User brendan foreman - MathOverflow

Answer by Brendan Foreman for Exponential (or other) families of distributions on manifolds.

2013-03-11T10:22:39Z

There are several subtle aspects to these questions that I am sure have emerged for you since you first posted this question. The primary one is of course that probability spaces are assumed to have algebraic structures so that moments and other elements for statistical investigation of a random variable can be found -- or at least defined -- through direct analytic methods.

For example, the kth moment of a random variable with values in $(0,1)$ and probability distribution $f(x)$ is $\int_0^1 x^k f(x) dx.$ Note that you need to be able to multiply the values of the random variables to themselves and a real-valued function ($f$). Even multivariate statistics (which lacks multiplication) utilizes the vector-algebraic structure of $R^N$ to calculate moments etc.

All of this is lacking on a general manifold. Embedding the manifold into Euclidean space will not help, since you would typically get expected values, etc. living outside of the manifold, although it would be rather entertaining to tell someone that the expected location of the crash site of a satellite with decaying orbit is the center of the Earth.

Anyway, that's a long-winded introduction to point out that you should check out the following papers:
Pennec, X. "Probabilities and statistics on Riemannian manifolds: a geometric approach" http://hal.inria.fr/inria-00071490/

Pennec, X. "Probabilities and statistics on Riemannian manifolds: Basic tools for geometric measurements." IEEE Workshop on Nonlinear Signal and Image Processing. Vol. 4. 1999.

Bochner in his text ${\it Harmonic\ Analysis\ and\ the\ Theory\ of\ Probability}$ describes stable distributions for general probability spaces, which is picked up for hyperbolic spaces in Getoor, R. K. "Infinitely divisible probabilities on the hyperbolic plane." Pacific Journal of Mathematics 11.4 (1961): 1287-1308.

Answer by Brendan Foreman for Interpolating a "manifold" between two points

2012-10-16T15:12:25Z

This isn't really a formula so much as a conceptual construction.

First suppose that $d=n-1.$

Let $T_1=span(\phi_1,..., \phi_{n-1})$ and $T_2=span(\tau_1,...,\tau_{n-1})$ be the tangent subspaces of $T{\mathbb R}^N$ at $p_1$ and $p_2,$ respectively. We can think of these as hyperplanes in real N-space so that there is an intersection of half-spaces created by these hyperplanes containing the line segment $l=p_1 p_2.$ Choose $r>0$ small enough that the spheres tangent to $T_1$ and $T_2$ at $p_1$ and $p_2$ with radii equal to $r$ and centers located on $l$ do not intersect. Call these spheres $S_1$ and $S_2$. Define a path of spheres by $S_{1+\lambda}=(1-\lambda)S_1 + \lambda S_2$ for $0\leq \lambda \leq 1,$ where addition is given by Minkowski addition of subsets in $N$-space.

The envelope of this path of spheres should be the boundary of a tubular neighborhood of a line segment, which is tangent to $T_1$ and $T_2.$

Second suppose that $d$ is strictly less than $n-1.$ Fill in the given orthonormal bases so that we have the above case and make the above construction. Then we let $T_1$ and $T_2$ be the subspaces at $p_1$ and $p_2$ spanned by the original given orthonormal bases and set $T_{1+\lambda}=(1-\lambda)T_1 + \lambda T_2$. The intersection of $T_{1+\lambda}$ with the corresponding $(1+\lambda)$-element from the envelope as constructed above should determine a $d$-dimensional submanifold satisfying the desired conditions.

Answer by Brendan Foreman for Unique symplectic form in an adapted complex structure

2012-10-09T11:00:09Z

This is a rather tough question to answer.

Check out: D. Burns & R. Hind, Symplectic geometry and the uniqueness of Grauert tubes, Geom. Func. Anal. Vol. 11 (2001), 1-10.

In this paper, they prove that, if two Grauert tubes of $M$ in $TM$ with respect to two Riemannian metrics $g_1,\ g_2$ are biholomorphic, then the biholomorphism restricts to an isometry from $(M,g_1)$ to $(M,g_2)$. Thus, if no such isometry exists, then the resulting Grauert tubes are not biholomorphic. (If I am interpreting this theorem correctly).

In general, though, the Kaehler form is derived from the complex structure composed with the given Riemannian metric $g$. Thus, if we change the metric, we will be necessarily changing the Kaehler form. So, I would conclude that the Kaehler form is not unique.

It would be interesting to see how this symplectic structure compares with the standard symplectic structure of $T^*M.$

Answer by Brendan Foreman for College (Euclidean) geometry textbook recommendations

2012-07-12T00:10:11Z

I would recommend Alfred Posamentier's Advanced Euclidean Geometry (Key College Press, 2002). It covers much of the same topics as Geometry Revisited by Coxeter/Greitzer and Episodes... by Honsberger, and it also presents accompanying technology (namely, Sketchpad applications) that allow the students to play around with the results. That is, it gives the students more opportunity to learn how to think geometrically.

I love all of the texts mentioned (Altshiller-Court, Coxeter, Coxeter/Greitzer, Honsberger,...), but their approach to the material is very different from what undergraduates would be used to. And very different than what they will be teaching.

You might find yourself spending a lot of time helping them process the text material into concepts they would find more natural, particularly, if any one of these were the primary text of the course. This may be more work than you would have originally desired: not only teaching the mathematical content but also how to translate mathematical texts.

Then again... this is a good thing for a high school teacher to know how to do...

Either way, I strongly recommend that you look at the Common Core Standards for geometry (at http://www.corestandards.org/the-standards/) and familiarize yourself with what content these future teachers will expect to be teaching. From there, you can get a good idea of what type of thinking and content knowledge you feel that someone would need to teach this material excellently.

Reference for a dual isoperimetric problem and solution

2012-01-16T12:41:18Z

I am trying to track down the first published solution to the following problem:

What curves within the unit disc in the plane and endpoints on the unit circle, minimize their length (within the ball) while dividing the area of the ball into two regions of a given ratio?

The solution is, of course, circular arcs orthogonal to the unit circle.

The only complete solution and proof that I've been able to find is embedded in a much stronger theorem within "Stability for Hypersurfaces of Constant Mean Curvature with Free Boundary" by A. Ros and E. Vergasta, Geometriae Dedicata, volume 56, 1995. But I suspect that there is an earlier proof for this specific case.

If anyone can point me in the right direction, I would very much appreciate it.

Edited for clarity of problem.

Answer by Brendan Foreman for Relationship between the focal locus and the cut locus

2011-09-13T01:59:46Z

I've been trying to work myself through this very problem of late. A text you might find useful is Geometric Differentiation by Ian Porteous. Therein the author through his own idiosyncratic yet quite utile methodology discusses the various level of contact that spheres can have with a given surface and hence involving the so called ridge points.

As alluded to in previous posters, I think that one of the key differences between the two loci is that of local vs global phenomena on a surface. Focal loci are local in nature, whereas cut loci are global.

Edit: This paper -- J.J. Hebda, Cut Loci of Submanifolds in Space Forms and in the Geometries of Moebius and Lie, Geometriae Dedicata, 55, 75-93, 1995 -- gives the following characterization of cut loci of submanifolds (which he takes to mean the entire set of focal points of the given submanifold and the points with two or more closest points on the submanifold): The set of cut points of a properly embedded submanifold of a complete, connected space form (i.e., Euclidean space, hyperbolic space or the sphere) is the set of centers of maximal supporting balls of that submanifold in the larger space, where a supporting ball is an open ball in the space-form that does not intersect the submanifold but whose boundary sphere does.

Answer by Brendan Foreman for Most memorable titles

2011-08-23T10:34:41Z

I've always enjoyed the poetry of the title:

"Period three implies chaos" -- T.-Y. Li & J. A. Yorke

Answer by Brendan Foreman for Foliating R^3 with straight lines

2011-04-28T12:28:48Z

The leaves of the foliation are regular. So, the space of leaves $X$ induces a submersion $\pi: R^3\rightarrow X$ with leaves of $R.$ Using the standard homotopy sequence, we see that $X$ is simply-connected. Furthermore, $X$ is path-connected since any two leaves can be connected by a path of leaves in the foliation.

We can actually define an embedding $\phi: X\rightarrow R^3$ by letting $\phi(l)$ be the point on the leaf $l\in X$ that is closest to the origin in $R^3.$ This gives us two things: a vector space structure on each leaf by using $\phi(l)$ as the origin and an orientation of $X$ (since only orientable simply-connected surfaces are embeddable in $R^3$). We use the orientation of $X$ to give an orientation on each leaf.

Thus, we have a 1-dimensional orientable vector bundle over a simply-connected space $X$, which means that the vector bundle is trivial. Hence, all smooth foliations with leaves of lines on $R^3$ are diffeomorphic.

I rather wanted to say all continuous foliations of such sort are homeomorphic, but I'm nervous about the embedding of $X.$

Answer by Brendan Foreman for Reference for Almost-Kahler geometry

2011-03-17T01:25:27Z

A slightly more current paper can be found here:

Apostolov, Vestislav; Drăghici, Tedi The curvature and the integrability of almost-Kähler manifolds: a survey. Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001), 25–53, Fields Inst. Commun., 35, Amer. Math. Soc., Providence, RI, 2003.

This gives a review of the various approaches to Almost Kaehler Geometry that has been taken since the Gray/Hervella article.

"incidental" intersections of a complete graph in the plane

2010-10-21T09:12:21Z

Given a complete graph of n vertices (no three of which are no collinear) in the plane and straight edges, what is the maximal possible number of "incidental intersections" of edges, i.e., number of non-vertices at which two distinct edges intersect each other, not counting multiplicity?

This is a question that I pose to the students in my Mathematics for Elementary School Teachers as a way to understand mathematical conjecturing and proving -- and not always finding the solution. But it occurs to me that it might be handy to know whether the answer is actually known or not.

Answer by Brendan Foreman for an engineering Ph.D. teaching math in college

2010-09-09T11:55:34Z

Accreditation is most likely not the reason for job postings of this sort. The Higher Learning Commission and most State Bodies (such as Ohio Board of Regents) give universities and colleges a great deal of latitude regarding whom they hire and tenure. It's only the professional programs involving licenses that will usually nitpick about education background: Teacher Education, Counseling Education, Athletic Training, School Psychology, etc. etc.

From personal experience in hiring faculty members, I would say that the most likely reason why these job postings are specifying Ph.D.'s in Mathematics is that in those situations they would have a fairly good idea of what course of study the given candidate would have taken. A Mathematics Department usually wants someone who can teach a variety of courses from the rudimentary to the specialized, and a Ph.D. in Mathematics would guarantee that the candidate personally experienced this range as a student.

Furthermore, a Mathematics Department usually wants someone with a specific mathematical research agenda, optimally one that complements their assigned teaching well. (Actually, the interface of Teaching and Research for Faculty Positions will of course differ greatly from school to school) The easiest way to guarantee this is to hire someone with the most obvious educational background.

That all said, I would encourage your colleague to apply for any position that they think they would be a good fit for, whether the posting specifies a math degree or not. The worst that can happen is that they hire someone else.

Comment by Brendan Foreman

2012-07-25T10:53:24Z

Of late, I've seen more work on the sub-Riemannian geometry of the Heisenberg group than on the Riemannian geometry. Two good texts on this topic are: An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem by Capogna, Danielli, Pauls, Tyson Birkhauser, 2007 Geometric Analysis on the Heisenberg Group and Its Generalizations by Calin, Chang, Greiner, AMS/IP, 2007 I don't know if this counts as particularly "hot," but I have found these to be interesting approaches.

Comment by Brendan Foreman

2012-05-22T16:38:47Z

This equation is a bit problematic. If $\alpha$ is a $k$-form, then $d\alpha$ is a $(k+1)$-form and $\delta\alpha$ is a $(k-1)$-form. I suppose you could interpret this equation as living on the entire space of forms, i.e., $\alpha$ is of mixed type. Did you mean to solve $(d+\delta)^2\alpha=0$? Those are the harmonic forms.

Comment by Brendan Foreman

2012-03-20T12:23:59Z

This question is much too vague to be of use here. The papers are freely available at arxiv. So, the poster can find out what mathematics is used there.

Comment by Brendan Foreman

2012-02-22T12:51:01Z

Thanks,Joseph. The second text here is especially useful for me.

Comment by Brendan Foreman

2012-01-22T23:09:43Z

This is the topic of study of this paper: M. Atiyah, "Complex analytic connections in fibre bundles" Transactions of the AMS, 1957, vol. 85, no. 1, 181-207. ams.org/journals/tran/1957-085-01/…

Comment by Brendan Foreman

2012-01-22T22:53:51Z

I'm not really sure what you're asking. But, when it comes to Sasakian geometry, the first place to go is the text "Sasakian Geometry" by C. Boyer and K. Galicki. Try looking at Chapter 10.

Comment by Brendan Foreman

2012-01-16T23:11:53Z

Thanks, Liviu. Your solution and interest are both gratifying. The problem is even more interesting given the context in which it was given to me -- namely, cellular anticlinal division. This is a case where Nature is doing the optimization for us.

Comment by Brendan Foreman

2012-01-16T17:27:45Z

Hi, Liviu, It is an easily solved problem. I was just curious if anyone knew of a published version of the solution in order to give credit where credit is due.

Comment by Brendan Foreman

2012-01-16T15:19:19Z

Edited as per Willie's comments

Comment by Brendan Foreman

2011-11-29T02:05:34Z

Does it not occur to you that the most "elementary" proof might actually be geometric in nature? Or are we using Conservapedian definitions here?

Comment by Brendan Foreman

2011-04-28T02:07:13Z

This shouldn't be too tricky. Since the leaves of the foliation are regular, the foliation will induce a submersion from \R^3→X for some surface X (namely, the space of leaves of the foliation). π_1(X) and π_2(X) are both trivial. So, X is diffeomorphic to \R^2. From here, it's a matter of classifying submersions over the plane with 1-dimensional leaves with a given affine structure. If we can find a surface embedded in R^3 that intersects each leaf exactly once, then this can be used to give the submersion a line bundle structure which means it's trivial.

Comment by Brendan Foreman

2011-03-09T11:37:12Z

It looks like the paper you want to read is: Manifolds with Many Complex Structures by Dominic Joyce, in Quart. J. Math. Oxford, vol. 46 (1995), 169-184. It's available here: eprints.maths.ox.ac.uk/61/1/complex.pdf In it, he constructs examples of manifolds with a prescribed number of complex structures with the relations (and subsequent restrictions) as you desire.

Comment by Brendan Foreman

2010-10-22T11:28:54Z

Thanks for the citation -- I didn't know about this book, to my embarassment. I like the almost Zen-like redundancy of the tile.

Comment by Brendan Foreman

2010-10-22T11:23:11Z

Yes, I meant the edges to be line segments -- I have modified the question accordingly.