4
votes
6answers
929 views

Algebraic Geometry for non-mathematician [closed]

I think I sound stupid but I have heard a lot about Algebraic Geometry as a subject and wish to study it without actually studying abstract algebra. I have never studied abstract algebra since I am a ...
2
votes
1answer
135 views

How to obtain the Period matrix from the Igusa Invariants of a genus two curve?

I am looking for an algorithm to obtain the period matrix tau= (tau1 & tau12\ tau12 & tau2) from the Igusa invariants of a genus two curve. More precisely: I consider a family of genus two ...
7
votes
2answers
356 views

Kummer's quartic surface and the Dirac operator

The picture shows 1936 photograph by Man Ray of a Kummer surface plaster model in the collection of the Institute Henri Poincare (from Mathematics, Art and Science of the Pseudosphere by Kenneth ...
1
vote
2answers
478 views

Uniqueness on square root of complex Line Bundle

Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
0
votes
1answer
274 views

Symplectic structure on $Sym^kG^{\mathbb{C}} $

Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its complexification. I am looking for a symplectic structure (without use of coordinates) on $$ Sym^kG^{\mathbb{C}}, $$ PS:Here ...
2
votes
1answer
320 views

looking for an identity for higher jet bundle $J^kM$?

We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*M×\mathbb{R}$.i.e, ($J^1M=T^*M×\mathbb{R}$) Is there something like this identity for higher jet bundle $J^kM$? I editted ...
2
votes
1answer
357 views

A question about flag variety of $SL(n,\mathbb{C})$

We know that the flag variety $SL(2,\mathbb{C})/B$ which $B$ is Borel subgroup, can be identified with $\mathbb{P^1}$, What can we say about $SL(n,\mathbb{C})/B$ which $B$ is Borel subgroup of ...
10
votes
2answers
1k views

What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...
4
votes
0answers
264 views

Which orbits of a separable representation of the infinite unitary group are closed?

Consider a separable irreducible unitary representation of $U(\mathcal{H})$ in the Hilbert space $V$. Assume that $\mathcal{H}$ is separable. My question is the following: Is it true that all ...
1
vote
0answers
108 views

Felder Kazhdan classical master equation

Has there been any follow up by anyone to Giovanni Felder, David Kazhdan, The classical master equation (arXiv:1212.1631)? Other than on nlab, I haven't found any citations.
6
votes
1answer
484 views

Darboux like theorem for non-degenerate 3-forms in 6-manifolds

we know Darboux theorem for higher-symplectic geometry is not correct in general, but is there any Darboux like theorem for non-degenerate 3-forms in 6-manifolds?
2
votes
0answers
113 views

universal connection for SU(3)

In 1961, Narasimhan and Ramanan (Am J Math 83 563) showed that one could represent an arbitrary SU(3) connection as $ i t^c_{a b} A^c_\mu(x) = e^*_a \cdot e_{b, \mu}(x) $ in which the $ t^a $'s are ...
1
vote
1answer
180 views

Gaussian measures on non-separable spaces

Let $X$ be a topological affine space which is neither separable nor metrizable. There are plenty of trivial Gaussian measures: each Dirac point-mass $\delta_x$ are the Gaussian measure with zero ...
5
votes
1answer
188 views

Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...
0
votes
0answers
265 views

Moment of Inertia of a Polygon

I'm programming a game and I have to find the moment of inertia of a flat polygon (axis of rotation is perpendicular to the polygon, from now on I will call it the point of rotation). My idea was to ...
10
votes
1answer
778 views

decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$

Do anybody know , why can we write the following decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$ 1) $E_8=(V^{\star}\otimes \wedge^{8}V^{\star})\bigoplus ...
0
votes
0answers
77 views

Divisors, factorisations of matrix valued functions, and the Lorentz group

How to construct a complex projective variety with several classes of non-intersecting divisors? How to keep the answer concrete and simple, so that explicit calculations can be done? And the problem ...
10
votes
0answers
525 views

conformal blocks for beginners

I have given now a couple of talks that involve conformal blocks bundles on the moduli stack $\overline{\mathcal{M}}_{g,n}$, in front of a public of algebraic geometers but not specialists of the ...
29
votes
4answers
2k views

Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one? I've found some examples: 1) In MO-Q111339 ...
6
votes
0answers
279 views

Equivariant quantum cohomology and K-theory of vector bundles over a Grassmannian

Consider a sum of m copies of the tautological bundle over the Grassmannian of n-planes in complex k-dimensional vector space. There is an obvious action of an (m+k)-dimensional torus T on the total ...
3
votes
0answers
262 views

Motivic DT-Invariants for the Algebro-Geophobic

I am looking for as gentle of possible of an introduction to Kontsevich-Soibelman's theory of motivic DT-invariants. Specifically I am interested in the algebraic aspects of the theory and the ...
7
votes
1answer
457 views

what is the stringy Kähler moduli space?

I saw the stringy moduli space mentioned in a few papers but with little no explanation. I vaguely understand it is supposed to be the moduli space of complex structures on the mirror manifold. Could ...
4
votes
1answer
468 views

Random geometries

Let $M$ be a smooth $n$-dimensional manifold, and let $FM = GL(M)$ indicate its tangent frame bundle. Let $G$ be a fixed linear subgroup of $GL(n)$, and consider the space $\mathcal S$ of all ...
5
votes
0answers
247 views

Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties

The Question Asked Definition: the Second-Hand Lion trace distance $D_k$ Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...
-3
votes
1answer
2k views

Quantum dynamics on varieties and Salmon Prizes

Concluding Progressive Remarks A new finding is Bates and Oeding's preprint "Toward a salmon conjecture" (arXiv:1009.6181), with its reference to the Salmon Prize. The Salmon Prize (photo of the ...
5
votes
2answers
761 views

Places to learn about Landau-Ginzburg models

Here is what I know about Landau-Ginzburg models: It is an important player in the story of mirror symmetry. It involves "potentials" which are functions of complex varibles, which have isolated ...
8
votes
1answer
409 views

Multiple Hodge integrals and integrability

It is known that a generating function of the linear Hodge integrals is a tau function of the KP hierarchy, namely a one-parameter deformation of the Kontsevich-Witten tau-function (see Kazarian). ...
13
votes
1answer
2k views

unbounded self-adjoint operator as Killing vector fields

Hey, the following is well-known (e. g. Ashtekar/Schilling, Brody/Hughston): A bounded self-adjoint operator $A$ on a Hilbert space $H$ induces a globally defined vector field $X$ on the projective ...
2
votes
2answers
654 views

Projective Hilbert space: L^2

I am trying to get my head around the geometric formulation of Quantum Mechanics as a projective Hilbert space (see Ashtekar, http://arxiv.org/abs/gr-qc/9706069). So one identifies all the rays ...
0
votes
1answer
182 views

Which bundles does the character vareity parameterize?

For any Riemann surface with punctures $C$, and lie group $G$, the character variety is the space of maps $\mathrm{Hom}(\pi_1(C), G)$. I know that if $G= S_n$ (not a lie group), then ...
2
votes
1answer
556 views

(Anti)Canonical divisor of a blow up

This question may be utterly trivial, or not, but as someone with hardly any knowledge of algebraic geometry I thought there could be a chance I get lucky. Let X be a rational surface obtained by n ...
3
votes
3answers
730 views

Fractional Quantum Hall Effect - Mathematics

Just to include something that starts to answer my own question Topological Quantum Computation Lecture notes covers a lot of the Mathematics of the Fractional Quantum Hall effect, or topological ...
6
votes
1answer
706 views

Conformal Field Theory and Langlands

I'm a Mathematics masters student currently studying some aspects of TQFT. I'm interested in Langlands, mainly because it sounds oppressive! Is anyone familiar with any links between CFT and ...
19
votes
6answers
2k views

Concrete example of $\infty$-categories.

I've seen many different notion of $\infty$-categories, actual I've seen the operadic-globular ones of Batanin and Leinster and the opetopic too and eventually I'll see the simplicial ones too. ...
0
votes
2answers
2k views

Calculating moment of inertia in 2d planar polygon

I've derived equations for 2d polygon's moment of inertia using Green's Theorem (constant density \rho) $$I_y = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 + x_i x_{i+1} + x_{i+1}^2 ) ( x_i y_{i+1} - ...
10
votes
1answer
428 views

Is there a canonical map between the cohomology of orbifold Chiral de Rham on an orbifold and the cohomology of Chiral de Rham on a crepant resolution?

The two-variable elliptic genus is a topological invariant of almost-complex manifolds that takes values in power series. These power series turn out to describe weak Jacobi forms when the manifold ...
2
votes
0answers
456 views

relation between AGT- conjecture and CNV-correspondence

Is there any relation between AGT conjecture 0906.3219 and the 4d-2d correspondence of 1006.3435 ? For pure SYM of $\mathcal{N}=2$ , SU(2) guage group thoery, we know the explicit instanton ...
0
votes
0answers
238 views

Is an immersed Kronecker join always a multilinear variety on a Hilbert space?

The question asked is: Is the implicitization of an arbitrary-rank immersed Kronecker join always a multilinear variety on a Hilbert space? This is related to another MathOverflow question ...
7
votes
2answers
689 views

The affine Grassmannian and the Bogomolny equations

In "Electric-Magnetic Duality and The Geometric Langlands Program", Sections 9 and 10, Kapustin and Witten describe certain convolution varieties in the affine Grassmannian (and more generally, in the ...
7
votes
1answer
503 views

Non-algebraic curve visualisation

Is there any software which can automatically visualise a non-algebraic complex curve, I mean the structure of it's ramification points and sheet? I think a good test example would be the Lambert ...
15
votes
2answers
2k views

Wall Crossing in Physics and Mathematics

This question is motivated by the current interest of Mathematics and Physics community in Wall Crossing. My questions are : What is wall crossing in Physics, what are the reasons for current ...
7
votes
0answers
644 views

triangulated/derived categories in Physics and algebraic geometry

Why do physicists care about the triangulated/derived categories? I mean what are the problems we want to approach using the machinery of triangulated/derived categories. e.g. in homological mirror ...
15
votes
6answers
2k views

mirror symmetry with algebraic geometry?

Why is it that mirror symmetry has many relations with algebraic geometry, rather than with complex geometry or differential geometry? (In other words, how is it that solutions to polynomials become ...
4
votes
1answer
443 views

What does it mean to extend a 2d (topological) conformal field theory to Deligne-Mumford space?

For 2D (topological) conformal field theory the corresponding moduli space is the space of Riemann surface with boundary, right? What does it mean by extending the theory to Deligne-Mumford space? How ...
10
votes
3answers
345 views

Algebraic Curves and Phase Diagrams of Physical Systems

Lots of low degree curves arise naturally as the phase spaces of physical systems (that is, the curve parameterized by $(q,p)$ where $q$ is a generalized position variable and $p$ is a generalized ...
4
votes
1answer
432 views

Gromov-Witten and integrability 2.

This is a followup of my previous question Gromov-Witten and integrability. As I have learned from the answer (but guessed before), GW potentials of the point and $P^1$ (with different modifications) ...
7
votes
1answer
946 views

Gromov-Witten and integrability.

The generation function of the Gromow-Witten invariants (with descendants) of the point is known to be Kontsevich-Witten tau-function of KdV, partition functions of $P^1$ and equivariant $P^1$ are ...
8
votes
1answer
567 views

Virasoro constraints for the generating function of Hurwitz numbers.

Generating function of the simple Hurwitz numbers is known to be connected with Gromov-Witten potential of the point (Kontsevich $\tau$-function) (see e.g. Ian Goulden, David Jackson and Ravi Vakil). ...
6
votes
2answers
1k views

Matrix factorization categories beyond the isolated singularity case

In his really nice thesis, Tobias Dyckerhoff proved the following theorems about matrix factorizations(of possibly infinite rank) over a regular local k-algebra R with a function w and residue field k ...
3
votes
1answer
516 views

Computation of homology groups of $M_{g,n}$

First some definitions: $\bar{M_{g,n}}$ is Deligne-Mumford space, i.e., the moduli space of stable nodal complex projective curves of genus $g$ with $n$ marked points. It is a complex orbifold, ...