User jon yard - MathOverflow

Answer by Jon Yard for Is there a Dirichlet Unitary Unit Theorem?

2012-05-19T06:00:17Z

Yes. Let $K \subset \mathbb{C}$ be a number field that is closed under complex conjugation (not necessarily Galois). Modulo the roots of unity, the group of unitary units in $K$ is free of rank equal to the number of infinite primes of $K$ minus the number of infinite primes of the real subfield $K \cap \mathbb{R}$. Recall that the number of infinite primes of a field is the number of real embeddings + half the number of complex embeddings. This is proved nicely in this paper by Daileda.

Note that $K$ and its real subfield have the same number of infinite places iff either $K$ is totally real or $K$ is CM (meaning a totally imaginary quadratic extension of a totally real field). So for precisely these cases, the only unitary units in $K$ are roots of unity.

Answer by Jon Yard for Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator

2012-04-20T06:29:22Z

I believe the double cover you are looking for is just $\tilde{U}(n) = \tilde{U}(1) \times SU(n)$, where $$\tilde{U}(1) = \lbrace\exp(\phi T) : 0 \leq \phi < 4\pi\rbrace$$ is the double cover of $U(1)$ and $T$ is a formal generator of its Lie algebra. This cover factors through the usual cover $U(1)\times SU(n) \to U(n)$ as $$(\exp(\phi T),U) \mapsto (e^{i\phi/2},U) \mapsto e^{i\phi/2} U.$$ So the problem reduces to classifying the $\tilde{U}(1)$ representations and the $SU(n)$ ones, and the answer is straightforward. First decompose
$$L^2(\mathbb{R}^n) \simeq \bigoplus_{N=0}^\infty V_N,$$ where $V_N$ is the eigenspace of $H$ with eigenvalue $2N+1$. Then $\tilde{U}(1)\times SU(n)$ should act irreducibly on each $V_N$ as $$(\exp(\phi T),U) \mapsto e^{iN\phi/2} \cdot \mathrm{Sym}^N U.$$

It is easy to see this. The $\tilde{U}(1)$ action is the same as with the usual $L^2(\mathbb{R})$ case. You can easily derive the $SU(n)$ representations using the Fock space isomorphism of $L^2(\mathbb{R}^n)$ with a Hilbert space of analytic functions: $$L^2(\mathbb{R}^n) \simeq L^2_\mathrm{hol}(\mathbb{C}^n,\pi^{-n}e^{-||z||^2/2}dz).$$ Under this isomorphism, $V_N$ maps to the space of homogeneous polynomials of degree $N$, and $SU(n)$ acts on polynomials in the usual way as $(Uf)(z) = f(Uz)$.

This is probably all contained in Bargmann's classic paper.

What are the roots of unity in abelian extensions of imaginary quadratic fields?

2011-06-03T23:40:44Z

What roots of unity can be contained in the abelian extensions of an imaginary quadratic number field $K = \mathbb{Q}(\sqrt{-d})$? In particular, I would like to know:

Is $K(\zeta_n)/K$ an abelian extension for every $n$?
What are the roots of unity in the ray class field of $K$ with conductor $\mathfrak{c}$?
What are the roots of unity in the ring class field of the order $\mathcal{O} = \mathbb{Z} + f\mathcal{O}_K$ with conductor $f$?

Answer by Jon Yard for Unitary groups over number fields

2011-05-01T20:44:28Z

If $E$ is not CM, then the action of complex conjugation on $E$ depends on how it is embedded into $\mathbb{C}$. In particular, it could have different real subfields depending on which embedding you are using. When $E$ is CM, so that it is a totally imaginary quadratic extension of the totally real subfield $F$, then it has a unique complex conjugation that commutes with all automorphisms of $E$, such that $F$ contains precisely the elements that are fixed by complex conjugation in all embeddings. This lets you talk about unitarity and hermiticity for the abstract field $E$, and not just for some particular embedding, which I would imagine could be problematic if not impossible to say anything useful about.

Or, to put it another way, there can be more than one way to consistently define a complex conjugation on your field, unless it is CM, in which case there is a unique way to define it.

Answer by Jon Yard for Fast Matrix Multiplication

2011-03-04T05:25:49Z

"Geometry and the complexity of matrix multiplication", by J. Landsberg from the AMS bulletin is a very nice article. It describes an approach to this problem based on algebraic geometry, that of bounding the "border rank" of the sequence of bilinear maps defining matrix multiplication. I don't think it reproduces the state of the art yet (but I'm not an expert so maybe), but it is a well-defined mathematical program that should in principle be able to uncover the optimal exponent. I think at least the basics of the approach should be pretty understandable with a minimum of background, but the whole theory does go pretty deep and technical. I believe this is, however, the nature of the beast - it is a shockingly deep question.

Cyclotomic extensions with split Galois group

2010-11-25T00:11:40Z

$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q}$

Consider the set of all Galois extensions $E/\Q(\zeta_n)$ of a given cyclotomic field $\Q(\zeta_n)$ such that $$ \Gal(E/\Q) \simeq\Gal(E/\Q(\zeta_n)) \rtimes \Gal(\Q(\zeta_n)/\Q). $$
In other words, such that there is a homomorphism $$ \Gal(E/\Q) \leftarrow \Gal(\Q(\zeta_n)/\Q) $$ inverting the natural quotient map $$ \Gal(E/\Q) \to \frac{\Gal(E/\Q) }{\Gal(E/\Q(\zeta_n))}\simeq \Gal(\Q(\zeta_n)/\Q). $$

Are they classified? Is there a "largest" one? What can be said about them (or about their cohomology) in general? Are there any prominent examples of such extensions arising "in nature"?

Answer by Jon Yard for How do you make a good math research poster for a non-mathematical audience?

2010-04-15T01:20:27Z

You should seriously consider using Adobe Illustrator if you can get your hands on it. I'm all for latex + beamer for talks, but you might end up waste far too much time on the formatting if you try to use latex. If you do this, you'll need to copy in your equations (as pdf with outlined fonts) created with some tool like LaTeXit (for Mac) or otherwise some equivalent. On the plus side, the added cost of putting in math will help you focus on the graphical content.

Answer by Jon Yard for What algorithm in algebraic geometry should I work on implementing?

2010-03-17T01:21:41Z

Just a thought, but maybe you should have a look at sage. It's a big open source project that is currently under very active development. If you're interested in contributing, I would suggest that you post to the sage-devel Google group with this same question. Some thoughts for things to do would be to improve the support for relative extensions of number fields and for function fields.

Which p-adic numbers are also algebraic?

2010-03-04T00:44:36Z

What is $\mathbb{Q}_p \cap \overline{\mathbb{Q}}$ ?

For instance, we know that $\mathbb{Q}_p$ contains the $p-1$st roots of unity, so we might say that $\mathbb{Q}(\zeta) \subset \mathbb{Q}_p \cap \overline{\mathbb{Q}}$, where $\zeta$ is a primitive $p-1$st root.

As a more specific example, $x^2 - 6$ has 2 solutions in $\mathbb{Q}_5$, so we could also say that $\mathbb{Q}(\sqrt{6},\sqrt{-1})\subset \mathbb{Q}_p \cap \overline{\mathbb{Q}}$.

Edit: I removed the motivation for this question (which I think stands by itself), as it will be better as a separate question once I think it through a bit better.

Answer by Jon Yard for What is the best notation for this set?

2010-01-31T11:07:28Z

Set the variables $V_{11}, V_{21}, \ldots, V_{X1}$ equal to zero.

Answer by Jon Yard for Geometric Interpretation of Trace

2010-01-31T02:32:26Z

I'm surprised nobody has mentioned this yet, but the trace defines a Hermitian inner product on the space of linear operators from $\mathbb{C}^n$ to $\mathbb{C}^m$: $$\langle A, B\rangle = \text{Tr}\ A^\dagger B.$$ You can't get much more geometric than that.

Answer by Jon Yard for Are the Gell-Mann matrices extremal when used as Kraus operators for a quantum channel?

2010-01-29T19:35:25Z

Actually, I think the channel is not extremal because I suspect you are misquoting the Landau-Streater result. So I will state it here.

To be precise, for anyone unfamiliar with the field, a quantum channel is a trace-preserving, completely-positive linear map on density matrices (positive semidefinite matrices with unit trace), of potentially different sizes. A basic theorem in quantum information says that every quantum channel from $m\times m$-dimensional to $n\times n$-dimensional density matrices can be written in Kraus form: $$ \rho \mapsto \sum_{i=1}^N A_i \rho A_i^\dagger, \text{ for linear operators } A_k \colon \mathbb{C}^m \to \mathbb{C}^n \text{ satisfying } \sum_k A_k^\dagger A_k = I_m. $$

It is easy to show that the set of quantum channels between systems of fixed dimension is convex. It also easy to show that the set of channels that map $\frac{1}{m} I_m$ to a fixed density matrix $\sigma$ is convex. Now the theorem of Landau-Streater says that if $m = n$, a channel with Kraus form as above is extremal in this latter set if and only if the $N^2$ linear operators $A_i^\dagger A_j \oplus A_j A^\dagger_i$ (of size $2m \times 2m$) are linearly independent. It seems you have instead been working with $m\times m$ matrices. But I think that even if you were to continue and apply the theorem correctly, you would only prove or disprove extremality in the convex subset of unital channels, i.e. those for which $\frac 1m I$ is a fixed point. So potentially you could strengthen Ben-Or's conclusion by showing non-extremality in this subset, or otherwise you might conclude extremality there, which would tell you nothing about extremality in the entire set of channels.

Do finite places of a number field also correspond to embeddings?

2010-01-28T09:10:17Z

Something that seems to be pretty standard in every introductory treatment is that the infinite places correspond to embeddings into $\mathbb{C}$. Do the finite places correspond to embeddings as well? I can envision two possibilities. My first guess is that the primes sitting above $p \in \mathbb{Q}$ correspond to embeddings into $\overline{\mathbb{Q}_p}$, and thus also to embeddings into $\mathbb{C}$ by some messy non-canonical field isomorphism. My second guess, which I think would imply the first, is that the places of $\mathbb{Q}[\alpha]$ above $p \in \mathbb{Q}$ correspond to embeddings into $\mathbb{Q}_p[\alpha]$. I've never been able to find a precise statement about this in any of the texts I've been studying (mostly Milne's notes and Frohlich & Taylor) and would appreciate if anyone could let me know where to learn more about this -- or if I'm just plain wrong.

One other thing is that the embeddings into $\mathbb{C}$ play a central role in analyzing the basic structure of a number field by way of Minkowski theory. Is there some analog for the finite places, or does that even make any sense?

Answer by Jon Yard for How would you compute that "average" ?

2010-01-28T08:09:48Z

It sounds to me that you need to detect the tempo of the music, and not the pitch. If you are trying to use a pitch-detection algorithms, then these are going to fluctuate rapidly, as they will lock onto the high frequences in your music. It sounds like you need something that filters out all but the lowest frequencies and allows you to determine how many BPM (beat per minute) the music is, as well as the phase of the beat also, so that you can do beatmatching as you originally mention.

However, I don't think that anyone here is going to be able to give you a simple formula for doing this directly from the samples. Digital signal processing is, by its very nature, a fairly mathematical subject. I do think that if you try Googling for "beat-matching signal-processing", or "beat-matching matlab", you will be pointed in the right direction, as you might find a published algorithm for doing exactly what you need. For instance, I found the following paper by searching: Design of an Automatic Beat-Matching Algorithm for Portable Media Devices. It might be worth looking at if you can get it without paying, say through a university with a subscription. Otherwise, I'm sure there are 100's of similar papers you can find on this subject. Also, many universities teach an audio signal processing class, and often the notes from these classes are online. Beat-matching is a common project for students to try in such classes and I'm sure you will be able to find some examples where people have done it.

Sorry I couldn't give you more explicit advice, but I hope I understood your question correctly and have pointed you in the right direction. Good luck.

Answer by Jon Yard for Switching Research Fields

2010-01-23T10:59:12Z

If you are interested in developing quantum algorithms, there is a nice up-to-date survey by Childs and van Dam called Quantum algorithms for algebraic problems, available at http://arxiv.org/abs/0812.0380. For PDE's, you might look into hyperbolic polynomials and hyperbolicity cones. The latter are characterized algebraically, there are lots of open questions about them, and their study is also practially relevant to engineering applications like convex optimization (in case practical applications, or perhaps hot funding areas, are what you're looking for).

What are the prime ideals in rings of cyclotomic integers?

2010-01-02T03:58:26Z

Is a good characterization of Spec $\mathbb{Z}[\zeta_n]$ known? Same question for its unit group.

Diagonalizing matrices over cyclotomic fields with unitaries

2009-12-27T22:13:29Z

Let $F$ be a number field with a fixed embedding $F \hookrightarrow \mathbb{C}$ such that the restriction of complex conjugation from $\mathbb{C}$ to $F$ is in Gal$(F/\mathbb{Q})$ and fix a Hermitian inner product $\langle v,w \rangle = \overline{v_1}w_1 + \cdots + \overline{v_n}w_n$ on $\mathbb{C}^n$ (with respect to the standard basis of $\mathbb{C}^n$. In particular, this restricts to a Hermitian inner product on $F^n$.

Now suppose we are given a unitary matrix $U$ on $F^n$ with respect to that inner product. It is well known (independent of unitarity) that $U$ is diagonalizable over some extension $E/F$ - for instance, take $E$ to be a splitting field of the minimal polynomial of $U$. This means there is a matrix $M$ over $E$ such that $M U M^{-1}$ is diagonal in the standard basis.

What if we instead want a unitary $W$ such that $W U W^\dagger$ is diagonal? This can be accomplished by working over a bigger extension $E'/E$ that includes some extra square roots of elements of E. Namely, given any $M$ that diagonalizes $U$ over $E$, just add in the square roots of the eigenvalues of $M^\dagger M$.

Now for my question:

Is there any sort of intrisic (i.e. independent of a choice of $M$) understanding of the extension $E'$? By understanding, I mean things like: is there a nice way of describing its generators over E? Can anything be said about its Galois group in general? When is it a semidirect product?

My actual interest is in the case where $F$ is cyclotomic and $U$ has finite order (and thus has roots of unity as eigenvalues, so $E$ is another cyclotomic field). Any advice on what is known in this specific, or otherwise the general case, would be be much appreciated.

Answer by Jon Yard for Famous mathematical quotes

2009-11-29T21:20:57Z

Someone once told me that Grothendieck said "a sheaf of groups is a group of sheaves," although I have been unable to find a real reference. Can anyone substantiate this?

Answer by Jon Yard for A learning roadmap for Representation Theory

2009-10-28T05:01:54Z

My favorite book right now on representation theory is Claudio Procesi's Lie groups: an approach through invariants and representations. It is one of those rare books that manages to be just about as formal as needed without being overburdened by excessive pedantry. He gives a rather complete picture of both compact and algebraic groups and how they interplay, while doing a nice job of explaining the necessary background in algebraic geometry and functional analysis. He covers all the "standard" material on Young symmetrizers, Schur duality, representations of GL_n, semisimple Lie groups & algebras, as well as more advanced stuff like Schubert calculus and some basic geometric invariant theory. This book was the first place I started to feel like I was "getting" the big picture, after picking up bits and pieces from different places.

If your institution has a subscription to SpringerLink, you can probably download this book for free (legally) and purchase an on-demand print version for around $25 USD.

Since this question was about a "learning roadmap," and not just for a single textbook, let me mention my favorite book that fits in your back pocket: "Lectures on Lie groups and Lie algebras" by Carter, Segal and Macdonald. The section by Segal is especially nice.

Answer by Jon Yard for Examples of great mathematical writing

2009-10-16T02:25:06Z

"Algebraic curves and Riemann surfaces" by Rick Miranda is one of my favorite books. It is full of concrete examples and is full of very clear explanations for everything from the basics of Riemann surfaces and their projective embeddings though sheaf cohomology. Also, it assumes little more than elementary complex analysis.

Answer by Jon Yard for Examples of great mathematical writing

2009-10-16T02:14:31Z

This paper changed my life:

Vaughan Jones, "Hecke algebra representations of braid groups and link polynomials"

Comment by Jon Yard

2011-06-04T02:07:53Z

Thanks guys. So it is easier than I thought. I'm just learning this stuff on the fly, but I guess I should have thought it through a bit more before posting.

Comment by Jon Yard

2011-05-06T16:04:54Z

This book truly is great. Just checked it out yesterday and wish I had a year ago. Shame that it currently costs over $100 on Amazon though.

Comment by Jon Yard

2011-01-06T05:46:05Z

You should also have a look at Mumford's Lectures on Theta Ch. 1 and Ch. 2. They are extremely concrete.

Comment by Jon Yard

2010-12-04T19:30:31Z

Thanks Franz. After hours of searching, I cannot find the Wyman article (or really much at all from Scripta mathematica) online. Does anyone know where to find an electronic copy?

Comment by Jon Yard

2010-11-27T00:42:19Z

Not sure I saw this explicitly mentioned here, but it could be that the Fourier transform doesn't exist but the Laplace transform does, only on a subset of the complex plane (the so-called "region of convergence", or ROC). When the ROC contains the imaginary axis then you get back the Fourier transform by evaluating there.

Comment by Jon Yard

2010-11-25T08:45:09Z

Alex, good point about Kummer extensions, and thanks for expanding on this Chandan.

Comment by Jon Yard

2010-11-25T06:12:26Z

Tom, this is a good point. So I guess this means that if E is such an extension and if s generates (Z/p), then s^{(p-1}/2} needs to map to complex conjugation in E. These sort of facts are what I was after.

Comment by Jon Yard

2010-10-23T23:54:51Z

Well, if you want to know more Martin, have a look at arxiv.org/abs/quant-ph/0702225

Comment by Jon Yard

2010-10-23T17:30:43Z

This really does have everything to do with mixed state entanglement in quantum mechanics. A positive matrix is "separable" precisely when it can be written as a sum of tensor powers of positive matrices. These form a convex cone that is a strict subset of the cone of all positive matrices in the tensor product, and the complement of the separable operators in this cone are "entangled". The operators of unit trace correspond to (either separable or entangled) density matrices. There are close parallels to the classification of positive/completely positive linear maps on matrix algebras.

Comment by Jon Yard

2010-05-17T17:15:21Z

This article homepages.cwi.nl/~lex/files/perma5.pdf from a recent issue of the American Mathematical Monthly describes the recent - and completely elementary - proof of Van der Waerden's theorem due to Leonid Gurvits.

Comment by Jon Yard

2010-03-10T04:15:28Z

In quantum mechanics, it is common to say something like "the first component" or otherwise "the first tensor factor" of the tensor product. My preference is for "component" since it's less wordy.

Comment by Jon Yard

2010-02-10T01:18:17Z

It is perhaps a bit of a historical fluke that information theory lives in EE. This, of course, stems from its roots in communication theory. Pure information theory might be better characterized as a branch of statistics, or possibly even physics. That said, another good example along these lines is the paper Information theoretic inequalities by Dembo, Cover & Thomas, where, for instance, some classical determinantal inequalities are derived from consideration of entropies of multivariate normals.

Comment by Jon Yard

2010-02-01T05:08:31Z

Thanks Pete. I didn't know the formulation using prime ideals in the tensor product algebra. Looking forward to seeing your lecture notes when they're done.

Comment by Jon Yard

2010-02-01T04:35:36Z

Thanks. I hadn't realized that completing the field completes its ring of integers as well, but I guess this makes complete sense in hindsight.

Comment by Jon Yard

2010-02-01T04:33:37Z

Thanks to pointing me to Neukirch for the Arakelov theory. I knew (loosely) about the analog between number and function fields, but didn't know where to look to find out more. This was actually part of the motivation for my question; since places are the points of a topological space, it seems they should be treated on the same footing as much as possible.