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Milnor Conjecture (topology)
In knot theory, the Milnor conjecture says that the slice genus of the (p, q) torus knot is :(p-1)(q-1)/2. It is in a similar vein to the Thom conjecture. It was first proved by gauge theoretic methods by Peter Kronheimer and Tomasz Mrowka. Jacob Rasmussen later gave a purely combinatorial proof In mathematics, the term ''combinatorial proof'' is often used to mean either of two types of mathematical proof: * A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set in t ... using Khovanov homology, by means of the s-invariant.. References Geometric topology Knot theory 4-manifolds Conjectures that have been proved {{knottheory-stub ... [...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] |
Knot Theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, Unknot, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3 (in topology, a circle is not bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slice Genus
In mathematics, the slice genus of a smooth knot ''K'' in ''S''3 (sometimes called its Murasugi genus or 4-ball genus) is the least integer g such that ''K'' is the boundary of a connected, orientable 2-manifold ''S'' of genus ''g'' properly embedded in the 4-ball ''D''4 bounded by ''S''3. More precisely, if ''S'' is required to be smoothly embedded, then this integer ''g'' is the ''smooth slice genus'' of ''K'' and is often denoted gs(''K'') or g4(''K''), whereas if ''S'' is required only to be topologically locally flatly embedded then ''g'' is the ''topologically locally flat slice genus'' of ''K''. (There is no point considering ''g'' if ''S'' is required only to be a topological embedding, since the cone on ''K'' is a 2-disk with genus 0.) There can be an arbitrarily great difference between the smooth and the topologically locally flat slice genus of a knot; a theorem of Michael Freedman says that if the Alexander polynomial of ''K'' is 1, then the topologically ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus Knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers ''p'' and ''q''. A torus link arises if ''p'' and ''q'' are not coprime (in which case the number of components is gcd(''p, q'')). A torus knot is trivial (equivalent to the unknot) if and only if either ''p'' or ''q'' is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot. Geometrical representation A torus knot can be rendered geometrically in multiple ways which are topologically equivalent (see Properties below) but geometrically distinct. The convention used in this article and its figures is the following. The (''p'',''q'')-torus knot winds ''q'' times around a circle in the interior of the torus, and ''p'' times around its axis of rotational symmetry.. If ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thom Conjecture
In mathematics, a smooth algebraic curve C in the complex projective plane, of degree d, has Genus_(mathematics)#Topology, genus given by the genus–degree formula :g = (d-1)(d-2)/2. The Thom conjecture, named after French mathematician René Thom, states that if \Sigma is any smoothly embedded connected curve representing the same class in homology (mathematics), homology as C, then the genus g of \Sigma satisfies the inequality :g \geq (d-1)(d-2)/2. In particular, ''C'' is known as a ''genus minimizing representative'' of its homology class. It was first proved by Peter B. Kronheimer, Peter Kronheimer and Tomasz Mrowka in October 1994, using the then-new Seiberg–Witten invariants. Assuming that \Sigma has nonnegative self intersection number this was generalized to Kähler manifolds (an example being the complex projective plane) by John Morgan (mathematician), John Morgan, Zoltán Szabó (mathematician), Zoltán Szabó, and Clifford Taubes, also using the Seiberg–Wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups). The term ''gauge'' refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called ''gauge transformations'', form a Lie group—referred to as the ''symmetry group'' or the ''gauge group'' of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the ''gauge field''. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called ''gauge invariance''). When such a theory is quantized, the quanta of the gauge fields are called '' gauge bosons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Kronheimer
Peter Benedict Kronheimer (born 1963) is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is William Caspar Graustein Professor of Mathematics at Harvard University and former chair of the mathematics department. Kronheimer's early work was on gravitational instantons, in particular the classification of hyperkähler 4-manifolds with asymptotical locally Euclidean geometry (ALE spaces), leading to the papers "The construction of ALE spaces as hyper-Kähler quotients" and "A Torelli-type theorem for gravitational instantons." He and Hiraku Nakajima gave a construction of instantons on ALE spaces generalizing the Atiyah–Hitchin–Drinfeld– Manin construction. This constructions identified these moduli spaces as moduli spaces for certain quivers (see "Yang-Mills instantons on ALE gravitational instantons.") He was the initial recipient of the Oberwolfach prize in 1998 on the basis of some of this work. Kr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tomasz Mrowka
Tomasz Mrowka (born September 8, 1961) is an American mathematician specializing in differential geometry and gauge theory. He is the Singer Professor of Mathematics and former head of the Department of Mathematics at the Massachusetts Institute of Technology. Mrowka is the son of Polish mathematician and is married to MIT mathematics professor Gigliola Staffilani. Career A 1983 graduate of the Massachusetts Institute of Technology, he received the PhD from the University of California, Berkeley in 1988 under the direction of Clifford Taubes and Robion Kirby. He joined the MIT mathematics faculty as professor in 1996, following faculty appointments at Stanford University and at the California Institute of Technology (professor 1994–96). At MIT, he was the Simons Professor of Mathematics from 2007–2010. Upon Isadore Singer's retirement in 2010 the name of the chair became the Singer Professor of Mathematics which Mrowka held until 2017. He was named head of the Departm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Proof
In mathematics, the term ''combinatorial proof'' is often used to mean either of two types of mathematical proof: * A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set in two different ways to obtain the different expressions in the identity. Since those expressions count the same objects, they must be equal to each other and thus the identity is established. * A bijective proof. Two sets are shown to have the same number of members by exhibiting a bijection, i.e. a one-to-one correspondence, between them. The term "combinatorial proof" may also be used more broadly to refer to any kind of elementary proof in combinatorics. However, as writes in his review of (a book about combinatorial proofs), these two simple techniques are enough to prove many theorems in combinatorics and number theory. Example An archetypal double counting proof is for the well known formula for the number \tbinom nk of ''k''-combi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Khovanov Homology
In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail Khovanov, then at the University of California, Davis, now at Columbia University. Overview To any link diagram ''D'' representing a link ''L'', we assign the Khovanov bracket ''D''.html" ;"title="/nowiki>''D''">/nowiki>''D''/nowiki>, a cochain complex of graded vector spaces. This is the analogue of the Kauffman bracket in the construction of the Jones polynomial. Next, we normalise ''D''.html" ;"title="/nowiki>''D''">/nowiki>''D''/nowiki> by a series of degree shifts (in the graded vector spaces) and height shifts (in the cochain complex) to obtain a new cochain complex C(''D''). The cohomology of this cochain complex turns out to be an invariant of ''L'', and its graded Euler characteristic is the Jones polynomial of ''L''. Definition This defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of ''simple'' homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently. Differences between low-dimensional and high-dimensional topology Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimension 4 is special, in that in some respects (topologica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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