Questions and Answers in Geometry and Analysis
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Below are the 10 most recent journal entries recorded in
Questions and Answers in Geometry and Analysis' LiveJournal:
Tuesday, August 29th, 2006  11:45 am [pasha_m]

Is there a geometrical meaning in the following number associated to a finitedimensional semisimple Lie algebra g: an integral over g of the density rho(x)=Det(Sinh(ad_x)/ad_x) for x\in g, calculated on matrices of adjoint representation of g?  Tuesday, April 4th, 2006  1:35 pm [nekura_ca]

Questions about Terminology.
Hello, I'm doing some work, and am having some difficulty with terminology, so I want to get it right. I have two questions for now. 1. I have a surface in threespace, and am looking for a term to describe if it's possible to determine which side a viewer is on. I was thinking about the 33336 tiling, how viewed from one side it has a left twist, and the other side has a right twist, so I thought chiral vs. achiral would work, but if I have a surface that is all black on one side, and all white on the other, compared to one that is white on both sides, that terminology is wrong. So is there a term to describe if a surface looks the same from both sides, or has a different appearance. 2. In a tiling, if A and B both belong to the same Transitivity class, is there an equivalence term to say that. When I've been writing I've been saying that A and B are Transitive, but I'm not sure it that's valid, or correct, so I want to make sure before I use it any more. Thanks Nekura Current Mood: curious  Friday, January 13th, 2006  3:32 pm [bravchick]

Relationship between Lpolynomial and the StiefelWhitney class Let M be a closed manifold of odd dimension n=2k+1. Let L(p) denote the Hirzebruch Lpolynomial in the Pontrjagin classes of M. Denote by L_{2k}(p) the component of L(p) in H^{2k}(M,Z) and by w_{2k} Î H^{2k}(M,Z_{2}) the 2kth StiefelWhitney class of M. I have strong reasons to believe that the reduction of L_{2k}(p) modulo 2 is equal to to w_{2k. }. Is it really so? And, if it is, how to prove it?
Remark: if k is odd, so that 2k is not divisible by 4, then, of course, L_{2k}(p) = 0 by dimensional reason. In this case, Massey [Amer. Jornal of Math. 82, 92102] showed that w_{2k}=0. Thus, if k is odd, the answer to my question is positive.  Monday, October 10th, 2005  8:01 pm [_vopros_otvet] 
Dear ALL! Do you probably know how to: 1. Find furthestneighbor or furthestsite diagram out of Voronoi Diagram? (Fast and efficient way for each site to know which one is farthest site from it)? 2. Let P1...Pk be a collection of pairwise disjoint simple polygons with a total of n edges, all enclosed in a given square. Find in O(n*logn) time a largest disk that can be inscribed in this square so that it is disjoint from all the interiors of the polygons Pi  how do I solve this? Provided I know how to find largest empty circle in Voronoi diagram, how can I take care that it does not intersect the EDGES of polygons (staying disjoint with their interior)? Thanks a lot!!!  Saturday, August 13th, 2005  11:32 am [dhilbert83]

Question on Haar Measure/Integration
Ok well you guys said this is for analysis, so don't yell at me since this post is not even remotely geometric :p If G is a locally compact group and K subse G is compact, and lim n > inf int_K f_n(x) dx = 0, then does lim n > inf int_{K^1} f_n(x^1) dx = 0? (f_n subset L^p(G) for some p >= 1) I actually ultimately need this question asked for integration being Bochner integration, but is this true even for Lebesgue integrals? I don't think it is, since the modular function will screw things up even if it's a 'mild' function.  Monday, March 28th, 2005  10:53 pm [mostconducive]

NonMetrical Geometry
What is NonMetrical Geometry? It is mentioned in A. N. Whitehead's "Process and Reality" (pages 490s), in discussion of infinitesimals (Weierstrass), presentational immediacy, strain locus, projection (opposite of injection, I think), God and mentality. Intuitively, it brings me back to the 470s (pages); "Its is to be remembered that two points determine a complete straight line, that three noncolinear points determine a complete plane, and that four noncoplanar points determine a complete three dimensional flat locus." ("Process and Reality", page 472)  Monday, March 14th, 2005  4:17 pm [philosophking]

I'm really sorry, but I have questions that really aren't directly pertinent to the group. How are you using the special math text that you used in your first post to this community? Can you give me a link to how to use it? Would graph theory be included in your interests? Thank you very much!  Thursday, March 10th, 2005  8:45 pm [bbixob]

naive defitinion of normalisation of complex manifolds
I need a reference to these simple statements about normalisation of complex spaces; are they correct at all ? I am willing to assume everything lies a very nice ambient space, say holomorphically convex manifold. \def\hatZ{{\hat Z}} Let $\nnn:\hatZ \ra Z$ be the normalisation of a variety $Z$, and let $\Zz k(\C) = \{z\in Z(\C): \#\nnn\inv(z)\geq k\}$ be a closed subvariety of points which have $k$ preimages or more. \begin{fact}\label{fact:zknrm} $\Zz k$ is a closed subvariety of $Z$, for all $k$; $\Zz {\deg \nnn + 1}$ is empty, $\Zz 1 = Z$. \end{fact} \bp Reference!!!! \ep \begin{fact}\label{fact:214} Around each point $z\in Z(\C)$ there exists a sufficiently small neighbourhood $z\in V$ open in complex topology such that $$V\cap \Zz k= \bigcup\limits_{i_1<..<i_k} V_{i_1}\cap ...\cap
V_{i_k},$$ where
$$Z\cap V=V_1\cup ...\cup V_n$$ is the irreducible decomposition of
$Z\cap V$.\end{fact} \bp not [Grauert, ???] [Abhyankar, Local
Analytic Geometry, p.402, Cl 7] \ep  Saturday, March 5th, 2005  10:59 am [novichyok] 
Flat connection vs. representation of the fundamental group
Let M be a closed manifold. A complex representation α: π_{1}(M) → C^{n} of its fundamental group gives rise to a flat vector bundle E_{α} = M×_{α}C whose monodromy is isomorphic to α. Given a continuous family of connections α(t) the bundles E_{α(t) }are isomorphic (as vector bundles without a connection). Thus it is not difficult to see that there exists one bundle E → M and a continuous family of connections (t) such that the monodromy of (t) is isomorphic to α(t). Moreover if α(t) is differentiable then (t) can be chosen to be differentiable. I have 2 questions: </p> 1. Is there any book or paper where these simple facts are proven? 2. How far this relationship between families of representations and families of connections can be pushed? For example, if α(t) is analytic can one find an analytic family (t)?  Friday, March 4th, 2005  12:53 am [bravchick]

Reference for variational formula for the determinant of an elliptic operator
Let A be a (note necessarily selfadjoint) invertible elliptic operator. The following variational formula for its determinant is well known δ Det(A) := δ (∂_{s} Tr ( A ^{s} ) _{s=0})_{ } = ∂_{s} ( s Tr [( δ A ) A ^{s1}])_{s=0}cf., for example, Burghelea, Friedlander, Kappeler, MeyerVietoris type formula for determinants of elliptic differential operators. J. Funct. Anal. 107 (1992), 3465, or Kontsevich, Vishik , Geometry of determinants of elliptic operators. (of course, to define A ^{s} one uses a spectral cut, which I did not write explicitly to simplify the notation). Is it proven anywhere in the literature? (I do know how to prove it, but I need a reference) 
