Moreover, A. Kostrikin for the case of a prime exponent and Efim Zelmanov in general proved that, among the finite groups with given number of generators and exponent, there exists a largest one. Issai Schur showed that any finitely generated periodic group that was a subgroup of the group of invertible n x n complex matrices was finite; he used this theorem to prove the Jordan—Schur theorem.
Nevertheless, the general answer to Burnside's problem turned out to be negative.
In , Golod and Shafarevich constructed an infinite group of Burnside type without assuming that all elements have uniformly bounded order. In , Pyotr Novikov and Sergei Adian's supplied a negative solution to the bounded exponent problem for all odd exponents larger than In , A. Ol'shanskii found some striking counterexamples for sufficiently large odd exponents greater than , and supplied a considerably simpler proof based on geometric ideas. The case of even exponents turned out to be much harder to settle. In S.
Ivanov announced the negative solution for sufficiently large even exponents divisible by a large power of 2 detailed proofs were published in and occupied some pages. Later joint work of Ol'shanskii and Ivanov established a negative solution to an analogue of Burnside's problem for hyperbolic groups, provided the exponent is sufficiently large. By contrast, when the exponent is small and different from 2,3,4 and 6, very little is known. Clearly, every finite group is periodic. The general Burnside problem can be posed as follows: If G is a periodic group, and G is finitely generated, then must G necessarily be a finite group?
This question was answered in the negative in by Evgeny Golod and Igor Shafarevich, who gave an example of an infinite p-group that is finitely generated. However, the orders of the elements of this group are not a priori bounded by a single constant. Bounded Burnside problem Part of the difficulty with the general Burnside problem is that the requirements of being finitely generated and periodic give very little information about the possible structure of a group.
A group with this property is said to be periodic with bounded exponent n, or just a group with exponent n. Burnside problem for groups with bounded exponent asks: If G is a finitely generated group with exponent n, is G necessarily finite? It turns out that this problem can be restated as a question about the finiteness of groups in a particular family.
More precisely, the characteristic property of B m, n is that, given any group G with m generators g1,…,gm and of exponent n, there is a unique homomorphism from B m, n to G that maps the ith generator xi of B m, n into the ith generator gi of G.
The existence of the free Burnside group and its uniqueness up to an isomorphism are established by standard techniques of group theory. Thus if G is any finitely generated group of exponent n, then G a homomorphic image of B m, n , where m is the number of generators of G. Burnside's problem can now be restated as follows: For which positive integers m, n is the free Burnside group B m,n finite?
The full solution to Burnside's problem in this form is not known. B m, 2 is the direct product of m copies of the cyclic group of order 2. The following additional results are known Burnside, Sanov, M. The particular case of B 2, 5 remains open: as of , it was not known whether this group is finite. Adian later improved the bound on the odd exponent to The case of even exponent turned out to be considerably more difficult.
Both Novikov—Adian and Ivanov established considerably more precise results on the structure of the free Burnside groups. In the case of the odd exponent, all finite subgroups of the free Burnside groups were shown to be cyclic groups. In the even exponent case, each finite subgroup is contained in a product of two dihedral groups, and there exist noncyclic finite subgroups.
Moreover, the word and conjugacy problems were shown to be effectively solvable in B m, n both for the cases of odd and even exponents n. A famous class of counterexamples to Burnside's problem is formed by finitely generated non-cyclic infinite groups in which every nontrivial proper subgroup is a finite cyclic group, the so-called Tarski Monsters. First examples of such groups were constructed by A. Ol'shanskii in using geometric methods, thus affirmatively solving O. Schmidt's problem. In a paper published in , Ivanov and Ol'shanskii solved an analogue of Burnside's problem in an arbitrary hyperbolic group for sufficiently large exponents.
Restricted Burnside problem The restricted Burnside problem, formulated in the s, asks another, related, question: If it is known that a group G with m generators and exponent n is finite, can one conclude that the order of G is bounded by some constant depending only on m and n? Equivalently, are there only finitely many finite groups with m generators of exponent n, up to isomorphism?
This variant of the Burnside problem can also be stated in terms of certain universal groups with m generators and exponent n. By basic results of group theory, the intersection of two subgroups of finite index in any group is itself a subgroup of finite index. Let M be the intersection of all subgroups of the free Burnside group B m, n which have finite index, then M is a normal subgroup of B m, n otherwise, there exists a subgroup g -1Mg with finite index containing elements not in M.
Every finite group of exponent n with m generators is a homomorphic image of B0 m,n. The restricted Burnside problem then asks whether B0 m,n is a finite group. In the case of the prime exponent p, this problem was extensively studied by A. Kostrikin during the s, prior to the negative solution of the general Burnside problem. His solution, establishing the finiteness of B0 m,p , used a relation with deep questions about identities in Lie algebras in finite characteristic.
The case of arbitrary exponent has been completely settled in the affirmative by Efim Zelmanov, who was awarded the Fields Medal in for his work. Chapter 10 Resolution of Singularities Observe that the Strong desingularization of resolution does not stop after the first blowing-up, when the strict transform is smooth, but when it is simple normal crossings with the exceptional divisors. For varieties over fields of characteristic 0 this was proved in Hironaka , while for varieties over fields of characteristic p it is an open problem in dimensions at least 4.
More generally, it is often useful to resolve the singularities of a variety X embedded into a larger variety W. Suppose we have a closed embedding of X into a regular variety W. The map from the strict transform of X to X is an isomorphism away from the singular points of X.
It cannot be made functorial for all not necessarily smooth morphisms in any reasonable way. Or in general, the sequence of blowings up is functorial with respect to smooth morphisms. Hironaka showed that there is a strong desingularization satisfying the first three conditions above whenever X is defined over a field of characteristic 0, and his construction was improved by several authors see below so that it satisfies all conditions above. Resolution of singularities of curves Every algebraic curve has a unique nonsingular projective model, which means that all resolution methods are essentially the same because they all construct this model.
In higher dimensions this is no longer true: varieties can have many different nonsingular projective models. Kollar lists about 20 ways of proving resolution of singularities of curves. Newton's method Resolution of singularities of curves was essentially first proved by Newton , who showed the existence of Puiseux series for a curve from which resolution follows easily. Riemann's method Riemann constructed a smooth Riemann surface from the function field of a complex algebraic curve, which gives a resolution of its singularities.
This can be done over more general fields by using the set of discrete valuation rings of the field as a substitute for the Riemann surface. Albanese's method Albanese's method consists of taking a curve that spans a projective space of sufficiently large dimension more than twice the degree of the curve and repeatedly projecting down from singular points to projective spaces of smaller dimension.
This method extends to higher dimensional varieties, and shows that any n-dimensional variety has a projective model with singularities of multiplicity at most n! Normalization A one step method of resolving singularities of a curve is to take the normalization of the curve. Normalization removes all singularities in codimension 1, so it works for curves but not in higher dimensions. Valuation rings Another one-step method of resolving singularities of a curve is to take a space of valuation rings of the function field of the curve.
This space can be made into a nonsingular projective curve birational to the original curve. This only gives a weak resolution, because there is in general no morphism from this nonsingular projective curve to the original curve. Blowing up Repeatedly blowing up the singular points of a curve will eventually resolve the singularities. The main task with this method is to find a way to measure the complexity of a singularity and to show that blowing up improves this measure.
There are many ways to do this. For example, one can use the arithmetic genus of the curve. Noether's method Noether's method takes a plane curve and repeatedly applies quadratic transformations determined by a singular points and two points in general position. Eventually this produces a plane curve whose only singularities are ordinary multiple points all tangent lines have multiplicity 1. Bertini's method Bertini's method is similar to Noether's method. It starts with a plane curve, and repeatedly applies birational transformations to the plane to improve the curve.
The birational transformations are more complicated than the quadratic transformations used in Noether's method, but produce the better result that the only singularities are ordinary double points. Resolution of singularities of surfaces Surfaces have many different nonsingular projective models unlike the case of curves where the nonsingular projective model is unique.
However a surface still has a unique minimal resolution, that all others factor through all others are resolutions of it. In higher dimensions there need not be a minimal resolution. Resolution for surfaces over the complex numbers was given informal proofs by Levi , Chisini and Albanese A rigorous proof was first given by Walker , and an algebraic proof for all fields of characteristic 0 was given by Zariski Abhyankar gave a proof for surfaces of non-zero characteristic. Resolution of singularities has also been shown for all excellent 2-dimensional schemes including all arithmetic surfaces by Lipman Normalization and blowup The usual method of resolution of singularities for surfaces is to repeatedly alternate normalizing the surface which kills codimension 1 singularities with blowing up points which makes codimension 2 singularities better, but may introduce new codimension 1 singularities.
Jung's method By applying strong embedded resolution for curves, Jung reduces to a surface with only rather special singularities abelian quotient singularities which are then dealt with explicitly. The higher-dimensional version of this method is de Jong's method. Albanese method In general the analogue of Albanese's method for curves shows that for any variety one can reduce to singularities of order at most n!
For surfaces this reduces to the case of singularities of order 2, which are easy enough to do explicitly. Hironaka's method Hironaka's method for arbitrary characteristic 0 varieties gives a resolution method for surfaces, which involves repeatedly blowing up points or smooth curves in the singular set. Lipman's method Lipman showed that a surface Y a 2-dimensional reduced Noetherian scheme has a desingularization if and only if its normalization is finite over Y and analytically normal the completions of its singular points are normal and has only finitely many singular points.
In particular if Y is excellent then it has a desingularization. His method was to consider normal surfaces Z with a birational proper map to Y and show that there is a minimal one with minimal possible arithmetic genus. He then shows that all singularities of this minimal Z are pseudo rational, and shows that pseudo rational singularities can be resolved by repeatedly blowing up points.
Resolution of singularities in higher dimensions The problem of resolution of singularities in higher dimensions is notorious for many incorrect published proofs and announcements of proofs that never appeared. Zariski's method For 3-folds the resolution of singularities was proved in characteristic 0 by Zariski Abhyankar's method Abhyankar proved resolution of singularities in characteristic greater than 6. The restriction on the characteristic arises because Abhyankar shows that it is possible to resolve any singularity of a 3-fold of multiplicity less than the characteristic, and then uses Albanese's method to show that singularities can be reduced to those of multiplicity at most dimension!
Cossart and Piltant , proved resolution of singularities of 3-folds in all characteristics. Hironaka's method Resolution of singularities in characteristic 0 in all dimensions was first proved by Hironaka He proved that it was possible to resolve singularities of varieties over fields of characteristic 0 by repeatedly blowing up along non-singular subvarieties, using a very complicated argument by induction on the dimension.
Some of the recent proofs are about a tenth of the length of Hironaka's original proof, and are easy enough to give in an introductory graduate course. De Jong's method gave a weaker result for varieties of all dimensions in characteristic p, which was strong enough to act as a substitute for resolution for many purposes. De Jong proved that for any variety X over a field there is a dominant proper morphism which preserves the dimension from a regular variety onto X. This need not be a birational map, so is not a resolution of singularities, as it may be generically finite to one and so involves a finite extension of the function field of X.
De Jong's idea was to try to represent X as a fibration over a smaller space Y with fibers that are curves this may involve modifying X , then eliminate the singularities of Y by induction on the dimension, then eliminate the singularities in the fibers. Resolution for schemes and status of the problem It is easy to extend the definition of resolution to all schemes.
Not all schemes have resolutions of their singularities: Grothendieck , section 7. Grothendieck also suggested that the converse might hold: in other words, if a locally Noetherian scheme X is reduced and quasi excellent, then it is possible to resolve its singularities. When X is defined over a field of characteristic 0, this follows from Hironaka's theorem, and when X has dimension at most 2 it was prove by Lipman.
In general it would follow if it is possible to resolve the singularities of all integral complete local rings. Hauser gave a survey of work on the unsolved characteristic p resolution problem. Method of proof in characteristic zero There are many constructions of strong desingularization but all of them give essentially the same result. In every case the global object the variety to be desingularized is replaced by local data the ideal sheaf of the variety and those of the exceptional divisors and some orders that represents how much should be resolved the ideal in that step.
With this local data the centers of blowing-up are defined. The centers will be defined locally and therefore it is a problem to guarantee that they will match up into a global center. This can be done by defining what blowings-up are allowed to resolve each ideal. Done this appropriately will make the centers match automatically. Another way is to define a local invariant depending on the variety and the history of the resolution the previous local centers so that the centers consist of the maximum locus of the invariant.
The definition of this is made such that making this choice is meaningful, giving smooth centers transversal to the exceptional divisors. In either case the problem is reduced to resolve singularities of the tuple formed by the ideal sheaf and the extra data the exceptional divisors and the order, d, to which the resolution should go for that ideal. This tuple is called a marked ideal and the set of points in which the order of the ideal is larger than d is called its co-support. The proof that there is a resolution for the marked ideals is done by induction on dimension.
The induction breaks in two steps: 1. Functorial desingularization of marked ideals of maximal order of dimension n implies functorial desingularization of a general marked ideal of dimension n. Here we say that a marked ideal is of maximal order if at some point of its co-support the order of the ideal is equal to d. A key ingredient in the strong resolution is the use of the Hilbert—Samuel function of the local rings of the points in the variety. This is one of the components of the resolution invariant.
Examples Multiplicity need not decrease under blowup The most obvious invariant of a singularity is its multiplicity. However this need not decrease under blowup, so it is necessary to use more subtle invariants to measure the improvement. In the previous example it was fairly clear that the singularity improved since the degree of one of the monimials defining it got smaller.
This does not happen in general. It is not immediately obvious that this new singularity is better, as both singularities have multiplicity 2 and are given by the sum of monomials of degrees 2, 3, and 4. Blowing up the most singular points does not work Whitney umbrella A natural idea for improving singularities is to blow up the locus of the "worst" singular points. However blowing up the origin reproduces the same singularity on one of the coordinate charts. So blowing up the apparently "worst" singular points does not improve the singularity. Instead the singualrity can be resolved by blowing up along the z-axis.
After the resolution the total transform, the union of the strict transform, X, and the exceptional divisors, is a variety with singularities of the simple normal crossings type. Then it is natural to consider the possibility of resolving singularities without resolving this type of singularities, this is finding a resolution that is an isomorphism over the set of smooth and simple normal crossing points.
When X is a divisor, i. Whitney's umbrella shows that it is not possible to resolve singularities avoiding blowing-up the normal crossings singularities. Incremental resolution procedures need memory A natural way to resolve singularities is to repeatedly blow up some canonically chosen smooth subvariety. This runs into the following problem. The only reasonable varieties to blow up are the origin, one of these two axes, or the whole singular set both axes.
However the whole singular set cannot be used since it is not smooth, and choosing one of the two axes breaks the symmetry between them so is not canonical. This means we have to start by blowing up the origin, but this reproduces the original singularity, so we seem to be going round in circles. The solution to this problem is that although blowing up the origin does not change the type of the singularity, it does give a subtle improvement: it breaks the symmetry between the two singular axes because one of them is an exceptional divisor for a previous blowup, so it is now permissible to blow up just one of these.
However in order to exploit this the resolution procedure needs to treat these 2 singularities differently, even though they are locally the same. This is sometimes done by giving the resolution procedure some memory, so the center of the blowup at each step depends not only on the singularity, but on the previous blowups used to produce it.
However it is not possible to find a strong resolution functorial for all possibly nonsmooth morphisms. The XY-plane is already nonsingular so should not be changed by resolution, and any resolution of the conical singularity factorizes through the minimal resolution given by blowing up the singular point. However the rational map from the XY-plane to this blowup does not extend to a regular map. Minimal resolutions need not exist Minimal resolutions resolutions such that every resolution factors through them exist in dimensions 1 and 2, but not always in higher dimensions.
The Atiyah flop gives an example in 3 dimensions of a singularity with no minimal resolution.
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Resolutions should not commute with products Kollar , example 3. Singularities of toric varieties Singularities of toric varieties give examples of high dimensional singularities that are easy to resolve explicitly. A toric variety is defined by a fan, a collection of cones in a lattice.
The singularities can be resolved by subdividing each cone into a union of cones each of which is generated by a basis for the lattice, and taking the corresponding toric variety. Chapter 11 Happy Ending Problem The Happy Ending problem: every set of five points in general position contains the vertices of a convex quadrilateral. Any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral. This was one of the original results that led to the development of Ramsey theory.
The Happy Ending theorem can be proven by a simple case analysis: If four or more points are vertices of the convex hull, any four such points can be chosen. If on the other hand the point set has the form of a triangle with two points inside it, the two inner points and one of the triangle sides can be chosen. It remains unproven, but less precise bounds are known. Larger polygons A set of eight points in general position with no convex pentagon.
For any positive integer N, any sufficiently large finite set of points in the plane in general position has a subset of N points that form the vertices of a convex polygon. The original solution to the Happy Ending problem can be adapted to show that any five points in general position have an empty convex quadrilateral, as shown in the illustration, and any ten points in general position have an empty convex pentagon.
However, there exist arbitrarily large sets of points in general position that contain no empty heptagon. Valtr supplies a simplification of Gerken's proof that however requires more points, f 15 instead of f 9. At least 30 points are needed: there exists a set of 29 points in general position with no empty convex hexagon. Related problems The problem of finding sets of n points minimizing the number of convex quadrilaterals is equivalent to minimizing the crossing number in a straight-line drawing of a complete graph.
The number of quadrilaterals must be proportional to the fourth power of n, but the precise constant is not known. It is straightforward to show that, in higher dimensional Euclidean spaces, sufficiently large sets of points will have a subset of k points that forms the vertices of a convex polytope, for any k greater than the dimension: this follows immediately from existence of convex k-gons in sufficiently large planar point sets, by projecting the higher dimensional point set into an arbitrary two-dimensional subspace.
However, the number of points necessary to find k points in convex position may be smaller in higher dimensions than it is in the plane, and it is possible to find subsets that are more highly constrained. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies results about the distribution of prime numbers that are in some ways as good as possible. Along with suitable generalizations, it is considered by some mathematicians to be the most important unresolved problem in pure mathematics Bombieri The Riemann hypothesis is part of Problem 8, along with the Goldbach conjecture, in Hilbert's list of 23 unsolved problems, and is also one of the Clay Mathematics Institute Millennium Prize Problems.
Since it was formulated, it has withstood concentrated efforts from many outstanding mathematicians. In , Pierre Deligne proved an analogue of the Riemann Hypothesis for zeta functions of varieties defined over finite fields. The full version of the hypothesis remains unsolved, although modern computer calculations have shown that the first 10 trillion zeros lie on the critical line.
It has zeros at the negative even integers i. These are called the trivial zeros. There are several popular books on the Riemann hypothesis, such as Derbyshire , Rockmore , Sabbagh , du Sautoy The books Edwards , Patterson and Borwein et al. The Riemann zeta function The Riemann zeta function is given for complex s with real part greater than 1 by Leonhard Euler showed that it is given by the Euler product where the infinite product extends over all prime numbers p, and again converges for complex s with real part greater than 1.
The Riemann hypothesis discusses zeros outside the region of convergence of this series, so it needs to be analytically continued to all complex s. This can be done by expressing it in terms of the Dirichlet eta function as follows. If s has positive real part, then the zeta function satisfies where the series on the right converges whenever s has positive real part. If s is a positive even integer this argument does not apply because the zeros of sin are cancelled by the poles of the gamma function.
The functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros, so all non-trivial zeros lie in the critical strip where s has real part between 0 and 1. History "…es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the next objective of my investigation.
He was discussing a version of the zeta function, modified so that its roots are real rather than on the critical line. This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. Consequences of the Riemann hypothesis The practical uses of the Riemann hypothesis include many propositions which are known to be true under the Riemann hypothesis, and some which can be shown to be equivalent to the Riemann hypothesis.
Distribution of prime numbers Riemann's explicit formula for the number of primes less than a given number in terms of a sum over the zeros of the Riemann zeta function says that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. Von Koch proved that the Riemann hypothesis is equivalent to the "best possible" bound for the error of the prime number theorem. A precise version of Koch's result, due to Schoenfeld , says that the Riemann hypothesis is equivalent to Growth of arithmetic functions The Riemann hypothesis implies strong bounds on the growth of many other arithmetic functions, in addition to the primes counting function above.
The determinant of the order n Redheffer matrix is equal to M n , so the Riemann hypothesis can also be stated as a condition on the growth of these determinants.
Here the Farey sequence of order n. The Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip. Large prime gap conjecture The prime number theorem implies that on average, the gap between the prime p and its successor is log p. However, some gaps between primes may be much larger than the average. Criteria equivalent to the Riemann hypothesis Many statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in solving it.
Some typical examples are as follows. Consequences of the generalized Riemann hypothesis Several applications use the generalized Riemann hypothesis for Dirichlet L-series or zeta functions of number fields rather than just the Riemann hypothesis. Many basic properties of the Riemann zeta function can easily be generalized to all Dirichlet L-series, so it is plausible that a method that proves the Riemann hypothesis for the Riemann zeta function would also work for the generalized Riemann hypothesis for Dirichlet Lfunctions.
Several results first proved using the generalized Riemann hypothesis were later given unconditional proofs without using it, though these were usually much harder. Many of the consequences on the following list are taken from Conrad In , Hardy and Littlewood showed that the generalized Riemann hypothesis implies a conjecture of Chebyshev that which says that in some sense primes 3 mod 4 are more common than primes 1 mod 4.
In Deshouillers, Effinger, te Riele, and Zinoviev showed that the generalized Riemann hypothesis implies that every odd number greater than 5 is the sum of 3 primes. In , Chowla showed that the generalized Riemann hypothesis implies that the first prime in the arithmetic progression a mod m is at most Km2log m 2 for some fixed constant K.
In , Hooley showed that the generalized Riemann hypothesis implies Artin's conjecture on primitive roots. In , Weinberger showed that the generalized Riemann hypothesis implies that Euler's list of idoneal numbers is complete. In , G. Miller showed that the generalized Riemann hypothesis implies that one can test if a number is prime in polynomial times.
Odlyzko discussed how the generalized Riemann hypothesis can be used to give sharper estimates for discriminants and class numbers of number fields. Generalizations and analogues of the Riemann hypothesis Dirichlet L-series and other number fields The Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, but much more general, global L-functions.
It is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta function, which accounts for the true importance of the Riemann hypothesis in mathematics. The generalized Riemann hypothesis extends the Riemann hypothesis to all Dirichlet Lfunctions. The extended Riemann hypothesis extends the Riemann hypothesis to all Dedekind zeta functions of algebraic number fields. The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis.
The Riemann hypothesis can also be extended to the L-functions of Hecke characters of number fields. The grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Hecke eigenforms. Function fields and zeta functions of varieties over finite fields Artin introduced global zeta functions of quadratic function fields and conjectured an analogue of the Riemann hypothesis for them, which has been proven by Hasse in the genus 1 case and by Weil in general.
For instance, the fact that the Gauss sum, of the quadratic character of a finite field of size q with q odd , has absolute value is actually an instance of the Riemann hypothesis in the function field setting. This led Weil to conjecture a similar statement for all algebraic varieties; the resulting Weil conjectures were proven by Pierre Deligne , Selberg zeta functions Selberg introduced the Selberg zeta function of a Riemann surface. These are similar to the Riemann zeta function: they have a functional equation, and an infinite product similar to the Euler product but taken over closed geodesics rather than primes.
The Selberg trace formula is the analogue for these functions of the explicit formulas in prime number theory. Selberg proved that the Selberg zeta functions satisfy the analogue of the Riemann hypothesis, with the imaginary parts of their zeros related to the eigenvalues of the Laplacian operator of the Riemann surface.
Ihara zeta functions The Ihara zeta function of a finite graph is an analogue of the Selberg zeta function introduced by Yasutaka Ihara. A regular finite graph is a Ramanujan graph, a mathematical model of efficient communication networks, if and only if its Ihara zeta function satisfies the analogue of the Riemann hypothesis as was pointed out by T. Montgomery's pair correlation conjecture Montgomery suggested the pair correlation conjecture that the correlation functions of the suitably normalized zeros of the zeta function should be the same as those of the eigenvalues of a random hermitian matrix.
Odlyzko showed that this is supported by large scale numerical calculations of these correlation functions. Dedekind zeta functions of algebraic number fields, which generalize the Riemann zeta function, often do have multiple complex zeros. This is because the Dedekind zeta functions factorize as a product of powers of Artin L-functions, so zeros of Artin Lfunctions sometimes give rise to multiple zeros of Dedekind zeta functions. Other examples of zeta functions with multiple zeros are the L-functions of some elliptic curves: these can have multiple zeros at the real point of their critical line; the BirchSwinnerton-Dyer conjecture predicts that the multiplicity of this zero is the rank of the elliptic curve.
Other zeta functions There are many other examples of zeta functions with analogues of the Riemann hypothesis, some of which have been proved. Goss zeta functions of function fields have a Riemann hypothesis, proved by Sheats Attempts to prove the Riemann hypothesis Several mathematicians have addressed the Riemann hypothesis, but none of their attempts have yet been accepted as correct solutions. Watkins lists some incorrect solutions, and more are frequently announced. Some support for this idea comes from several analogues of the Riemann zeta functions whose zeros correspond to eigenvalues of some operator: the zeros of a zeta function of a variety over a finite field correspond to eigenvalues of a Frobenius element on an etale cohomology group, the zeros of a Selberg zeta function are eigenvalues of a Laplacian operator of a Riemann surface, and the zeros of a p-adic zeta function correspond to eigenvectors of a Galois action on ideal class groups.
Odlyzko showed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of random matrices drawn from the Gaussian unitary ensemble. In a connection with this Quantum mechanical problem Berry and Connes had proposed that the inverse of the potential of the Hamiltonian is connected to the halfderivative of the function then, in Berry-Connes approach Connes This yields to a Hamiltonian whose eigenvalues are the square of the imaginary part of the Riemann zeros, also the functional determinant of this Hamiltonian operator is just the Riemann Xi-function The analogy with the Riemann hypothesis over finite fields suggests that the Hilbert space containing eigenvectors corresponding to the zeros might be some sort of first cohomology group of the spectrum Spec Z of the integers.
Solving the heat equation or Fourier analysis? I recommend it. The title is the not-very promising "A personal view of average-case complexity". Gregory Grant Gregory Grant Additionally, even solutions of smaller problems could bring about respectable impacts.
And the world runs on encryption these days. I doubt it.
What Purpose Do the Great Mathematical Problems Serve?
BCLC 1. David G. Stork David G. Stork Maybe as avid19 said "some connection between current fields has not been established yet. The connection is part of mathematics, and not part of either component alone. We had geometry and algebra for instance , but geometric algebra is a full field of its own. Many such hyphenated fields are of this nature. Stork May 4 '15 at So it may not be the case if the connection is not necessary to answer a question?
Fermat's Last Theorem is a classic example. Whole branches of math grew out of trying to solve it.
However, it's not always the case. The RSA algorithm was a result that changed the world, many would call it revolutionary, but the proof was based on very simple math and provided no new methods. Active 4 years, 4 months ago. Viewed 2k times. Thank you in advance. Martin Sleziak Colbi Colbi 5 5 silver badges 18 18 bronze badges.
Like Fourier created Fourier series to solve the heat equation.
The Unsolvable Math Problem
And today what's more important? Solving the heat equation or Fourier analysis? I recommend it. The title is the not-very promising "A personal view of average-case complexity". Gregory Grant Gregory Grant Additionally, even solutions of smaller problems could bring about respectable impacts. And the world runs on encryption these days. I doubt it. BCLC 1. David G. Stork David G. Stork Maybe as avid19 said "some connection between current fields has not been established yet. The connection is part of mathematics, and not part of either component alone.
We had geometry and algebra for instance , but geometric algebra is a full field of its own.