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Matthias Flach (mathematician)
Matthias Flach is a German mathematician, professor and former executive officer for mathematics (department chair) at California Institute of Technology. Professional overview Research interests includes: *Arithmetic algebraic geometry (see Glossary of arithmetic and Diophantine geometry). *Special values of L-functions. *Conjectures of: ** Bloch ** Beilinson **Deligne ** Bloch–Kato conjecture (see also List of conjectures). *Galois module theory. *Motivic cohomology. Education overview *Ph.D. University of Cambridge UK 1991 Dissertation: Selmer groups for the Symmetric Square of an Elliptic Curve – Algebraic geometry *Diplom, Goethe University Frankfurt, Germany, 1986 Publications *Iwasawa Theory and Motivic L-functions (2009) – Flach, Matthias *On Galois structure invariants associated to Tate motives – Matthias Flach and D. Burns, King's College London *On the Equivariant Tamagawa Number Conjecture for Tate Motives, Part II. (2006) – Burns, David; Flach, M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated between the Baltic and North seas to the north, and the Alps to the south; it covers an area of , with a population of almost 84 million within its 16 constituent states. Germany borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, and France, Luxembourg, Belgium, and the Netherlands to the west. The nation's capital and most populous city is Berlin and its financial centre is Frankfurt; the largest urban area is the Ruhr. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity. A region named Germania was documented before AD 100. In 962, the Kingdom of Germany formed the bulk of the Holy Roman Empire. During the 16th ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices. K-theory involves the construction of families of ''K''-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyahâ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Tamagawa Number Conjecture
In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely :1 \,-\, \frac \,+\, \frac \,-\, \frac \,+\, \frac \,-\, \cdots \;=\; \frac,\! by the recognition that expression on the left-hand side is also ''L''(1) where ''L''(''s'') is the Dirichlet L-function for the Gaussian field. This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1, and also contains four roots of unity, so accounting for the factor ¼. Conjectures There are two families of conjectures, formulated for general classes of ''L''-functions (the very general setting being for ''L''-functions ''L''(''s'') associated to Chow motives over number fields), the division into two reflecting the questions of: how to replace π in the Leibniz formula by some other "transcendental" number (whether or not it is yet possible fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
King's College London
King's College London (informally King's or KCL) is a public research university located in London, England. King's was established by royal charter in 1829 under the patronage of King George IV and the Duke of Wellington. In 1836, King's became one of the two founding colleges of the University of London. It is one of the oldest university-level institutions in England. In the late 20th century, King's grew through a series of mergers, including with Queen Elizabeth College and Chelsea College of Science and Technology (in 1985), the Institute of Psychiatry (in 1997), the United Medical and Dental Schools of Guy's and St Thomas' Hospitals and the Florence Nightingale School of Nursing and Midwifery (in 1998). King's has five campuses: its historic Strand Campus in central London, three other Thames-side campuses (Guy's, St Thomas' and Waterloo) nearby and one in Denmark Hill in south London. It also has a presence in Shrivenham, Oxfordshire, for its professional mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate Motive
In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring , its representation space is generally denoted by (that is, it is a representation ). ''p''-adic cyclotomic character Fix a prime, and let denote the absolute Galois group of the rational numbers. The roots of unity \mu_ = \left\ form a cyclic group of order p^n, generated by any choice of a primitive th root of unity . Since all of the primitive roots in \mu_ are Galois conjugate, the Galois group G_\mathbf acts on \mu_ by automorphisms. After fixing a primitive root of unity \zeta_ generating \mu_, any element of \mu_ can be written as a power of \zeta_, where the exponent is a unique element in (\mathbf/p^n\mathbf)^\times. One can thus write \sigma.\zeta := \sigma(\zeta) = \zeta_^ where a(\sigma,n) \in (\mathbf/p^n \mathbf)^\times is the unique element as above, depending on both \sigma and p. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is ''solvable by radicals'' if its roots may be expressed by a formula involving only integers, th roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals. Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iwasawa Theory
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently (early 1990s), Ralph Greenberg has proposed an Iwasawa theory for motives. Formulation Iwasawa worked with so-called \Z_p-extensions - infinite extensions of a number field F with Galois group \Gamma isomorphic to the additive group of p-adic integers for some prime ''p''. (These were called \Gamma-extensions in early papers.) Every closed subgroup of \Gamma is of the form \Gamma^, so by Galois theory, a \Z_p-extension F_\infty/F is the same thing as a tower of fields :F=F_0 \subset F_1 \subset F_2 \subset \cdots \subset F_\infty such that \operatorname(F_n/F)\cong \Z/p^n\Z. Iwasawa studied classical Galois modules over F_n by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal property: for every linear map from to a commutative algebra , there is a unique algebra homomorphism such that , where is the inclusion map of in . If is a basis of , the symmetric algebra can be identified, through a canonical isomorphism, to the polynomial ring , where the elements of are considered as indeterminates. Therefore, the symmetric algebra over can be viewed as a "coordinate free" polynomial ring over . The symmetric algebra can be built as the quotient of the tensor algebra by the two-sided ideal generated by the elements of the form . All these definitions and properties extend naturally to the case where is a module (not necessarily a free one) over a commutative ring. Construction From tensor algebra It is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selmer Group
In arithmetic geometry, the Selmer group, named in honor of the work of by , is a group constructed from an isogeny of abelian varieties. The Selmer group of an isogeny The Selmer group of an abelian variety ''A'' with respect to an isogeny ''f'' : ''A'' → ''B'' of abelian varieties can be defined in terms of Galois cohomology as :\operatorname^(A/K)=\bigcap_v\ker(H^1(G_K,\ker(f))\rightarrow H^1(G_,A_v /\operatorname(\kappa_v)) where ''A''v 'f''denotes the ''f''-torsion of ''A''v and \kappa_v is the local Kummer map B_v(K_v)/f(A_v(K_v))\rightarrow H^1(G_,A_v . Note that H^1(G_,A_v /\operatorname(\kappa_v) is isomorphic to H^1(G_,A_v) /math>. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have ''K''v-rational points for all places ''v'' of ''K''. The Selmer group is finite. This implies that the part of the Tate–Shafarevich group killed by ''f'' is finite due to the following exact sequence : 0 → ''B''(''K'')/ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
