Source: Quanta Magazine, Jul 2016 Feynman diagrams look to be pictures of processes that happen in space and time, and in a sense they are, but they should not be interpreted too literally. What they show are not rigid geometric trajectories, but more flexible, “topological” constructions, reflecting quantum uncertainty. In other words, you can be […]

via Feynman Diagrams – Visual Depictions of Physics — Delightful & Distinctive COLRS

A Comprehensive Course in Analysis by Barry Simon

In the second half of 2015, the American Math Society will publish a five volume (total about 3200 pages) set of books that is a graduate analysis text with lots of additional bonus material. Included are hundreds of problems and copious notes which extend the text and provide historical background.  Efforts have been made to find simple and elegant proofs and to keeping the writing style clear

Part 1 Real Analysis:

Point Set Topology, Banach and Hilbert Space, Measure Theory, Fourier Series and Transforms, Distribution Theory, Locally Convex Spaces, Basics of Probability Theory, Hausdorff Measure and Dimension.

Part 2a Basic Complex Analysis:

Cauchy Integral Theorem, Consequences of the Cauchy Integral Theorem (including holomorphic iff analytic, Local Behavior, Phragmén-Lindelöf, Reflection Principle, Calculation of Integrals), Montel, Vitali and Hurwitz’s Theorems, Fractional Linear Transformations, Conformal Maps, Zeros and Product Formulae, Elliptic Functions, Global Analytic Functions, Picard’s Theorem.

Part 2b Advanced Complex Analysis:

Conformal metric methods, topics in analytic number theory,  Fuchsian ODEs and associated special functions, asymptotic methods, univalent functions, Nevanlinna theory.

Part 3 Harmonic Analysis:

Maximal functions and pointwise limits, harmonic functions and potential theory, phase space analysis, Hp spaces, more inequalities.

Part 4 Operator Theory:

Eigenvalue Perturbation Theory, Operator Basics, Compact Operators, Orthogonal Polynomials, Spectral Theory, Banach Algebras, Unbounded Self-Adjoint Operators.

Reference: Click Here

 

Maryam Mirzakhani: the first woman and the first Iranian honored with the Fields Medal

In 2014, Mirzakhani became both the first woman and the first Iranian honored with the Fields Medal, the most prestigious award in mathematics. The award committee cited her work in “the dynamics and geometry of Riemann surfaces and their moduli spaces.”

Her research topics include Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry.

More Information: https://en.wikipedia.org/wiki/Maryam_Mirzakhani

Mirzakhani_original.jpg

Mathematics is In Everything — Delightful & Distinctive COLRS

Source: Asian Scientist, Jan 2016 These days, Lamport wants to impart the following message to programmers: using mathematics allows you to approach coding in the way an architect would approach designing a building—both precisely and abstractly. “Science requires precise thinking, and mathematics is what has developed over a couple of millennia as our best way […]

via Mathematics is In Everything — Delightful & Distinctive COLRS

The Mathematics Of Dan’s Inferno — Gödel's Lost Letter and P=NP

A possible error with mathematical ramifications Non-technical fact-check source Dan Brown is the bestselling author of the novel The Da Vinci Code. His most recent bestseller, published in 2013, is Inferno. Like two of his earlier blockbusters it has been made into a movie. It stars Tom Hanks and Felicity Jones and is slated for […]

via The Mathematics Of Dan’s Inferno — Gödel’s Lost Letter and P=NP

How to type LaTeX in WordPress without using "$latex"

Singapore Maths Tuition

Currently, LaTeX is well supported in WordPress, however there is one practical issue when typing LaTeX in WordPress, the need to type “$latex” for every single math expression! It gets pretty troublesome after a while.

For those familiar with LaTeX, one would know that for ordinary LaTeX typesetting, typing double $ will do, there is no need to type the word LaTeX. If I remember correctly, for Blogger there is no need to type “$latex”, hence it is a uniquely WordPress issue.

So far, I have not found any solution to this issue. (I am using WordPress.com hosted WordPress). Any readers who happen to know a solution, please enlighten me by dropping a comment! It will be greatly appreciated.

WordPress vs Blogger LaTeX:

http://mathtuition88.com/2014/12/25/math-formula-in-wordpress-vs-blogspot/


LaTeX Beginner’s Guide

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Math Formula in WordPress vs Blogspot

Singapore Maths Tuition

Refering to How to write Math Formulas on Blogspot / Blogger, Blogspot is also capable of rendering beautiful Math Equations based on LaTeX.

Blogger uses MathJax for rendering the Math Formulas, which look like the ones shown above.

Let’s try the WordPress version and readers can judge for themselves which is better.

$latex \displaystyle a^p \equiv a \pmod p$

$latex \displaystyle\int_0^{\frac{\pi}{2}} \sin x\ dx=1$

My personal opinion is: they look the same to be honest. However, the Blogger way of typing is much more convenient, just using $ and $$, as opposed to WordPress requiring “$latex” and “$latex\displaystyle”, which is more cumbersome especially for long texts. However, experts like Terence Tao have opted for WordPress, which shows that WordPress does probably have some advantages.


Featured Book:

The LaTeX Companion (Tools and Techniques for Computer Typesetting)

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Curiosity & Creativity Benefit Humanity — Delightful & Distinctive COLRS

Source: Straits Times, Jan 2016 Other vivid examples in his lecture bore witness to the power of beautiful things in inspiring the mind of the mathematician. Like the beautiful and complex shapes of corals in the sea that actually follow very simple mathematical rules. Drawing connections between seemingly far-flung subjects is a hallmark of Prof […]

via Curiosity & Creativity Benefit Humanity — Delightful & Distinctive COLRS

Lusin’s Theorem and Egorov’s Theorem — Singapore Maths Tuition

Lusin’s Theorem and Egorov’s Theorem are the second and third of Littlewood’s famous Three Principles. There are many variations and generalisations, the most basic of which I think are found in Royden’s book. Lusin’s Theorem: Informally, “every measurable function is nearly continuous.” (Royden) Let be a real-valued measurable function on . Then for each , […]

via Lusin’s Theorem and Egorov’s Theorem — Singapore Maths Tuition

Iterative Methods — Mathematics, Learning and Technology

Looking at the new content for UK GCSE Mathematics a completely new entry on the specification is “find approximate solutions to equations numerically using iteration”. Teaching the new specification this year I needed to work on this and wrote this post; GCSE New Content – Iterative Methods for Numerical Solution of Equations. I see from my […]

via Iterative Methods — Mathematics, Learning and Technology

What is the fundamental group of a topological space

What is the fundamental group of a topological space (a so-called manifold)? For example, a knotted loop?

To understand a knot K, we want to understand how it behaves, what we can (or can’t) do to it.

Since the knot has to stay intact, and since we’re on the outside, it’s better to look at the knot’s complement. That should be okay, because it has exactly the same shape as the thing itself, right? The Gordon-Luecke theorem is there to tell us that you can’t weirdly wrap the complement in ways not possible with the knot itself.

https://en.wikipedia.org/wiki/Gordon%E2%80%93Luecke_theorem

And besides, a knot’s inside has no room to maneuver loops in. So instead, we play lasso in the space left after we have carved out our knot!

https://en.wikipedia.org/wiki/Knot_complement

Our test lasso shouldn’t tangle up with itself, because we already have a knot to study and wouldn’t want to complicate things. So you should think of it as being made of that mythical topological material, that can stretch indefinitely and pass through itself (but not the knot), in first approximation: a rubber band.

Then a lasso configuration A is different from another one B if we cannot deform A and B to match up positionally. For example, if A doesn’t wrap around K at all, and B does so once, they’re different!

Finding out the set G of all essentially different ways to tie a string around our knot K will tell us something fundamental about K. There’ll be an element for each number of times we can wrap our test lace around K. If that were all, our story would end here, but interestingly, sometimes there are more elements, sometimes less!

Before we get to these let me show you that the set G is actually a group. First we need an operation between lasso paths, a kind of addition to combine two of them into another one:

To add two paths A and B, first pick a base point P, then deform A and B to touch P. Cut both of them up there, and stick them together the other way. If you draw arrows on your paths, giving them an orientation, you’ll notice that only one of the ways to reglue the ends into a single loop makes their orientations agree. That’s the one we want!

Putting arrows on your paths is generally a good idea, because now we can have both, left and right windings around a piece of knot: all our elements have inverses, there’s a B for every (signed) integer. And A is a zero-element, adding nothing to any path!

So, there’s an operation: check, inverses: check, neutral element: check – we have a group! That’s the fundamental group!

https://en.wikipedia.org/wiki/Knot_group

It’s also called homotopy group because our rubber lasso material is what your innocent-looking topologist neighbour likes to tie together his proofs with, that two particular manifolds he brought with him last night, are indeed homotopy equivalent.

So, it is a knot invariant, you can use it to prove that two knots are not equivalent under isotopy (in R³). knot group isn’t a complete invariant, so sometimes it cannot distinguish different knots. There are many more knot invariants:

https://en.wikipedia.org/wiki/Knot_invariant

Example: Picture a line’s complement (in R³). It is homotopic (in R³-L) to a circle, and its homotopy group is Z: the integers with addition. R³ without two lines gives the free group over two generators. Every group is a subgroup of a free group and also a quotient of one.

So we can expect that any knot’s complement’s fundamental group can be written as a quotient of a free group over some equalities that describe the new freedom to move the lasso we get when linking line segments among each other instead with the infinite far away. Here’s the one for the trefoil knot:

<x,y|x²=y³> or
<a,b|aba=bab>
also known as Artin’s braid group B_3.

If you find that unconvincing, maybe this page is for you:

http://math.stackexchange.com/questions/143728/why-is-the-knot-group-of-the-trefoil-isomorphic-to-the-group-of-3-braids

It’s where I stumbled over this post, where +John Baez entangles the knot groups with other fancy stuff:

http://math.ucr.edu/home/baez/week261.html

Finally a word of warning: only the abelianization of the first homotopy group is the first homology group… Also, Milnor’s link group here is a bit but not quite similar, be careful:

https://en.wikipedia.org/wiki/Link_group

I’m posting this as a warm-up for some more topological things I have in my queue. Stay tuned!
reference: https://plus.google.com/u/0/+RefurioAnachro/posts/ha8N3bSch3R

Pics From: https://lifethroughamathematicianseyes.wordpress.com/2016/02/16/drawing-mathematical-concepts/
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A Great Book in Mathematical Analysis

Was plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is pure mathematics, and I hope it appeals to you, the budding pure mathematician.

032168

http://www.springer.com/us/book/9783319177700#reviews

http://press.ut.ac.ir/book_901_آنالیز+ریاضی+حقیقی+3403.html

Noncommutative spectral geometry of Riemannian foliations: some results and open problems

We review some applications of noncommutative geometry to the study of transverse geometry of Riemannian foliations and discuss open problems.

Comments: 35 pages, final version, minor changes, to appear in Proceedings of the Conference “Foliations-2005” (Lodz, Poland), World Scientific, Singapore, 2006
Subjects: Differential Geometry (math.DG); Spectral Theory (math.SP)
Cite as: arXiv:math/0601093 [math.DG]
(or arXiv:math/0601093v2 [math.DG] for this version)

Submission history

From: Yuri A. Kordyukov [view email]
[v1] Thu, 5 Jan 2006 13:46:18 GMT (32kb)
[v2] Sat, 6 May 2006 10:24:38 GMT (32kb)

The most Striking Theorem in Real Analysis — Singapore Maths Tuition

Lebesgue’s Theorem (see below) has been called one of the most striking theorems in real analysis. Indeed it is a very surprising result. Lebesgue’s Theorem (Monotone functions) If the function is monotone on the open interval , then it is differentiable almost everywhere on . Absolutely Continuous Functions Definition A real-valued function on a closed, […]

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The Success of French New Math Education in Secondary Schools — Math Online Tom Circle

New Math education is successful in France but failed miserably in the U. S. (1960s) because of ONE simple reason: Math teacher’s quality and quantity. In France, each year the best of the undergraduate Math students are selected via the CONCOURS for Ecole Normale Supérieure – the Top university in France, and the World’s BEST […]

via The Success of French New Math Education in Secondary Schools — Math Online Tom Circle