Question

members..this question,is one from "Hodge conjecture".
the guy should be really appreciated if any among us can solve out this one.
first let me give intro..about the hodge conjecture,,,

The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the homology classes of subvarieties.

Answer 1

now the question is :

The strongest evidence in favor of the Hodge conjecture is the algebraicity result of. Suppose that we vary the complex structure of X over a simply connected base. Then the topological cohomology of X does not change, but the Hodge decomposition does change. It is known that if the Hodge conjecture is true, then the locus of all points on the base where the cohomology of a fiber is a Hodge class is in fact an algebraic subset, that is, it is cut out by polynomial equations. Cattani, Deligne & Kaplan (1995) proved that this is always true, without assuming the Hodge conjecture.

Answer 2

what you want to know just tell me about that from this one..i will surely elaborate it.

Answer 3

Actually, I don't want to know. It hurts my eyes to look at words I've never seen before. For example "cohomology".

Answer 4

okay let me xeplain it..it's better to know about anything you don't know then to get shocked.

Answer 5

specifically in algebra, cohomology is a general term for a sequence of abelian groups[ an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order] defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries

Answer 6

List of unfamiliar words and phrases:

"Hodge Conjecture"
"algebraic geometry" (why do they have to describe geometry as algebraic)
"algebraic topology" (remind me what topology is again. seen it before but idkwit means)
"non-singular complex algebraic variety" (what?)
"de Rham cohomology classes" (again, what?)
"Poincaré duals" (excuse mwah?)
"homology classes of subvarieties" ( it just gets sadder)
"co-chain complex" (even your explanations have never before seen expressions in them)
"cochains, cocycles, and coboundaries" (this is the point of no return)

Answer 7

lol point of no return =))))

Answer 8

guys thats why i said that noone can do this one..but let me help in these points till my best.

Answer 9

By all means. You can teach the whole class while you're at it.

Answer 10

This is advanced (pure) mathematics.

Answer 11

""""""Algebraic geometry"""""
is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology[Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation.] and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

Answer 12

yes throught that i'am as a tutorial..elaborating all those points..which needs to be explained.

Answer 13

in algebraic geometry i have written about TOPOLOGY..

Answer 14

Considering your other question I am not too sure if you should attempt solving this one right now :) or you are just trying to understand? In either case this is not something that attracts attention of an applied mathematician ;)

Answer 15

I recognize the word 'commutative'
i have been reading about the commutator and the anti-commutator ,
i do not recognize any other other words

Answer 16

okay..if you wana know then do tell me.

Answer 17

OK well i just said ive been reading about the commutator and the anti-commutator, i dont really understand what they represent in reality or what contexts they are useful in, can you tell me something about them?

Answer 18

let me try it out.

Answer 19

The commutator of A and B
\[[A,B]_−=AB-BA\]
The anti-commutator of A and B
\[[A,B]_+=AB+BA\]

Answer 20

this one seem from matrix isn't it ?
but i can't tell you much about that coz its of uppergrade.

Answer 21

i dont think A or B have to be matricies

Answer 22

In mathematics, the"""""""""""""""POINCARE DUALITY """"""" theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the (n − k)th homology group of M, for all integers k

Answer 23

i cant really say i understand what a manifold is

Answer 24

a manifold is a subset of Euclidean space which is locally the graph of a smooth (perhaps vector-valued) function.