List of unsolved problems in mathematics explained

Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention.

This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative. Although this list may never be comprehensive, the problems listed here vary widely in both difficulty and importance.

Lists of unsolved problems in mathematics

Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions.

List Number of
problems
Number unsolved
or incompletely solved
Proposed by Proposed
in
23 15 1900
Landau's problems[1] 4 4 1912
Taniyama's problems[2] 36 - 1955
Thurston's 24 questions[3] [4] 24 - 1982
18 14 1998
7 6[5] 2000
15 <12[6] [7] 2000
Unsolved Problems on Mathematics for the 21st Century[8] 22 - Jair Minoro Abe, Shotaro Tanaka 2001
DARPA's math challenges[9] [10] 23 - 2007
Erdős's problems[11] >850 588 Over six decades of Erdős' career, from the 1930s to 1990s

Millennium Prize Problems

Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:[5]

The seventh problem, the Poincaré conjecture, was solved by Grigori Perelman in 2003.[12] However, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is unsolved.[13]

Notebooks

Unsolved problems

Algebra

See main article: Algebra.

f

applied to a complex matrix

A

is at most twice the supremum of

|f(z)|

over the field of values of

A

.

K(G,1)

.

k

, a Hadamard matrix of order

4k

exists.

\{0\}

, then it has no nil one-sided ideal other than

\{0\}

.

f:RnR

is the maximum of a finite set of minimums of finite collections of polynomials.

n

with

n

disjoint bases

Bi

, it is possible to create an

n x n

matrix whose rows are

Bi

and whose columns are also bases.

G

is a simply connected semisimple algebraic group over a perfect field of cohomological dimension at most

2

, then the Galois cohomology set

H1(F,G)

is zero.

R

is a commutative regular local ring, and

P,Q

are prime ideals of

R

, then

\dim(R/P)+\dim(R/Q)=\dim(R)

implies

\chi(R/P,R/Q)>0

.

g\geq2

over number fields

K

have at most some bounded number

N(K,g)

of

K

-rational points?

n x n

matrices under simultaneous conjugation.

V

with coordinate ring

R

, if the derivations of

R

are a free module over

R

, then

V

is smooth.

X

is a mixed Shimura variety or semiabelian variety defined over

C

, and

V\subseteqX

is a subvariety, then

V

contains only finitely many atypical subvarieties.

Group theory

See main article: Group theory.

G

form a partition of

G

, then the finite indices of said subgroups cannot be distinct.

Representation theory

G

, the number of irreducible complex characters of degree not divisible by a prime number

p

is equal to the number of irreducible complex characters of the normalizer of any Sylow

p

-subgroup
within

G

.

Analysis

See main article: Mathematical analysis.

C

R

and

R2

are spectral if and only if they tile by translation.

f

of degree

d\ge2

and a complex number

z

, is there a critical point

c

of

f

such that

|f(z)-f(c)|\le|f'(z)||z-c|

?

2

has all roots in the closed unit disk, then each root is within distance

1

from some critical point.

C

with analytic capacity

0

Transcendental numbers and diophantine approximation

\gamma

(the Euler–Mascheroni constant),

\pi+e,\pi-e,\pie,\pi/e,\pie,\pi\sqrt{2

}, \pi^, e^, \ln\pi, 2^e, e^e, Catalan's constant, or Khinchin's constant rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers?[18] [19]

\zeta(3)

transcendental?

Combinatorics

See main article: Combinatorics.

k

runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance

1/k

from each other runner) at some time?[21]

n x n

grid so that no three of them lie on a line?

k

size sets required for the existence of a sunflower of

r

sets be bounded by an exponential function in

k

for every fixed

r>2

?

M(n)

for

n\ge10

[23]

R(5,5)

Dynamical systems

See main article: Dynamical system.

3n+1

conjecture)

x 2, x 3

action on the circle either Lebesgue or atomic?

Games and puzzles

See main article: Game theory.

Combinatorial games

See main article: Combinatorial game theory.

Games with imperfect information

Geometry

See main article: Geometry.

Algebraic geometry

See main article: Algebraic geometry.

A[t1,\ldots,tn]

.

KML

constructed from a positive holomorphic line bundle

L

on a compact complex manifold

M

and the canonical line bundle

KM

of

M

X

is a smooth algebraic surface and

L

is an ample line bundle on

X

of degree

d

, then for sufficiently large

r

, the Seshadri constant satisfies

\varepsilon(p1,\ldots,pr;X,L)=d/\sqrt{r}

.

k

to the Galois group of

k

.

p

Covering and packing

n

is a triangular number, packing

n-1

circles in an equilateral triangle requires a triangle of the same size as packing

n

circles

Differential geometry

See main article: Differential geometry.

Sn+1

is

n

.

Discrete geometry

See main article: Discrete geometry.

n

2d

points can be equidistant in

L1

spaces

Euclidean geometry

See main article: Euclidean geometry.

n

-by-

n

matrix depending on

n

points in

R3

K

in

n

dimensions containing a single lattice point in its interior as its center of mass cannot have volume greater than

(n+1)n/n!

d/2

in

Rd

must have a distance set of nonzero Lebesgue measure

n

-dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to

n

?

n

mutually-repelling particles on a unit sphere?

Graph theory

See main article: Graph theory.

Algebraic graph theory

Games on graphs

O(\sqrtn)

Graph coloring and labeling

n

is odd or even and

k\geqn,n-1

respectively, then a

k

-regular graph with

2n

vertices is 1-factorable.

\Delta(G)\geqn/3

is class 2 if and only if it has an overfull subgraph

S

satisfying

\Delta(S)=\Delta(G)

.

Graph drawing and embedding

\ell1

embeddings with bounded distortion

Restriction of graph parameters

d,k

, what is the largest graph of diameter

k

such that all vertices have degrees at most

d

?

Subgraphs

\sqrt{3}/2

n

-dimensional hypercube graph?

(2n-2)

-vertex tournament contain as a subgraph every

n

-vertex oriented tree?[62]

n

-dimensional doubly-directed hypercube graph can be routed with edge-disjoint paths.

\nu

, can all triangles be hit by a set of at most

2\nu

edges?[63]

Word-representation of graphs

Miscellaneous graph theory

Model theory and formal languages

See main article: Model theory and formal languages.

\aleph0

is a simple algebraic group over an algebraically closed field.

\aleph1

-saturated models of a countable theory.[77]
L
\omega1,\omega
: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.

λ

there exists a cardinal

\mu(λ)

such that if an AEC K with LS(K)<=

λ

is categorical in a cardinal above

\mu(λ)

then it is categorical in all cardinals above

\mu(λ)

.[78]

\aleph0

, or
\aleph0
2
.
\aleph
\omega1
does it have a model of cardinality continuum?[82]

\alephn

, is it categorical in every cardinal?[83] [84]

Zp

decidable? of the field of polynomials over

C

?

Probability theory

See main article: Probability theory.

Number theory

See main article: articles.

See also: Number theory.

General

n!+1=m2

other than

n=4,5,7

?

1

?

d

defined over a field

K

of characteristic

0

has a factor in common with its first through

d-1

-th derivative, then must

f

be the

d

-th power of a linear polynomial?

\epsilon>0

, is the pair

(\epsilon,1/2+\epsilon)

an exponent pair?

do the nontrivial zeros of all automorphic L-functions lie on the critical line

1/2+it

with real

t

?

do the nontrivial zeros of all Dirichlet L-functions lie on the critical line

1/2+it

with real

t

?

do the nontrivial zeros of the Riemann zeta function lie on the critical line

1/2+it

with real

t

?

\epsilon>0

, there is some constant

c(\epsilon)

such that either

y2=x3

or

|y2-x3|>c(\epsilon)x1/2

.

k

-th order in a general algebraic number field, where

k

is a power of a prime.

Q

to any base number field.

if

\phi(n)

divides

n-1

, must

n

be prime?

\epsilon>0

,

\zeta(1/2+it)=o(t\epsilon)

\alpha,\beta

,

\liminfnn\Vertn\alpha\Vert\Vertn\beta\Vert=0

, where

\Vertx\Vert

is the distance from

x

to the nearest integer.

x

has the property that the fractional parts of

x(3/2)n

are less than

1/2

for all positive integers

n

.

\epsilon>0

,

rad(abc)1+\epsilon<c

is true for only finitely many positive

a,b,c

such that

a+b=c

.

\epsilon>0

, there is some constant

C(\epsilon)

such that, for any elliptic curve

E

defined over

Q

with minimal discriminant

\Delta

and conductor

f

, we have

|\Delta|\leqC(\epsilon)f6+\epsilon

.

\Deltak(x)=Dk(x)-xPk(log(x))

k=1

2n-1

is at most

n-1

plus the length of the shortest addition chain producing

n

.

\pi

a normal number (i.e., is each digit 0–9 equally frequent)?[90]

Additive number theory

See main article: Additive number theory.

See also: Problems involving arithmetic progressions.

B

is an additive basis of order

2

, then the number of ways that positive integers

n

can be expressed as the sum of two numbers in

B

must tend to infinity as

n

tends to infinity.

2

is the sum of two prime numbers.

m

k

-th powers of positive integers is equal to a different sum of

n

k

-th powers of positive integers, then

m+n\geqk

.

5

can be represented as the sum of an odd prime number and an even semiprime.

\{1,\ldots,2n\}

Algebraic number theory

See main article: Algebraic number theory.

n

, to assign a sequence of natural numbers to each real number such that the sequence for

x

is eventually periodic if and only if

x

is algebraic of degree

n

?

p

do not divide the class number of the maximal real subfield of the

p

-th cyclotomic field.

X

is within a constant multiple of

\sqrt{X}/ln{X}

1/4

.

Computational number theory

See main article: Computational number theory.

Diophantine equations

Ax+By=Cz

where

x,y,z>2

, all three numbers

A,B,C

must share some prime factor.

11+21=31

the only solution to the Erdős–Moser equation?

n\geq2

, there are positive integers

x,y,z

such that

4/n=1/x+1/y+1/z

.

(am,bn,ck)

to the equation

am+bn=ck

with

a,b,c

being positive coprime integers and

m,n,k

being positive integers satisfying

1/m+1/n+1/k<1

.

(xm-1)/(x-1)=(yn-1)/(y-1)

where

x>y>1

and

m,n>2

.

A,B,C

, the equation

Axm-Byn=C

has finitely many solutions when

m,n

are not both

2

.

Prime numbers

See main article: Prime numbers.

p

is prime if and only if

pBp-1\equiv-1\pmodp

n

and

r

, if

(X-1)n\equivXn-1\pmod{n,Xr-1}

, then either

n

is prime or

n2\equiv1\pmod{r}

-1

, then it is a primitive root modulo infinitely many prime numbers

p

4

prime numbers between consecutive squares of prime numbers, aside from

22

and

32

.

f

has a positive leading coefficient, is irreducible over the integers, and has no common factors over all

f(x)

where

x

is a positive integer, then

f(x)

is prime infinitely often.

a1+b1n,\ldots,ak+bkn

with each

bi\geq1

, there are infinitely many

n

for which all forms are prime, unless there is some congruence condition preventing it.

4208

is the sum of two primes which both have a twin.

p

and

q

,

(pq-1)/(p-1)

does not divide

(qp-1)/(q-1)

2

are the sum of two prime numbers.

n

, there is a prime between

n2

and

(n+1)2

.

n2+1

?

p

, if any two of the three conditions

p=2k\pm1

or

p=4k\pm3

,

2p-1

is prime, and

(2p+1)/3

is prime are true, then the third condition is also true.

n

, there are infinitely many prime gaps of size

n

.

\{f1,\ldots,fk\}

of nonconstant irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers

n

for which

f1(n),\ldots,fk(n)

are all primes, or there is some fixed divisor

m>1

which, for all

n

, divides some

fi(n)

.

e-1/2

?

2p-1\equiv1\pmod{p2}

and

3p-1\equiv1\pmod{p2}

simultaneously?[93]

k\geq1,b\geq2,c0

, with and are there infinitely many primes of the form

(k x bn+c)/gcd(k+c,b-1)

with integer n ≥ 1?
2n
2

+1

composite for

n>4

?

Set theory

See main article: Set theory.

Note: These conjectures are about models of Zermelo-Frankel set theory with choice, and may not be able to be expressed in models of other set theories such as the various constructive set theories or non-wellfounded set theory.

λ

?
\aleph\omega
2

<

\aleph
\omega1
(see Singular cardinals hypothesis)? The best bound, ℵω4, was obtained by Shelah using his PCF theory.
\aleph0
2

>\aleph2

?

Topology

See main article: Topology.

n

-dimensional homogeneous absolute neighborhood retract is a topological manifold.

K

, is the space

K x [0,1]

collapsible?

Problems solved since 1995

Algebra

Analysis

KSr

and

KS'r

conjectures, Bourgain-Tzafriri conjecture and

R\epsilon

-conjecture)

Combinatorics

Dynamical systems

Game theory

Geometry

21st century

20th century

Graph theory

K2n+1

can be decomposed into

2n+1

copies of any tree with

n

edges (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020)[138] [139]

Group theory

Number theory

21st century

20th century

Ramsey theory

Theoretical computer science

Topology

Uncategorised

2010s

2000s

See also

Further reading

Books discussing problems solved since 1995

Books discussing unsolved problems

External links

Notes and References

  1. .
  2. Shimura . G. . Goro Shimura . Yutaka Taniyama and his time . Bulletin of the London Mathematical Society . 21 . 2 . 186–196 . 1989 . 10.1112/blms/21.2.186 .
  3. Friedl . Stefan . 10.1365/s13291-014-0102-x . 4 . Jahresbericht der Deutschen Mathematiker-Vereinigung . 3280572 . 223–241 . Thurston's vision and the virtual fibering theorem for 3-manifolds . 116 . 2014. 56322745 .
  4. Thurston . William P. . 10.1090/S0273-0979-1982-15003-0 . 3 . Bulletin of the American Mathematical Society . 648524 . 357–381 . New Series . Three-dimensional manifolds, Kleinian groups and hyperbolic geometry . 6 . 1982.
  5. Web site: Millennium Problems . https://web.archive.org/web/20170606121331/http://claymath.org/millennium-problems . 2017-06-06 . 2015-01-20 . claymath.org.
  6. Web site: Fields Medal awarded to Artur Avila . . 2014-08-13 . 2018-07-07 . https://web.archive.org/web/20180710010437/http://www2.cnrs.fr/en/2435.htm?debut=8&theme1=12 . 2018-07-10 .
  7. Web site: Fields Medals 2014: the maths of Avila, Bhargava, Hairer and Mirzakhani explained . . Bellos . Alex . 2014-08-13 . 2018-07-07 . https://web.archive.org/web/20161021115900/https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/aug/13/fields-medals-2014-maths-avila-bhargava-hairer-mirzakhani . 2016-10-21 . live .
  8. Book: Abe . Jair Minoro . Tanaka . Shotaro . Unsolved Problems on Mathematics for the 21st Century . IOS Press . 2001 . 978-90-5199-490-2.
  9. Web site: DARPA invests in math . . 2008-10-14 . 2013-01-14 . https://web.archive.org/web/20090304121240/http://edition.cnn.com/2008/TECH/science/10/09/darpa.challenges/index.html . 2009-03-04.
  10. Web site: Broad Agency Announcement (BAA 07-68) for Defense Sciences Office (DSO) . DARPA . 2007-09-10 . 2013-06-25 . https://web.archive.org/web/20121001111057/http://www.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html . 2012-10-01.
  11. https://www.erdosproblems.com/
  12. Web site: Poincaré Conjecture . https://web.archive.org/web/20131215120130/http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture . 2013-12-15 . Clay Mathematics Institute.
  13. Web site: rybu . November 7, 2009 . Smooth 4-dimensional Poincare conjecture . live . https://web.archive.org/web/20180125203721/http://www.openproblemgarden.org/?q=op%2Fsmooth_4_dimensional_poincare_conjecture . 2018-01-25 . 2019-08-06 . Open Problem Garden.
  14. Book: RSFSR . MV i SSO . Свердловская тетрадь: нерешенные задачи теории подгрупп . Russie) . Uralʹskij gosudarstvennyj universitet im A. M. Gorʹkogo (Ekaterinbourg . 1969 . S. l. . ru.
  15. Book: Свердловская тетрадь: Сб. нерешённых задач по теории полугрупп . . 1979 . .
  16. Book: Свердловская тетрадь: Сб. нерешённых задач по теории полугрупп . . 1989 . .
  17. Dowling . T. A. . A class of geometric lattices based on finite groups. . Series B . February 1973 . 14 . 1 . 61–86 . 10.1016/S0095-8956(73)80007-3 . free .
  18. For some background on the numbers in this problem, see articles by Eric W. Weisstein at Wolfram MathWorld (all articles accessed 15 December 2014):
  19. Waldschmidt . Michel . 2008 . An introduction to irrationality and transcendence methods. . 2008 Arizona Winter School . https://web.archive.org/web/20141216004531/http://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture5.pdf . 16 December 2014 . 15 December 2014.
  20. .
  21. Tao . Terence . Terence Tao. Some remarks on the lonely runner conjecture. Contributions to Discrete Mathematics. 13. 2. 1–31. 2018. 1701.02048. 10.11575/cdm.v13i2.62728 . free.
  22. LMS Journal of Computation and Mathematics. 17. 1. 2014. 58–76. On a conjecture of Rudin on squares in arithmetic progressions. González-Jiménez, Enrique. Xarles, Xavier. 10.1112/S1461157013000259. 1301.5122. 11615385 .
  23. Web site: Dedekind Numbers and Related Sequences . 2020-04-30 . 2015-03-15 . https://web.archive.org/web/20150315021125/http://www.sfu.ca/~tyusun/ThesisDedekind.pdf .
  24. Liśkiewicz. Maciej. Ogihara. Mitsunori. Toda. Seinosuke. 2003-07-28. The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes. Theoretical Computer Science. 304. 1. 129–156. 10.1016/S0304-3975(03)00080-X. 33806100 .
  25. S. M. Ulam, Problems in Modern Mathematics. Science Editions John Wiley & Sons, Inc., New York, 1964, page 76.
  26. Kaloshin . Vadim . Vadim Kaloshin . Sorrentino . Alfonso . On the local Birkhoff conjecture for convex billiards . 10.4007/annals.2018.188.1.6 . 188 . 1 . 2018 . 315–380 . Annals of Mathematics. 1612.09194 . 119171182 .
  27. Paul Halmos, Ergodic theory. Chelsea, New York, 1956.
  28. Kari . Jarkko . Jarkko Kari . 2009 . Structure of Reversible Cellular Automata . International Conference on Unconventional Computation . . Springer . 5715 . 6 . 2009LNCS.5715....6K . 10.1007/978-3-642-03745-0_5 . 978-3-642-03744-3 . free . Structure of reversible cellular automata.
  29. Web site: Open Q - Solving and rating of hard Sudoku . https://web.archive.org/web/20171110030932/http://english.log-it-ex.com/2.html . 10 November 2017 . english.log-it-ex.com.
  30. Web site: Higher-Dimensional Tic-Tac-Toe . . . 2017-09-21 . 2018-07-29 . https://web.archive.org/web/20171011000653/https://www.youtube.com/watch?v=FwJZa-helig . 2017-10-11 . live .
  31. On two conjectures of Hartshorne's . Barlet . Daniel . Peternell . Thomas . Schneider . Michael . 10.1007/BF01453563 . . 1990 . 286 . 1–3 . 13–25. 122151259 .
  32. Zariski . Oscar . Oscar Zariski . Some open questions in the theory of singularities . . 77 . 4 . 1971 . 481–491 . 10.1090/S0002-9904-1971-12729-5 . 0277533. free .
  33. Musin . Oleg R. . Tarasov . Alexey S. . The Tammes Problem for N = 14 . Experimental Mathematics . 2015 . 24 . 4 . 460–468 . 10.1080/10586458.2015.1022842. 39429109 .
  34. .
  35. .
  36. Moreno . José Pedro . Prieto-Martínez . Luis Felipe . 10486/705416 . 1 . La Gaceta de la Real Sociedad Matemática Española . es . 4225268 . 111–130 . El problema de los triángulos de Kobon . The Kobon triangles problem . 24 . 2021.
  37. .
  38. . See in particular Conjecture 23, p. 327.
  39. Mahler. Kurt. Ein Minimalproblem für konvexe Polygone . Mathematica (Zutphen) B. 118–127. 1939.
  40. Ghomi . Mohammad . 2018-01-01 . Dürer's Unfolding Problem for Convex Polyhedra . Notices of the American Mathematical Society . 65 . 1 . 25–27 . 10.1090/noti1609 . 0002-9920 . free.
  41. .
  42. .
  43. .
  44. , Problem G10.
  45. .
  46. .
  47. .
  48. .
  49. .
  50. .
  51. .
  52. .
  53. .
  54. .
  55. .
  56. .
  57. .
  58. .
  59. .
  60. Book: Babai, László . Handbook of Combinatorics . June 9, 1994 . Automorphism groups, isomorphism, reconstruction . PostScript . László Babai . https://web.archive.org/web/20070613201449/http://www.cs.uchicago.edu/research/publications/techreports/TR-94-10 . 13 June 2007.
  61. Schwenk . Allen . 2012 . Some History on the Reconstruction Conjecture . Joint Mathematics Meetings . https://web.archive.org/web/20150409233306/http://faculty.nps.edu/rgera/Conjectures/jmm2012/Schwenk,%20%20Some%20History%20on%20the%20RC.pdf . 2015-04-09 . 2018-11-26.
  62. .
  63. Tuza . Zsolt . 10.1007/BF01787705 . 4 . Graphs and Combinatorics . 1092587 . 373–380 . A conjecture on triangles of graphs . 6 . 1990. 38821128 .
  64. .
  65. Book: Kitaev . Sergey . Sergey Kitaev. Words and Graphs . Lozin . Vadim . 2015 . 978-3-319-25857-7 . Monographs in Theoretical Computer Science. An EATCS Series . 10.1007/978-3-319-25859-1 . link.springer.com . 7727433.
  66. Kitaev . Sergey . 2017-05-16 . A Comprehensive Introduction to the Theory of Word-Representable Graphs . . en . 10.1007/978-3-319-62809-7_2. 1705.05924v1 .
  67. Word-Representable Graphs: a Survey. S. V.. Kitaev. A. V.. Pyatkin. April 1, 2018. Journal of Applied and Industrial Mathematics. 12. 2. 278–296. Springer Link. 10.1134/S1990478918020084. 125814097 .
  68. Kitaev . Sergey V. . Pyatkin . Artem V. . 2018 . Графы, представимые в виде слов. Обзор результатов . Word-representable graphs: A survey . Дискретн. анализ и исслед. опер. . ru . 25 . 2 . 19–53 . 10.17377/daio.2018.25.588.
  69. 1605.01688. Marc Elliot Glen. Colourability and word-representability of near-triangulations. math.CO. 2016.
  70. Kitaev . Sergey . 2014-03-06 . On graphs with representation number 3 . math.CO . 1403.1616v1 .
  71. On the representation number of a crown graph . 10.1016/j.dam.2018.03.013 . 244 . 2018 . Discrete Applied Mathematics . 89–93 . Glen . Marc . Kitaev . Sergey . Pyatkin . Artem. 1609.00674 . 46925617 .
  72. .
  73. Web site: Seymour's 2nd Neighborhood Conjecture . live . https://web.archive.org/web/20190111175310/https://faculty.math.illinois.edu/~west/openp/2ndnbhd.html . 11 January 2019 . 17 August 2022 . faculty.math.illinois.edu.
  74. Web site: mdevos . May 4, 2007 . 5-flow conjecture . live . https://web.archive.org/web/20181126134833/http://www.openproblemgarden.org/op/5_flow_conjecture . November 26, 2018 . Open Problem Garden.
  75. Web site: mdevos . March 31, 2010 . 4-flow conjecture . live . https://web.archive.org/web/20181126134908/http://www.openproblemgarden.org/op/4_flow_conjecture . November 26, 2018 . Open Problem Garden.
  76. Hrushovski . Ehud . 1989 . Kueker's conjecture for stable theories . Journal of Symbolic Logic . 54 . 1. 207–220 . 10.2307/2275025. 2275025 . 41940041.
  77. Book: Shelah S . Classification Theory . North-Holland . 1990.
  78. Book: Shelah , Saharon . Classification theory for abstract elementary classes. College Publications. 2009. 978-1-904987-71-0.
  79. Peretz . Assaf . 2006 . Geometry of forking in simple theories . Journal of Symbolic Logic. 71 . 1. 347–359 . 10.2178/jsl/1140641179. math/0412356. 9380215 .
  80. Cherlin . Gregory . Shelah . Saharon . Saharon Shelah. May 2007 . Universal graphs with a forbidden subtree . . Series B . math/0512218 . 10.1016/j.jctb.2006.05.008 . free . 97 . 3 . 293–333. 10425739 .
  81. Džamonja, Mirna, "Club guessing and the universal models." On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
  82. Shelah . Saharon . Saharon Shelah . 1999 . Borel sets with large squares . . math/9802134 . 159 . 1 . 1–50. 1998math......2134S . 10.4064/fm-159-1-1-50 . 8846429 .
  83. Book: Baldwin, John T. . July 24, 2009 . Categoricity . . 978-0-8218-4893-7 . February 20, 2014 . https://web.archive.org/web/20100729073738/http://www.math.uic.edu/%7Ejbaldwin/pub/AEClec.pdf . July 29, 2010 . live .
  84. Shelah . Saharon . Introduction to classification theory for abstract elementary classes . 2009 . math.LO . 0903.3428 .
  85. Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
  86. Makowsky J, "Compactness, embeddings and definability," in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
  87. Keisler . HJ . 1967 . Ultraproducts which are not saturated . J. Symb. Log. . 32 . 1. 23–46 . 10.2307/2271240. 2271240 . 250345806 .
  88. 1208.2140 . math.LO . Maryanthe . Malliaris . Saharon . Shelah . Maryanthe Malliaris . Saharon Shelah . A Dividing Line Within Simple Unstable Theories . 10 August 2012. A Dividing Line within Simple Unstable Theories . 1208.2140 . Malliaris . M. . Shelah . S. . 2012 . math.LO .
  89. .
  90. Web site: Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key. 2016-03-18. https://web.archive.org/web/20160327035021/http://www2.lbl.gov/Science-Articles/Archive/pi-random.html. 2016-03-27. live.
  91. Robertson . John P. . 1996-10-01 . Magic Squares of Squares . Mathematics Magazine . 69 . 4 . 289–293 . 10.1080/0025570X.1996.11996457 . 0025-570X.
  92. 1604.07746. Huisman . Sander G.. Newer sums of three cubes. math.NT. 2016.
  93. Dobson . J. B. . 1 April 2017 . On Lerch's formula for the Fermat quotient . 1103.3907v6. 23. cs2. math.NT .
  94. Book: Ribenboim, P. . Paulo Ribenboim . 2006 . Die Welt der Primzahlen . 2nd . de . Springer . 10.1007/978-3-642-18079-8 . 978-3-642-18078-1 . 242–243 . Springer-Lehrbuch .
  95. Vessilin . Dimitrov . Ziyang . Gao . Philipp . Habegger . Uniformity in Mordell–Lang for curves . . 194 . 2021 . 237–298 . 10.4007/annals.2021.194.1.4 . 2001.10276 . 210932420 .
  96. 24523356 . Guan . Qi'an . Zhou . Xiangyu . Xiangyu Zhou . A solution of an

    L2

    extension problem with optimal estimate and applications . Annals of Mathematics . 2015 . 181 . 3 . 1139–1208 . 10.4007/annals.2015.181.3.6 . 56205818 . 1310.7169.
  97. Merel . Loïc . 1996 . "Bornes pour la torsion des courbes elliptiques sur les corps de nombres" [Bounds for the torsion of elliptic curves over number fields] . Inventiones Mathematicae . 124 . 1 . 437–449 . 10.1007/s002220050059 . 1369424 . 1996InMat.124..437M . 3590991 .
  98. Book: Casazza. Peter G.. Fickus. Matthew. Tremain. Janet C.. Weber. Eric. Han. Deguang. Jorgensen. Palle E. T.. Larson. David Royal. The Kadison-Singer problem in mathematics and engineering: A detailed account. Contemporary Mathematics. 2006. 414. 299–355. https://books.google.com/books?id=9b-4uqEGJdoC&pg=PA299. 24 April 2015. Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7–12, 2005, University of Central Florida, Florida. American Mathematical Society.. 978-0-8218-3923-2. 10.1090/conm/414/07820.
  99. News: Mackenzie. Dana. Kadison–Singer Problem Solved. 24 April 2015. SIAM News. January/February 2014. Society for Industrial and Applied Mathematics. https://web.archive.org/web/20141023120958/http://www.siam.org/pdf/news/2123.pdf. 23 October 2014. live.
  100. math/0405568. Agol . Ian. Tameness of hyperbolic 3-manifolds. 2004.
  101. math/9906212. Kurdyka . Krzysztof. Mostowski . Tadeusz. Parusiński . Adam. Proof of the gradient conjecture of R. Thom. Annals of Mathematics. 763–792. 152. 2000. 3. 10.2307/2661354. 2661354 . 119137528 .
  102. Moreira . Joel . Richter . Florian K. . Robertson . Donald . A proof of a sumset conjecture of Erdős . . 10.4007/annals.2019.189.2.4 . 189 . 2 . 605–652 . en-US. 2019 . 1803.00498 . 119158401 .
  103. . See in particular p. 316.
  104. Web site: Kalai . Gil . Amazing: Karim Adiprasito proved the g-conjecture for spheres! . 2019-02-15 . https://web.archive.org/web/20190216031650/https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/ . 2019-02-16 . live . 2018-12-25 .
  105. Santos . Franciscos . 2012 . A counterexample to the Hirsch conjecture . Annals of Mathematics . 176 . 1 . 383–412 . 10.4007/annals.2012.176.1.7 . 1006.2814 . 15325169 .
  106. Ziegler . Günter M. . 2012 . Who solved the Hirsch conjecture? . Documenta Mathematica . Documenta Mathematica Series . Extra Volume "Optimization Stories" . 75–85 . 10.4171/dms/6/13 . 978-3-936609-58-5 .
  107. Kauers . Manuel . Manuel Kauers . Koutschan . Christoph . Christoph Koutschan . Zeilberger . Doron . Doron Zeilberger . Proof of Ira Gessel's lattice path conjecture . Proceedings of the National Academy of Sciences . 106 . 28 . 2009-07-14 . 0027-8424 . 10.1073/pnas.0901678106 . 11502–11505 . 2710637 . 0806.4300 . 2009PNAS..10611502K . free .
  108. Chung . Fan . Greene . Curtis . Hutchinson . Joan . April 2015 . Herbert S. Wilf (1931–2012) . . 62 . 4 . 358 . 1088-9477 . 34550461 . The conjecture was finally given an exceptionally elegant proof by A. Marcus and G. Tardos in 2004. . 10.1090/noti1247 . free .
  109. Kemnitz' conjecture revisited . 10.1016/j.disc.2005.02.018 . free. 297. 1–3 . Discrete Mathematics. 196–201. 2005 . Savchev . Svetoslav.
  110. Green . Ben . Ben J. Green . math.NT/0304058 . 10.1112/S0024609304003650 . 6 . The Bulletin of the London Mathematical Society . 2083752 . 769–778 . The Cameron–Erdős conjecture . 36 . 2004. 119615076 .
  111. Web site: News from 2007 . . 31 December 2007 . American Mathematical Society . AMS . 2015-11-13 . The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the Cameron-Erdős conjecture..." . https://web.archive.org/web/20151117030726/http://www.ams.org/news?news_id=155 . 17 November 2015 . live .
  112. Brown . Aaron . Fisher . David . Hurtado . Sebastian . 2017-10-07 . Zimmer's conjecture for actions of . 1710.02735 . math.DS.
  113. Noncollision Singularities in a Planar Four-body Problem. Xue. Jinxin. 2014. math.DS . 1409.0048.
  114. Non-collision singularities in a planar 4-body problem. Xue. Jinxin. 2020. Acta Mathematica. 224. 2. 253–388. 10.4310/ACTA.2020.v224.n2.a2. 226420221.
  115. Web site: Known Historical Beggar-My-Neighbour Records . Richard P Mann . 2024-02-10 .
  116. Web site: The angel game in the plane . Brian H. . Bowditch. 2006. . warwick.ac.uk Warwick University. 2016-03-18 . https://web.archive.org/web/20160304185616/http://homepages.warwick.ac.uk/~masgak/papers/bhb-angel.pdf . 2016-03-04 . live .
  117. Web site: A Solution to the Angel Problem . Oddvar . Kloster . SINTEF ICT . Oslo, Norway. 2016-03-18 . https://web.archive.org/web/20160107125925/http://home.broadpark.no/~oddvark/angel/Angel.pdf . 2016-01-07 .
  118. The Angel of power 2 wins . Andras . Mathe . 2007. . 16 . 3. 363–374. 10.1017/S0963548306008303 . 16892955 . 2016-03-18 . https://web.archive.org/web/20161013034302/http://homepages.warwick.ac.uk/~masibe/angel-mathe.pdf . 2016-10-13 . live .
  119. Web site: Gacs . Peter . June 19, 2007 . THE ANGEL WINS . https://web.archive.org/web/20160304030433/http://www.cs.bu.edu/~gacs/papers/angel.pdf . 2016-03-04 . 2016-03-18.
  120. 2303.10798v2 . Smith . David . Myers . Joseph Samuel . Kaplan . Craig S. . Goodman-Strauss . Chaim . An aperiodic monotile . 2023 . math.CO.
  121. 1711.04906 . Larson . Eric . The Maximal Rank Conjecture . 2017 . math.AG .
  122. Web site: Existence of infinitely many minimal hypersurfaces in closed manifolds. . Song, Antoine . www.ams.org . "..I will present a solution of the conjecture, which builds on min-max methods developed by F. C. Marques and A. Neves.." . 19 June 2021.
  123. Web site: Antoine Song | Clay Mathematics Institute . "...Building on work of Codá Marques and Neves, in 2018 Song proved Yau's conjecture in complete generality".
  124. Marques . Fernando C.. André. Neves. Min-max theory and the Willmore conjecture. Annals of Mathematics . 2013. 1202.6036. 10.4007/annals.2014.179.2.6. 179. 2. 683–782. 50742102.
  125. 1011.4105. Guth . Larry. Katz . Nets Hawk. On the Erdos distinct distance problem in the plane. Annals of Mathematics. 155–190. 181. 2015. 1. 10.4007/annals.2015.181.1.2 . free.
  126. Web site: Squaring the Plane . Frederick V. . Henle . James M. . Henle . 2016-03-18 . www.maa.org Mathematics Association of America. https://web.archive.org/web/20160324074609/http://www.ww.amc12.org/sites/default/files/pdf/pubs/SquaringThePlane.pdf . 2016-03-24 . live .
  127. math/0412006. Brock . Jeffrey F.. Canary . Richard D.. Minsky . Yair N. . Yair Minsky. The classification of Kleinian surface groups, II: The Ending Lamination Conjecture. 2012. Annals of Mathematics. 176. 1. 1–149. 10.4007/annals.2012.176.1.1 . free.
  128. Shestakov . Ivan P. . Umirbaev . Ualbai U. . 10.1090/S0894-0347-03-00440-5 . 1 . Journal of the American Mathematical Society . 2015334 . 197–227 . The tame and the wild automorphisms of polynomial rings in three variables . 17 . 2004.
  129. Hutchings . Michael . Morgan . Frank . Ritoré . Manuel . Ros . Antonio . 10.2307/3062123 . 2 . Annals of Mathematics . 1906593 . 459–489 . Second Series . Proof of the double bubble conjecture . 155 . 2002. 3062123 . math/0406017 . 10481/32449 .
  130. math/9906042. Hales . Thomas C. . Thomas Callister Hales. The Honeycomb Conjecture. Discrete & Computational Geometry. 25. 1–22. 2001. 10.1007/s004540010071 . free.
  131. Teixidor i Bigas . Montserrat . Montserrat Teixidor i Bigas . Barbara . Russo . On a conjecture of Lange . alg-geom/9710019 . 1689352 . 1999 . Journal of Algebraic Geometry . 1056-3911 . 8 . 3 . 483–496 . 1997alg.geom.10019R .
  132. Ullmo . E . 1998 . Positivité et Discrétion des Points Algébriques des Courbes . Annals of Mathematics . 147 . 1. 167–179 . 10.2307/120987 . 0934.14013. 120987 . alg-geom/9606017 . 119717506 .
  133. Zhang . S.-W. . 1998 . Equidistribution of small points on abelian varieties . Annals of Mathematics . 147 . 1. 159–165 . 10.2307/120986 . 120986 .
  134. 1501.02155. Hales . Thomas. Adams . Mark. Bauer . Gertrud. Dang . Dat Tat. Harrison . John. Hoang . Le Truong. Kaliszyk . Cezary. Magron . Victor. McLaughlin . Sean. Nguyen . Tat Thang. Nguyen . Quang Truong. Nipkow . Tobias. Obua . Steven. Pleso . Joseph. Rute . Jason. Solovyev . Alexey. Ta . Thi Hoai An. Tran . Nam Trung. Trieu . Thi Diep. Urban . Josef. Ky . Vu. Zumkeller . Roland. A formal proof of the Kepler conjecture. Forum of Mathematics, Pi. 5. 2017. e2. 10.1017/fmp.2017.1 . free.
  135. math/9811079. Hales . Thomas C.. McLaughlin . Sean. The dodecahedral conjecture. Journal of the American Mathematical Society. 23. 2010. 2 . 299–344. 10.1090/S0894-0347-09-00647-X . 2010JAMS...23..299H . free.
  136. Park . Jinyoung . Pham . Huy Tuan . 2022-03-31 . A Proof of the Kahn-Kalai Conjecture . math.CO . 2203.17207 .
  137. Dujmović . Vida . Vida Dujmović . Eppstein . David . David Eppstein . Hickingbotham . Robert . Morin . Pat . Pat Morin . Wood . David R. . David Wood (mathematician) . 2011.04195 . August 2021 . 10.1007/s00493-021-4585-7 . . Stack-number is not bounded by queue-number. 42 . 2 . 151–164 . 226281691 .
  138. Huang . C.. Further results on tree labellings . Utilitas Mathematica . 21 . 31–48 . 1982. 668845. Kotzig. A.. Rosa. A.. Anton Kotzig. .
  139. Web site: Rainbow Proof Shows Graphs Have Uniform Parts. Hartnett . Kevin. Quanta Magazine. 19 February 2020. en. 2020-02-29.
  140. Shitov . Yaroslav . 2019-09-01 . dmy-all . Counterexamples to Hedetniemi's conjecture . Annals of Mathematics . 190 . 2 . 663–667 . 1905.02167 . 10.4007/annals.2019.190.2.6 . 10.4007/annals.2019.190.2.6 . 3997132 . 1451.05087 . 146120733 . 2021-07-19.
  141. He . Dawei . Wang . Yan . Yu . Xingxing . 2019-12-11 . The Kelmans-Seymour conjecture I: Special separations . Journal of Combinatorial Theory, Series B . 144 . 197–224 . 10.1016/j.jctb.2019.11.008 . 0095-8956 . 1511.05020 . 29791394.
  142. He . Dawei . Wang . Yan . Yu . Xingxing . 2019-12-11 . The Kelmans-Seymour conjecture II: 2-Vertices in K4− . Journal of Combinatorial Theory, Series B . 144 . 225–264 . 10.1016/j.jctb.2019.11.007 . 0095-8956 . 1602.07557. 220369443 .
  143. He . Dawei . Wang . Yan . Yu . Xingxing . 2019-12-09 . The Kelmans-Seymour conjecture III: 3-vertices in K4− . Journal of Combinatorial Theory, Series B . 144 . 265–308 . 10.1016/j.jctb.2019.11.006 . 0095-8956 . 1609.05747 . 119625722.
  144. He . Dawei . Wang . Yan . Yu . Xingxing . 2019-12-19 . The Kelmans-Seymour conjecture IV: A proof . Journal of Combinatorial Theory, Series B . 144 . 309–358 . 10.1016/j.jctb.2019.12.002 . 0095-8956 . 1612.07189 . 119175309.
  145. Zang . Wenan . Jing . Guangming . Chen . Guantao . 2019-01-29 . Proof of the Goldberg–Seymour Conjecture on Edge-Colorings of Multigraphs . math.CO . en . 1901.10316v1.
  146. Zallaghi M.. Abdollahi A. . 2015 . Communications in Algebra . Character sums for Cayley graphs . 43. 12. 5159–5167 . 10.1080/00927872.2014.967398 . 117651702 .
  147. Huh . June . June Huh . Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs . 1008.4749 . Journal of the American Mathematical Society . 25 . 2012 . 3 . 907–927 . 10.1090/S0894-0347-2012-00731-0 . free.
  148. Chalopin . Jérémie . Gonçalves . Daniel . Mitzenmacher . Michael . Every planar graph is the intersection graph of segments in the plane: extended abstract . 10.1145/1536414.1536500 . 631–638 . ACM . Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009 . 2009.
  149. math/0509397. Aharoni . Ron . Ron Aharoni. Berger . Eli. Menger's theorem for infinite graphs. Inventiones Mathematicae. 176. 1–62. 2009. 1 . 10.1007/s00222-008-0157-3 . 2009InMat.176....1A . free.
  150. News: Seigel-Itzkovich . Judy . Russian immigrant solves math puzzle . The Jerusalem Post . 2008-02-08 . 2015-11-12.
  151. Book: Diestel, Reinhard . 2005 . Minors, Trees, and WQO . Electronic Edition 2005 . 326–367 . Springer . Graph Theory . http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/preview/Ch12.pdf.
  152. The strong perfect graph theorem . Chudnovsky . Maria . Robertson . Neil . Seymour . Paul . Thomas . Robin . Annals of Mathematics . 2002 . 164 . 51–229 . math/0212070 . 10.4007/annals.2006.164.51 . 2002math.....12070C . 119151552.
  153. Klin, M. H., M. Muzychuk and R. Poschel: The isomorphism problem for circulant graphs via Schur ring theory, Codes and Association Schemes, American Math. Society, 2001.
  154. Harary's conjectures on integral sum graphs . 10.1016/0012-365X(95)00163-Q . free . . 160 . 1–3 . 241–244 . 1996 . Chen . Zhibo.
  155. Friedman . Joel . January 2015 . Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: with an Appendix by Warren Dicks . Memoirs of the American Mathematical Society . en . 233 . 1100 . 0 . 10.1090/memo/1100 . 117941803 . 0065-9266.
  156. Mineyev . Igor . 10.4007/annals.2012.175.1.11 . 1 . Annals of Mathematics . 2874647 . 393–414 . Second Series . Submultiplicativity and the Hanna Neumann conjecture . 175 . 2012.
  157. Non-realizability and ending laminations: Proof of the density conjecture. 10.1007/s11511-012-0088-0. Acta Mathematica. 209. 2. 323–395. 2012. Namazi. Hossein. Souto. Juan. free.
  158. Pila . Jonathan . Shankar . Ananth . Tsimerman . Jacob . Esnault . Hélène . Groechenig . Michael . 2021-09-17 . Canonical Heights on Shimura Varieties and the André-Oort Conjecture . math.NT . 2109.08788.
  159. Bourgain . Jean. Demeter. Ciprian. Guth. Larry. Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three. Annals of Mathematics . 2015. 10.4007/annals.2016.184.2.7. 184. 2. 633–682. 1721.1/115568. 2015arXiv151201565B. 1512.01565. 43929329.
  160. 1305.2897 . Major arcs for Goldbach's theorem. Helfgott. Harald A. . math.NT . 2013.
  161. 1205.5252 . Minor arcs for Goldbach's problem . Helfgott. Harald A.. math.NT . 2012.
  162. 1312.7748 . The ternary Goldbach conjecture is true. Helfgott. Harald A. . math.NT . 2013.
  163. Zhang. Yitang. 2014-05-01. Bounded gaps between primes. Annals of Mathematics. 179. 3. 1121–1174. 10.4007/annals.2014.179.3.7. 0003-486X.
  164. Web site: Bounded gaps between primes - Polymath Wiki. 2021-08-27. asone.ai. 2020-12-08. https://web.archive.org/web/20201208045925/https://asone.ai/polymath/index.php?title=Bounded_gaps_between_primes.
  165. Maynard. James. 2015-01-01. Small gaps between primes. Annals of Mathematics. 383–413. 10.4007/annals.2015.181.1.7. 1311.4600. 55175056. 0003-486X.
  166. Generalized Sidon sets. 10.1016/j.aim.2010.05.010 . 225. 5. Advances in Mathematics. 2786–2807. 2010 . Cilleruelo . Javier. 10261/31032. 7385280. free. free.
  167. . 2011 Cole Prize in Number Theory . . 58 . 4 . 610–611 . 1088-9477 . 34550461 . 2015-11-12 . https://web.archive.org/web/20151106051835/http://www.ams.org/notices/201104/rtx110400610p.pdf . 2015-11-06 . live .
  168. . May 2010 . Bombieri and Tao Receive King Faisal Prize . . 57 . 5 . 642–643 . 1088-9477 . 34550461 . Working with Ben Green, he proved there are arbitrarily long arithmetic progressions of prime numbers—a result now known as the Green–Tao theorem. . 2016-03-18 . https://web.archive.org/web/20160304063504/http://www.ams.org/notices/201005/rtx100500642p.pdf . 2016-03-04 . live .
  169. Metsänkylä . Tauno . 5 September 2003 . Catalan's conjecture: another old diophantine problem solved . . 41 . 1 . 43–57 . 0273-0979 . The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu. . 10.1090/s0273-0979-03-00993-5 . 13 November 2015 . https://web.archive.org/web/20160304082755/http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf . 4 March 2016 . live .
  170. Book: Croot, Ernest S. III . Ernest S. Croot III . University of Georgia, Athens . Ph.D. thesis . Unit Fractions . 2000. Croot . Ernest S. III . Ernest S. Croot III . math.NT/0311421 . 10.4007/annals.2003.157.545 . 2 . . 545–556 . On a coloring conjecture about unit fractions . 157 . 2003. 2003math.....11421C . 13514070 .
  171. Wiles. Andrew. Andrew Wiles. 1995. Modular elliptic curves and Fermat's Last Theorem. Annals of Mathematics. 141. 3. 443–551. 37032255. 10.2307/2118559. 2118559. 10.1.1.169.9076. 2016-03-06. https://web.archive.org/web/20110510062158/http://math.stanford.edu/%7Elekheng/flt/wiles.pdf. 2011-05-10. live.
  172. . 1995 . Ring theoretic properties of certain Hecke algebras . Annals of Mathematics . 141 . 3 . 553–572 . 10.1.1.128.531 . 10.2307/2118560 . 2118560 . 37032255 . https://web.archive.org/web/20000916161311/http://www.math.harvard.edu/~rtaylor/hecke.ps . 16 September 2000.
  173. Lee . Choongbum . 2017 . Ramsey numbers of degenerate graphs . Annals of Mathematics . 185 . 3. 791–829 . 10.4007/annals.2017.185.3.2 . 1505.04773 . 7974973 .
  174. Lamb . Evelyn . 26 May 2016 . Two-hundred-terabyte maths proof is largest ever . Nature . 10.1038/nature.2016.19990 . 534 . 7605 . 17–18 . 27251254 . 2016Natur.534...17L. free .
  175. Book: Heule . Marijn J. H. . Marijn Heule . Kullmann . Oliver . Marek . Victor W. . Victor W. Marek . Creignou . N. . Le Berre . D. . 1605.00723 . Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer . 10.1007/978-3-319-40970-2_15 . 3534782 . 228–245 . Springer, [Cham] . Lecture Notes in Computer Science . Theory and Applications of Satisfiability Testing – SAT 2016 . 9710 . 2016. 978-3-319-40969-6 . 7912943 .
  176. Web site: Linkletter, David . 27 December 2019 . The 10 Biggest Math Breakthroughs of 2019 . 20 June 2021 . Popular Mechanics.
  177. Piccirillo . Lisa . 2020 . The Conway knot is not slice . . 191 . 2 . 581–591 . 10.4007/annals.2020.191.2.5. 52398890 .
  178. Web site: Klarreich . Erica . Erica Klarreich . 2020-05-19 . Graduate Student Solves Decades-Old Conway Knot Problem . 2022-08-17 . . en.
  179. 1204.2810v1. Agol . Ian. The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves, and Jason Manning). Documenta Mathematica. 18. 2013. 1045–1087. 10.4171/dm/421 . 255586740 .
  180. 1203.6597. Brendle . Simon . Simon Brendle. Embedded minimal tori in

    S3

    and the Lawson conjecture. Acta Mathematica. 211. 2. 177–190. 2013. 10.1007/s11511-013-0101-2 . free.
  181. 1101.1330. Kahn . Jeremy . Jeremy Kahn. Markovic . Vladimir . Vladimir Markovic. The good pants homology and the Ehrenpreis conjecture. Annals of Mathematics. 1–72. 182. 2015. 1. 10.4007/annals.2015.182.1.1 . free.
  182. 0909.2360. Austin . Tim. Rational group ring elements with kernels having irrational dimension. Proceedings of the London Mathematical Society. 107. 6. 1424–1448. December 2013. 10.1112/plms/pdt029 . 2009arXiv0909.2360A. 115160094.
  183. Lurie . Jacob . 2009 . On the classification of topological field theories . Current Developments in Mathematics . 2008 . 129–280 . 10.4310/cdm.2008.v2008.n1.a3. 2009arXiv0905.0465L . 0905.0465 . 115162503 .
  184. . March 18, 2010 . Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman . November 13, 2015 . The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman. . https://web.archive.org/web/20100322192115/http://www.claymath.org/poincare/ . March 22, 2010 . live .
  185. 0809.4040. Morgan . John . Completion of the Proof of the Geometrization Conjecture. Tian. Gang. math.DG. 2008.
  186. M.E. . Rudin . Mary Ellen Rudin. Nikiel's Conjecture. Topology and Its Applications. 116. 2001. 3 . 305–331. 10.1016/S0166-8641(01)00218-8 . free.
  187. Web site: Ganea's Conjecture on Lusternik-Schnirelmann Category. Norio Iwase. 1 November 1998. ResearchGate.
  188. 1509.05363v5. Tao. Terence . Terence Tao. The Erdős discrepancy problem. math.CO. 2015.
  189. Proof of the umbral moonshine conjecture. John F. R.. Duncan. Michael J.. Griffin. Ken. Ono. 1 December 2015. Research in the Mathematical Sciences. 2. 1. 26. 10.1186/s40687-015-0044-7. 2015arXiv150301472D. 1503.01472. 43589605 . free .
  190. 1406.6534. Cheeger . Jeff. Naber . Aaron. Regularity of Einstein Manifolds and the Codimension 4 Conjecture. Annals of Mathematics. 1093–1165. 182. 3. 2015. 10.4007/annals.2015.182.3.5 . free.
  191. Wolchover . Natalie . March 28, 2017 . A Long-Sought Proof, Found and Almost Lost . live . . https://web.archive.org/web/20170424133433/https://www.quantamagazine.org/20170328-statistician-proves-gaussian-correlation-inequality/ . April 24, 2017 . May 2, 2017.
  192. 1104.2922. Newman . Alantha . Nikolov . Aleksandar . A counterexample to Beck's conjecture on the discrepancy of three permutations. cs.DM. 2011.
  193. Web site: On motivic cohomology with Z/l-coefficients . Voevodsky . Vladimir . 2016-03-18 . https://web.archive.org/web/20160327035457/http://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p11-p.pdf . Princeton, NJ . annals.math.princeton.edu . . 1 July 2011. 174. 1. 401–438. 2016-03-27 . live .
  194. Geisser . Thomas . Levine . Marc . 10.1515/crll.2001.006 . Journal für die Reine und Angewandte Mathematik . 1807268 . 55–103 . The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky . 2001 . 2001. 530 .
  195. Web site: Kahn . Bruno . Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry . live . https://web.archive.org/web/20160327035553/https://webusers.imj-prg.fr/~bruno.kahn/preprints/kcag.pdf . 2016-03-27 . 2016-03-18 . webusers.imj-prg.fr.
  196. Web site: motivic cohomology – Milnor–Bloch–Kato conjecture implies the Beilinson-Lichtenbaum conjecture – MathOverflow. 2016-03-18 .
  197. 0906.1612. Mattman . Thomas W.. Solis . Pablo. A proof of the Kauffman-Harary Conjecture. Algebraic & Geometric Topology. 9. 4. 2027–2039. 2009. 10.2140/agt.2009.9.2027 . 2009arXiv0906.1612M . 8447495.
  198. 0910.5501. Kahn . Jeremy. Markovic . Vladimir. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. Annals of Mathematics. 1127–1190. 175. 3. 2012. 10.4007/annals.2012.175.3.4 . free.
  199. Zhiqin . Lu. 2007. Normal Scalar Curvature Conjecture and its applications. 0711.3510. Journal of Functional Analysis. 261. 5. September 2011. 1284–1308. 10.1016/j.jfa.2011.05.002 . free.
  200. Web site: Research Awards . . 2019-04-07 . https://web.archive.org/web/20190407160116/https://www.claymath.org/research . 2019-04-07 . live .
  201. Lewis . A. S. . Parrilo . P. A. . Ramana . M. V. . 10.1090/S0002-9939-05-07752-X . 9 . Proceedings of the American Mathematical Society . 2146191 . 2495–2499 . The Lax conjecture is true . 133 . 2005. 17436983 .
  202. Web site: Fields Medal – Ngô Bảo Châu . . 19 August 2010 . International Congress of Mathematicians 2010 . ICM . 2015-11-12 . Ngô Bảo Châu is being awarded the 2010 Fields Medal for his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods. . https://web.archive.org/web/20150924032610/http://www.icm2010.in/prize-winners-2010/fields-medal-ngo-bao-chau . 24 September 2015 . live .
  203. Reduced power operations in motivic cohomology . 1–57. Publications Mathématiques de l'IHÉS . 98 . 2003 . Voevodsky . Vladimir . 10.1007/s10240-003-0009-z . 10.1.1.170.4427 . 2016-03-18 . live . https://web.archive.org/web/20170728114725/http://archive.numdam.org/item/PMIHES_2003__98__1_0 . 2017-07-28 . math/0107109 . 8172797.
  204. Baruch . Ehud Moshe . 10.4007/annals.2003.158.207 . 1 . Annals of Mathematics . 1999922 . 207–252 . Second Series . A proof of Kirillov's conjecture . 158 . 2003.
  205. Haas . Bertrand . 2002 . A Simple Counterexample to Kouchnirenko's Conjecture . live . Beiträge zur Algebra und Geometrie . 43 . 1 . 1–8 . https://web.archive.org/web/20161007091417/http://www.emis.de/journals/BAG/vol.43/no.1/b43h1haa.pdf . 2016-10-07 . 2016-03-18.
  206. Haiman . Mark . 10.1090/S0894-0347-01-00373-3 . 4 . Journal of the American Mathematical Society . 1839919 . 941–1006 . Hilbert schemes, polygraphs and the Macdonald positivity conjecture . 14 . 2001. 9253880 .
  207. Auscher . Pascal . Hofmann . Steve . Lacey . Michael . McIntosh . Alan . Tchamitchian . Ph. . 10.2307/3597201 . 2 . Annals of Mathematics . 1933726 . 633–654 . Second Series . The solution of the Kato square root problem for second order elliptic operators on

    Rn

    . 156 . 2002. 3597201 .
  208. math/0102150. Barbieri-Viale . Luca. Rosenschon . Andreas. Saito . Morihiko. Deligne's Conjecture on 1-Motives. Annals of Mathematics. 593–633. 158. 2003. 2. 10.4007/annals.2003.158.593 . free.
  209. On a conjecture of Erdős and Stewart. 10.1090/s0025-5718-00-01178-9. Mathematics of Computation. 70. 234. 893–897. 2000. Luca. Florian. 2016-03-18. https://web.archive.org/web/20160402030443/http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01178-9/S0025-5718-00-01178-9.pdf. 2016-04-02. live. 2001MaCom..70..893L.
  210. Book: Atiyah, Michael . Michael Atiyah . Yau . Shing-Tung . Shing-Tung Yau . The geometry of classical particles . 10.4310/SDG.2002.v7.n1.a1 . 1919420 . 1–15 . International Press . Somerville, Massachusetts . Surveys in Differential Geometry . Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer . 7 . 2000.