Why is the Langlands program so important?
As an analogue to the possible exact distribution of primes, the Langlands program allows a potential general tool for resolution of invariance at generalized algebraic structures. This in turn permits a somewhat unified analysis of arithmetic objects through their automorphic functions.
What are the main features of Langlands?
They are usually identified with the active, contemplative, and “mixed” religious life, but the allegory of the poem is often susceptible to more than one interpretation, and some critics have related it to the traditional exegetical way of interpreting the Scriptures historically, allegorically, anagogically, and …
Is Hodge conjecture solved?
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.
What is algebraic geometry used for?
In algebraic statistics, techniques from algebraic geometry are used to advance research on topics such as the design of experiments and hypothesis testing [1]. Another surprising application of algebraic geometry is to computational phylogenetics [2,3].
What is a Galois representation kisin?
a Galois Representation? Mark Kisin. Let Q be the field of algebraic numbers. The Galois group GQ = Gal(Q/Q) is the group of automor- phisms of the field Q. A Galois representation is simply a representation of this group, or indeed of any Galois group.
Is the ABC conjecture proved?
Various attempts to prove the abc conjecture have been made, but none are currently accepted by the mainstream mathematical community and as of 2020, the conjecture is still regarded as unproven.
What is the hardest math problem?
But those itching for their Good Will Hunting moment, the Guinness Book of Records puts Goldbach’s Conjecture as the current longest-standing maths problem, which has been around for 257 years. It states that every even number is the sum of two prime numbers: for example, 53 + 47 = 100.
Is algebraic geometry useful for physics?
In recent years the interaction between algebraic geometry and theoretical physics has been particularly fruitful. “In recent years algebraic geometry and mathematical physics have begun to interact very deeply mostly because of string theory and mirror symmetry,” said Migliorini.
Is an algebraic tool for studying the geometry?
Coordinate geometry has been developed as an Algebraic tools for studying geometric figures.
What is the Galois group of a polynomial?
Definition (Galois Group): If F is the splitting field of a polynomial p(x) then G is called the Galois group of the polynomial p(x), usually written \mathrm{Gal}(p). So, taking the polynomial p(x)=x^2-2, we have G=\mathrm{Gal}(p)=\{f,g\} where f(a+b\sqrt{2})=a-b\sqrt{2} and g(x)=x.
Has the abc conjecture been solved?
What kind of program is the Langlands program?
The Langlands program consists of some very complicated theoretical abstractions, which can be difficult even for specialist mathematicians to grasp.
Is the Langlands program a number theory setting?
The Langlands program is incredibly vast and far-reaching. The deepest aspect of it, as far as we know, involves the number theory setting where Langlands started close to forty years ago. However, the Langlands program has all kinds of…
How is the reciprocity conjecture related to the Langlands program?
Roughly speaking, the reciprocity conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an L -group. There are numerous variations of this, in part because the definitions of Langlands group and L -group are not fixed.
Where did Robert Langlands live as a child?
Born on October 6, 1936, in British Columbia, Robert Langlands grew up in a small town where his father owned a building supply store. “When I was a child I liked to add and subtract,” says Langlands. “In our store, my mother worked.