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
Hilbert's problems[1] 23 15 David Hilbert 1900
Landau's problems[2] 4 4 Edmund Landau 1912
Taniyama's problems[3] 36 - Yutaka Taniyama 1955
Thurston's 24 questions[4][5] 24 - William Thurston 1982
Smale's problems 18 14 Stephen Smale 1998
Millennium Prize Problems 7 6[6] Clay Mathematics Institute 2000
Simon problems 15 <12[7][8] Barry Simon 2000
Unsolved Problems on Mathematics for the 21st Century[9] 22 - Jair Minoro Abe, Shotaro Tanaka 2001
DARPA's math challenges[10][11] 23 - DARPA 2007

### Millennium Prize Problems

Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:[6]

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

• The Kourovka Notebook (Russian: Коуровская тетрадь) is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.[14]
• The Sverdlovsk Notebook (Russian: Свердловская тетрадь) is a collection of unsolved problems in semigroup theory, first published in 1969 and updated many times since.[15][16][17]
• The Dniester Notebook (Russian: Днестровская тетрадь) lists several hundred unsolved problems in algebra, particularly ring theory and modulus theory.[18][19]
• The Erlagol Notebook (Russian: Эрлагольская тетрадь) lists unsolved problems in algebra and model theory.[20]

## Unsolved problems

### Algebra

 Main article: Algebra

#### Group theory

 Main article: Group theory

### Analysis

 Main article: Mathematical analysis
• The Brennan conjecture: estimating the integral of powers of the moduli of the derivative of conformal maps into the open unit disk, on certain subsets of ${\displaystyle \mathbb {C} }$
• Fuglede's conjecture on whether nonconvex sets in ${\displaystyle \mathbb {R} }$ and ${\displaystyle \mathbb {R} ^{2))$ are spectral if and only if they tile by translation.
• Goodman's conjecture on the coefficients of multivalent functions
• Invariant subspace problem – does every bounded operator on a complex Banach space send some non-trivial closed subspace to itself?
• Kung–Traub conjecture on the optimal order of a multipoint iteration without memory[23]
• Lehmer's conjecture on the Mahler measure of non-cyclotomic polynomials[24]
• The mean value problem: given a complex polynomial ${\displaystyle f}$ of degree ${\displaystyle d\geq 2}$ and a complex number ${\displaystyle z}$, is there a critical point ${\displaystyle c}$ of ${\displaystyle f}$ such that ${\displaystyle |f(z)-f(c)|\leq |f'(z)||z-c|}$?
• The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy[25]
• Sendov's conjecture: if a complex polynomial with degree at least ${\displaystyle 2}$ has all roots in the closed unit disk, then each root is within distance ${\displaystyle 1}$ from some critical point.
• Vitushkin's conjecture on compact subsets of ${\displaystyle \mathbb {C} }$ with analytic capacity ${\displaystyle 0}$
• What is the exact value of Landau's constants, including Bloch's constant?

#### Transcendental numbers and diophantine approximation

 Further information: Transcendental number and Transcendental number theory
 Further information: Diophantine approximation

### Combinatorics

 Main article: Combinatorics
• The 1/3–2/3 conjecture – does every finite partially ordered set that is not totally ordered contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?[30]
• The Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition
• Problems in Latin squares – open questions concerning Latin squares
• The lonely runner conjecture – if ${\displaystyle k}$ runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance ${\displaystyle 1/k}$ from each other runner) at some time?[31]
• Map folding – various problems in map folding and stamp folding.
• No-three-in-line problem – how many points can be placed in the ${\displaystyle n\times n}$ grid so that no three of them lie on a line?
• Rudin's conjecture on the number of squares in finite arithmetic progressions[32]
• The sunflower conjecture – can the number of ${\displaystyle k}$ size sets required for the existence of a sunflower of ${\displaystyle r}$ sets be bounded by an exponential function in ${\displaystyle k}$ for every fixed ${\displaystyle r>2}$?
• Frankl's union-closed sets conjecture – for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets[33]

### Dynamical systems

 Main article: Dynamical system

### Games and puzzles

 Main article: Game theory

#### Combinatorial games

 Main article: Combinatorial game theory

### Geometry

 Main article: Geometry

#### Algebraic geometry

 Main article: Algebraic geometry
• Are infinite sequences of flips possible in dimensions greater than 3?
• Resolution of singularities in characteristic ${\displaystyle p}$

#### Covering and packing

• Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a bounded n-dimensional set.
• The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?[47]
• The Erdős–Oler conjecture: when ${\displaystyle n}$ is a triangular number, packing ${\displaystyle n-1}$ circles in an equilateral triangle requires a triangle of the same size as packing ${\displaystyle n}$ circles[48]
• The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24[49]
• Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets[50]
• Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.
• Square packing in a square: what is the asymptotic growth rate of wasted space?[51]
• Ulam's packing conjecture about the identity of the worst-packing convex solid[52]

#### Differential geometry

 Main article: Differential geometry

#### Discrete geometry

 Main article: Discrete geometry

#### Euclidean geometry

 Main article: Euclidean geometry

### Graph theory

 Main article: Graph theory

### Model theory and formal languages

 Main articles: Model theory and formal languages
• The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in ${\displaystyle \aleph _{0))$ is a simple algebraic group over an algebraically closed field.
• Generalized star height problem: can all regular languages be expressed using generalized regular expressions with limited nesting depths of Kleene stars?
• For which number fields does Hilbert's tenth problem hold?
• Kueker's conjecture[135]
• The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for ${\displaystyle \aleph _{1))$-saturated models of a countable theory.[136]
• Shelah's categoricity conjecture for ${\displaystyle L_{\omega _{1},\omega ))$: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[136]
• Shelah's eventual categoricity conjecture: For every cardinal ${\displaystyle \lambda }$ there exists a cardinal ${\displaystyle \mu (\lambda )}$ such that if an AEC K with LS(K)<= ${\displaystyle \lambda }$ is categorical in a cardinal above ${\displaystyle \mu (\lambda )}$ then it is categorical in all cardinals above ${\displaystyle \mu (\lambda )}$.[136][137]
• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
• The stable forking conjecture for simple theories[138]
• Tarski's exponential function problem: is the theory of the real numbers with the exponential function decidable?
• The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?[139]
• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[140]
• Vaught conjecture: the number of countable models of a first-order complete theory in a countable language is either finite, ${\displaystyle \aleph _{0))$, or ${\displaystyle 2^{\aleph _{0))}$.
• Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality ${\displaystyle \aleph _{\omega _{1))}$ does it have a model of cardinality continuum?[141]
• Do the Henson graphs have the finite model property?
• Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
• Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
• If the class of atomic models of a complete first order theory is categorical in the ${\displaystyle \aleph _{n))$, is it categorical in every cardinal?[142][143]
• Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
• Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?[144]
• Is the theory of the field of Laurent series over ${\displaystyle \mathbb {Z} _{p))$ decidable? of the field of polynomials over ${\displaystyle \mathbb {C} }$?
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[145]
• Determine the structure of Keisler's order.[146][147]

### Probability theory

 Main article: Probability theory

### Number theory

#### General

• Beilinson's conjectures
• Brocard's problem: are there any integer solutions to ${\displaystyle n!+1=m^{2))$ other than ${\displaystyle n=4,5,7}$?
• Büchi's problem on sufficiently large sequences of square numbers with constant second difference.
• Carmichael's totient function conjecture: do all values of Euler's totient function have multiplicity greater than ${\displaystyle 1}$?
• Casas-Alvero conjecture: if a polynomial of degree ${\displaystyle d}$ defined over a field ${\displaystyle K}$ of characteristic ${\displaystyle 0}$ has a factor in common with its first through ${\displaystyle d-1}$-th derivative, then must ${\displaystyle f}$ be the ${\displaystyle d}$-th power of a linear polynomial?
• Catalan–Dickson conjecture on aliquot sequences: no aliquot sequences are infinite but non-repeating.
• Erdős–Moser problem: is ${\displaystyle 1^{1}+2^{1}=3^{1))$ the only solution to the Erdős–Moser equation?
• Erdős–Ulam problem: is there a dense set of points in the plane all at rational distances from one-another?
• Exponent pair conjecture: for all ${\displaystyle \epsilon >0}$, is the pair ${\displaystyle (\epsilon ,1/2+\epsilon )}$ an exponent pair?
• The Gauss circle problem: how far can the number of integer points in a circle centered at the origin be from the area of the circle?
• Grand Riemann hypothesis: do the nontrivial zeros of all automorphic L-functions lie on the critical line ${\displaystyle 1/2+it}$ with real ${\displaystyle t}$?
• Generalized Riemann hypothesis: do the nontrivial zeros of all Dirichlet L-functions lie on the critical line ${\displaystyle 1/2+it}$ with real ${\displaystyle t}$?
• Riemann hypothesis: do the nontrivial zeros of the Riemann zeta function lie on the critical line ${\displaystyle 1/2+it}$ with real ${\displaystyle t}$?
• Grimm's conjecture: each element of a set of consecutive composite numbers can be assigned a distinct prime number that divides it.
• Hall's conjecture: for any ${\displaystyle \epsilon >0}$, there is some constant ${\displaystyle c(\epsilon )}$ such that either ${\displaystyle y^{2}=x^{3))$ or ${\displaystyle |y^{2}-x^{3}|>c(\epsilon )x^{1/2-\epsilon ))$.
• Hardy–Littlewood zeta-function conjectures
• Hilbert–Pólya conjecture: the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator.
• Hilbert's eleventh problem: classify quadratic forms over algebraic number fields.
• Hilbert's ninth problem: find the most general reciprocity law for the norm residues of ${\displaystyle k}$-th order in a general algebraic number field, where ${\displaystyle k}$ is a power of a prime.
• Hilbert's twelfth problem: extend the Kronecker–Weber theorem on Abelian extensions of ${\displaystyle \mathbb {Q} }$ to any base number field.
• Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function[148]
• Lehmer's totient problem: if ${\displaystyle \phi (n)}$ divides ${\displaystyle n-1}$, must ${\displaystyle n}$ be prime?
• Leopoldt's conjecture: a p-adic analogue of the regulator of an algebraic number field does not vanish.
• Lindelöf hypothesis that for all ${\displaystyle \epsilon >0}$, ${\displaystyle \zeta (1/2+it)=o(t^{\epsilon })}$
• Littlewood conjecture: for any two real numbers ${\displaystyle \alpha ,\beta }$, ${\displaystyle \liminf _{n\rightarrow \infty }n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0}$, where ${\displaystyle \Vert x\Vert }$ is the distance from ${\displaystyle x}$ to the nearest integer.
• Mahler's 3/2 problem that no real number ${\displaystyle x}$ has the property that the fractional parts of ${\displaystyle x(3/2)^{n))$ are less than ${\displaystyle 1/2}$ for all positive integers ${\displaystyle n}$.
• Montgomery's pair correlation conjecture: the normalized pair correlation function between pairs of zeros of the Riemann zeta function is the same as the pair correlation function of random Hermitian matrices.
• n conjecture: a generalization of the abc conjecture to more than three integers.
• abc conjecture: for any ${\displaystyle \epsilon >0}$, ${\displaystyle {\text{rad))(abc)^{1+\epsilon } is true for only finitely many positive ${\displaystyle a,b,c}$ such that ${\displaystyle a+b=c}$.
• Szpiro's conjecture: for any ${\displaystyle \epsilon >0}$, there is some constant ${\displaystyle C(\epsilon )}$ such that, for any elliptic curve ${\displaystyle E}$ defined over ${\displaystyle \mathbb {Q} }$ with minimal discriminant ${\displaystyle \Delta }$ and conductor ${\displaystyle f}$, we have ${\displaystyle |\Delta |\leq C(\epsilon )\cdot f^{6+\epsilon ))$.
• Newman's conjecture: the partition function satisfies any arbitrary congruence infinitely often.
• Piltz divisor problem on bounding ${\displaystyle \Delta _{k}(x)=D_{k}(x)-xP_{k}(\log(x))}$
• Dirichlet's divisor problem: the specific case of the Piltz divisor problem for ${\displaystyle k=1}$
• Ramanujan–Petersson conjecture: a number of related conjectures that are generalizations of the original conjecture.
• Sato–Tate conjecture: also a number of related conjectures that are generalizations of the original conjecture.
• Scholz conjecture: the length of the shortest addition chain producing ${\displaystyle 2^{n}-1}$ is at most ${\displaystyle n-1}$ plus the length of the shortest addition chain producing ${\displaystyle n}$.
• Do Siegel zeros exist?
• Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?[149]
• Vojta's conjecture on heights of points on algebraic varieties over algebraic number fields.

• Erdős conjecture on arithmetic progressions that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long arithmetic progressions.
• Erdős–Heilbronn conjecture that ${\displaystyle |2^{\wedge }A|\geq \min\{p,2|A|-3\))$ if ${\displaystyle p}$ is a prime and ${\displaystyle A}$ is a nonempty subset of the field ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$.
• Erdős–Turán conjecture on additive bases: if ${\displaystyle B}$ is an additive basis of order ${\displaystyle 2}$, then the number of ways that positive integers ${\displaystyle n}$ can be expressed as the sum of two numbers in ${\displaystyle B}$ must tend to infinity as ${\displaystyle n}$ tends to infinity.
• Gilbreath's conjecture on consecutive applications of the unsigned forward difference operator to the sequence of prime numbers.
• Goldbach's conjecture: every even natural number greater than ${\displaystyle 2}$ is the sum of two prime numbers.
• Lander, Parkin, and Selfridge conjecture: if the sum of ${\displaystyle m}$ ${\displaystyle k}$-th powers of positive integers is equal to a different sum of ${\displaystyle n}$ ${\displaystyle k}$-th powers of positive integers, then ${\displaystyle m+n\geq k}$.
• Lemoine's conjecture: all odd integers greater than ${\displaystyle 5}$ can be represented as the sum of an odd prime number and an even semiprime.
• Minimum overlap problem of estimating the minimum possible maximum number of times a number appears in the termwise difference of two equally large sets partitioning the set ${\displaystyle \{1,\ldots ,2n\))$
• Pollock's conjectures
• Does every nonnegative integer appear in Recamán's sequence?
• Skolem problem: can an algorithm determine if a constant-recursive sequence contains a zero?
• The values of g(k) and G(k) in Waring's problem

#### Algebraic number theory

 Main article: Algebraic number theory
• Characterize all algebraic number fields that have some power basis.

#### Computational number theory

 Main article: Computational number theory

#### Diophantine equations

 Further information: Diophantine equation
• Beal's conjecture: for all integral solutions to ${\displaystyle A^{x}+B^{y}=C^{z))$ where ${\displaystyle x,y,z>2}$, all three numbers ${\displaystyle A,B,C}$ must share some prime factor.
• Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem): determine precisely what rational numbers are congruent numbers.
• Erdős–Straus conjecture: for every ${\displaystyle n\geq 2}$, there are positive integers ${\displaystyle x,y,z}$ such that ${\displaystyle 4/n=1/x+1/y+1/z}$.
• Fermat–Catalan conjecture: there are finitely many distinct solutions ${\displaystyle (a^{m},b^{n},c^{k})}$ to the equation ${\displaystyle a^{m}+b^{n}=c^{k))$ with ${\displaystyle a,b,c}$ being positive coprime integers and ${\displaystyle m,n,k}$ being positive integers satisfying ${\displaystyle 1/m+1/n+1/k<1}$.
• Goormaghtigh conjecture on solutions to ${\displaystyle (x^{m}-1)/(x-1)=(y^{n}-1)/(y-1)}$ where ${\displaystyle x>y>1}$ and ${\displaystyle m,n>2}$.
• The uniqueness conjecture for Markov numbers[153] that every Markov number is the largest number in exactly one normalized solution to the Markov Diophantine equation.
• Pillai's conjecture: for any ${\displaystyle A,B,C}$, the equation ${\displaystyle Ax^{m}-By^{n}=C}$ has finitely many solutions when ${\displaystyle m,n}$ are not both ${\displaystyle 2}$.
• Which integers can be written as the sum of three perfect cubes?[154]
• Can every integer be written as a sum of four perfect cubes?

#### Prime numbers

 Main article: Prime numbers
• Agoh–Giuga conjecture on the Bernoulli numbers that ${\displaystyle p}$ is prime if and only if ${\displaystyle pB_{p-1}\equiv -1{\pmod {p))}$
• Agrawal's conjecture that given coprime positive integers ${\displaystyle n}$ and ${\displaystyle r}$, if ${\displaystyle (X-1)^{n}\equiv X^{n}-1{\pmod {n,X^{r}-1))}$, then either ${\displaystyle n}$ is prime or ${\displaystyle n^{2}\equiv 1{\pmod {r))}$
• Artin's conjecture on primitive roots that if an integer is neither a perfect square nor ${\displaystyle -1}$, then it is a primitive root modulo infinitely many prime numbers ${\displaystyle p}$
• Brocard's conjecture: there are always at least ${\displaystyle 4}$ prime numbers between consecutive squares of prime numbers, aside from ${\displaystyle 2^{2))$ and ${\displaystyle 3^{2))$.
• Bunyakovsky conjecture: if an integer-coefficient polynomial ${\displaystyle f}$ has a positive leading coefficient, is irreducible over the integers, and has no common factors over all ${\displaystyle f(x)}$ where ${\displaystyle x}$ is a positive integer, then ${\displaystyle f(x)}$ is prime infinitely often.
• Catalan's Mersenne conjecture: some Catalan–Mersenne number is composite and thus all Catalan–Mersenne numbers are composite after some point.
• Dickson's conjecture: for a finite set of linear forms ${\displaystyle a_{1}+b_{1}n,\ldots ,a_{k}+b_{k}n}$ with each ${\displaystyle b_{i}\geq 1}$, there are infinitely many ${\displaystyle n}$ for which all forms are prime, unless there is some congruence condition preventing it.
• Dubner's conjecture: every even number greater than ${\displaystyle 4208}$ is the sum of two primes which both have a twin.
• Elliott–Halberstam conjecture on the distribution of prime numbers in arithmetic progressions.
• Erdős–Mollin–Walsh conjecture: no three consecutive numbers are all powerful.
• Feit–Thompson conjecture: for all distinct prime numbers ${\displaystyle p}$ and ${\displaystyle q}$, ${\displaystyle (p^{q}-1)/(p-1)}$ does not divide ${\displaystyle (q^{p}-1)/(q-1)}$
• Fortune's conjecture that no Fortunate number is composite.
• The Gaussian moat problem: is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded?
• Gillies' conjecture on the distribution of prime divisors of Mersenne numbers.
• Landau's problems
• Goldbach conjecture: all even natural numbers greater than ${\displaystyle 2}$ are the sum of two prime numbers.
• Legendre's conjecture: for every positive integer ${\displaystyle n}$, there is a prime between ${\displaystyle n^{2))$ and ${\displaystyle (n+1)^{2))$.
• Twin prime conjecture: there are infinitely many twin primes.
• Are there infinitely many primes of the form ${\displaystyle n^{2}+1}$?
• Problems associated to Linnik's theorem
• New Mersenne conjecture: for any odd natural number ${\displaystyle p}$, if any two of the three conditions ${\displaystyle p=2^{k}\pm 1}$ or ${\displaystyle p=4^{k}\pm 3}$, ${\displaystyle 2^{p}-1}$ is prime, and ${\displaystyle (2^{p}+1)/3}$ is prime are true, then the third condition is true.
• Polignac's conjecture: for all positive even numbers ${\displaystyle n}$, there are infinitely many prime gaps of size ${\displaystyle n}$.
• Schinzel's hypothesis H that for every finite collection ${\displaystyle \{f_{1},\ldots ,f_{k}\))$ of nonconstant irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers ${\displaystyle n}$ for which ${\displaystyle f_{1}(n),\ldots ,f_{k}(n)}$ are all primes, or there is some fixed divisor ${\displaystyle m>1}$ which, for all ${\displaystyle n}$, divides some ${\displaystyle f_{i}(n)}$.
• Selfridge's conjecture: is 78,557 the lowest Sierpiński number?
• Does the converse of Wolstenholme's theorem hold for all natural numbers?

### Set theory

 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.

### Topology

 Main article: Topology

## Notes

1. ^ An aperiodic monotile has been discovered and the formal proof is awaiting publication. A preprint of the proof is available.[72]
2. ^ A disproof has been announced, with a preprint made available on arXiv.[159]

## References

1. ^ Thiele, Rüdiger (2005), "On Hilbert and his twenty-four problems", in Van Brummelen, Glen (ed.), Mathematics and the historian's craft. The Kenneth O. May Lectures, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 21, pp. 243–295, ISBN 978-0-387-25284-1
2. ^ Guy, Richard (1994), Unsolved Problems in Number Theory (2nd ed.), Springer, p. vii, ISBN 978-1-4899-3585-4, archived from the original on 2019-03-23, retrieved 2016-09-22.
3. ^ Shimura, G. (1989). "Yutaka Taniyama and his time". Bulletin of the London Mathematical Society. 21 (2): 186–196. doi:10.1112/blms/21.2.186.
4. ^ Friedl, Stefan (2014). "Thurston's vision and the virtual fibering theorem for 3-manifolds". Jahresbericht der Deutschen Mathematiker-Vereinigung. 116 (4): 223–241. doi:10.1365/s13291-014-0102-x. MR 3280572. S2CID 56322745.
5. ^ Thurston, William P. (1982). "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. New Series. 6 (3): 357–381. doi:10.1090/S0273-0979-1982-15003-0. MR 0648524.
6. ^ a b "Millennium Problems". claymath.org. Archived from the original on 2017-06-06. Retrieved 2015-01-20.
7. ^ "Fields Medal awarded to Artur Avila". Centre national de la recherche scientifique. 2014-08-13. Archived from the original on 2018-07-10. Retrieved 2018-07-07.
8. ^ Bellos, Alex (2014-08-13). "Fields Medals 2014: the maths of Avila, Bhargava, Hairer and Mirzakhani explained". The Guardian. Archived from the original on 2016-10-21. Retrieved 2018-07-07.
9. ^ Abe, Jair Minoro; Tanaka, Shotaro (2001). Unsolved Problems on Mathematics for the 21st Century. IOS Press. ISBN 978-90-5199-490-2.
10. ^ "DARPA invests in math". CNN. 2008-10-14. Archived from the original on 2009-03-04. Retrieved 2013-01-14.
11. ^ "Broad Agency Announcement (BAA 07-68) for Defense Sciences Office (DSO)". DARPA. 2007-09-10. Archived from the original on 2012-10-01. Retrieved 2013-06-25.
12. ^ "Poincaré Conjecture". Clay Mathematics Institute. Archived from the original on 2013-12-15.
13. ^ rybu (November 7, 2009). "Smooth 4-dimensional Poincare conjecture". Open Problem Garden. Archived from the original on 2018-01-25. Retrieved 2019-08-06.
14. ^ Khukhro, Evgeny I.; Mazurov, Victor D. (2019), Unsolved Problems in Group Theory. The Kourovka Notebook, arXiv:1401.0300v16
15. ^ RSFSR, MV i SSO; Russie), Uralʹskij gosudarstvennyj universitet im A. M. Gorʹkogo (Ekaterinbourg (1969). Свердловская тетрадь: нерешенные задачи теории подгрупп (in Russian). S. l.
16. ^ Свердловская тетрадь: Сб. нерешённых задач по теории полугрупп. Свердловск: Уральский государственный университет. 1979.
17. ^ Свердловская тетрадь: Сб. нерешённых задач по теории полугрупп. Свердловск: Уральский государственный университет. 1989.
18. ^ ДНЕСТРОВСКАЯ ТЕТРАДЬ [DNIESTER NOTEBOOK] (PDF) (in Russian), The Russian Academy of Sciences, 1993
19. ^ "DNIESTER NOTEBOOK: Unsolved Problems in the Theory of Rings and Modules" (PDF), University of Saskatchewan, retrieved 2019-08-15
20. ^ Эрлагольская тетрадь [Erlagol notebook] (PDF) (in Russian), The Novosibirsk State University, 2018
21. ^ Dowling, T. A. (February 1973). "A class of geometric lattices based on finite groups". Journal of Combinatorial Theory. Series B. 14 (1): 61–86. doi:10.1016/S0095-8956(73)80007-3.
22. ^ Aschbacher, Michael (1990), "On Conjectures of Guralnick and Thompson", Journal of Algebra, 135 (2): 277–343, doi:10.1016/0021-8693(90)90292-V
23. ^ Kung, H. T.; Traub, Joseph Frederick (1974), "Optimal order of one-point and multipoint iteration", Journal of the ACM, 21 (4): 643–651, doi:10.1145/321850.321860, S2CID 74921
24. ^ Smyth, Chris (2008), "The Mahler measure of algebraic numbers: a survey", in McKee, James; Smyth, Chris (eds.), Number Theory and Polynomials, London Mathematical Society Lecture Note Series, vol. 352, Cambridge University Press, pp. 322–349, ISBN 978-0-521-71467-9
25. ^ Berenstein, Carlos A. (2001) [1994], "Pompeiu problem", Encyclopedia of Mathematics, EMS Press
26. ^ a b Waldschmidt, Michel (2013), Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables, Springer, pp. 14, 16, ISBN 978-3-662-11569-5
27. ^ For some background on the numbers in this problem, see articles by Eric W. Weisstein at Wolfram MathWorld (all articles accessed 15 December 2014):
28. ^ Waldschmidt, Michel (2008). An introduction to irrationality and transcendence methods (PDF). 2008 Arizona Winter School. Archived from the original (PDF) on 16 December 2014. Retrieved 15 December 2014.
29. ^ Albert, John, Some unsolved problems in number theory (PDF), archived from the original (PDF) on 17 January 2014, retrieved 15 December 2014
30. ^ Brightwell, Graham R.; Felsner, Stefan; Trotter, William T. (1995), "Balancing pairs and the cross product conjecture", Order, 12 (4): 327–349, CiteSeerX 10.1.1.38.7841, doi:10.1007/BF01110378, MR 1368815, S2CID 14793475.
31. ^ Tao, Terence (2018). "Some remarks on the lonely runner conjecture". Contributions to Discrete Mathematics. 13 (2): 1–31. arXiv:1701.02048. doi:10.11575/cdm.v13i2.62728.
32. ^ González-Jiménez, Enrique; Xarles, Xavier (2014). "On a conjecture of Rudin on squares in arithmetic progressions". LMS Journal of Computation and Mathematics. 17 (1): 58–76. arXiv:1301.5122. doi:10.1112/S1461157013000259. S2CID 11615385.
33. ^ Bruhn, Henning; Schaudt, Oliver (2015), "The journey of the union-closed sets conjecture" (PDF), Graphs and Combinatorics, 31 (6): 2043–2074, arXiv:1309.3297, doi:10.1007/s00373-014-1515-0, MR 3417215, S2CID 17531822, archived (PDF) from the original on 2017-08-08, retrieved 2017-07-18
34. ^ Murnaghan, F. D. (1938), "The Analysis of the Direct Product of Irreducible Representations of the Symmetric Groups", American Journal of Mathematics, 60 (1): 44–65, doi:10.2307/2371542, JSTOR 2371542, MR 1507301, PMC 1076971, PMID 16577800
35. ^ "Dedekind Numbers and Related Sequences" (PDF). Archived from the original (PDF) on 2015-03-15. Retrieved 2020-04-30.
36. ^ 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. doi:10.1016/S0304-3975(03)00080-X. S2CID 33806100.
37. ^ S. M. Ulam, Problems in Modern Mathematics. Science Editions John Wiley & Sons, Inc., New York, 1964, page 76.
38. ^ Kaloshin, Vadim; Sorrentino, Alfonso (2018). "On the local Birkhoff conjecture for convex billiards". Annals of Mathematics. 188 (1): 315–380. arXiv:1612.09194. doi:10.4007/annals.2018.188.1.6. S2CID 119171182.
39. ^ Sarnak, Peter (2011), "Recent progress on the quantum unique ergodicity conjecture", Bulletin of the American Mathematical Society, 48 (2): 211–228, doi:10.1090/S0273-0979-2011-01323-4, MR 2774090
40. ^ Paul Halmos, Ergodic theory. Chelsea, New York, 1956.
41. ^ Kari, Jarkko (2009). "Structure of reversible cellular automata". Structure of Reversible Cellular Automata. International Conference on Unconventional Computation. Lecture Notes in Computer Science. Vol. 5715. Springer. p. 6. Bibcode:2009LNCS.5715....6K. doi:10.1007/978-3-642-03745-0_5. ISBN 978-3-642-03744-3.
42. ^ a b c "Open Q - Solving and rating of hard Sudoku". english.log-it-ex.com. Archived from the original on 10 November 2017.
43. ^ "Higher-Dimensional Tic-Tac-Toe". PBS Infinite Series. YouTube. 2017-09-21. Archived from the original on 2017-10-11. Retrieved 2018-07-29.
44. ^ Barlet, Daniel; Peternell, Thomas; Schneider, Michael (1990). "On two conjectures of Hartshorne's". Mathematische Annalen. 286 (1–3): 13–25. doi:10.1007/BF01453563. S2CID 122151259.
45. ^ Maulik, Davesh; Nekrasov, Nikita; Okounov, Andrei; Pandharipande, Rahul (2004-06-05), Gromov–Witten theory and Donaldson–Thomas theory, I, arXiv:math/0312059, Bibcode:2003math.....12059M
46. ^
47. ^ Bereg, Sergey; Dumitrescu, Adrian; Jiang, Minghui (2010), "On covering problems of Rado", Algorithmica, 57 (3): 538–561, doi:10.1007/s00453-009-9298-z, MR 2609053, S2CID 6511998
48. ^ Melissen, Hans (1993), "Densest packings of congruent circles in an equilateral triangle", American Mathematical Monthly, 100 (10): 916–925, doi:10.2307/2324212, JSTOR 2324212, MR 1252928
49. ^ Conway, John H.; Neil J.A. Sloane (1999), Sphere Packings, Lattices and Groups (3rd ed.), New York: Springer-Verlag, pp. 21–22, ISBN 978-0-387-98585-5
50. ^ Hales, Thomas (2017), The Reinhardt conjecture as an optimal control problem, arXiv:1703.01352
51. ^ Brass, Peter; Moser, William; Pach, János (2005), Research Problems in Discrete Geometry, New York: Springer, p. 45, ISBN 978-0387-23815-9, MR 2163782
52. ^ Gardner, Martin (1995), New Mathematical Diversions (Revised Edition), Washington: Mathematical Association of America, p. 251
53. ^ Barros, Manuel (1997), "General Helices and a Theorem of Lancret", Proceedings of the American Mathematical Society, 125 (5): 1503–1509, doi:10.1090/S0002-9939-97-03692-7, JSTOR 2162098
54. ^ Katz, Mikhail G. (2007), Systolic geometry and topology, Mathematical Surveys and Monographs, vol. 137, American Mathematical Society, Providence, RI, p. 57, doi:10.1090/surv/137, ISBN 978-0-8218-4177-8, MR 2292367
55. ^ Rosenberg, Steven (1997), The Laplacian on a Riemannian Manifold: An introduction to analysis on manifolds, London Mathematical Society Student Texts, vol. 31, Cambridge: Cambridge University Press, pp. 62–63, doi:10.1017/CBO9780511623783, ISBN 978-0-521-46300-3, MR 1462892
56. ^ Ghosh, Subir Kumar; Goswami, Partha P. (2013), "Unsolved problems in visibility graphs of points, segments, and polygons", ACM Computing Surveys, 46 (2): 22:1–22:29, arXiv:1012.5187, doi:10.1145/2543581.2543589, S2CID 8747335
57. ^ Boltjansky, V.; Gohberg, I. (1985), "11. Hadwiger's Conjecture", Results and Problems in Combinatorial Geometry, Cambridge University Press, pp. 44–46.
58. ^ Morris, Walter D.; Soltan, Valeriu (2000), "The Erdős-Szekeres problem on points in convex position—a survey", Bull. Amer. Math. Soc., 37 (4): 437–458, doi:10.1090/S0273-0979-00-00877-6, MR 1779413; Suk, Andrew (2016), "On the Erdős–Szekeres convex polygon problem", J. Amer. Math. Soc., 30 (4): 1047–1053, arXiv:1604.08657, doi:10.1090/jams/869, S2CID 15732134
59. ^ Kalai, Gil (1989), "The number of faces of centrally-symmetric polytopes", Graphs and Combinatorics, 5 (1): 389–391, doi:10.1007/BF01788696, MR 1554357, S2CID 8917264.
60. ^ Moreno, José Pedro; Prieto-Martínez, Luis Felipe (2021). "El problema de los triángulos de Kobon" [The Kobon triangles problem]. La Gaceta de la Real Sociedad Matemática Española (in Spanish). 24 (1): 111–130. hdl:10486/705416. MR 4225268.
61. ^ Guy, Richard K. (1983), "An olla-podrida of open problems, often oddly posed", American Mathematical Monthly, 90 (3): 196–200, doi:10.2307/2975549, JSTOR 2975549, MR 1540158
62. ^ Matoušek, Jiří (2002), Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, p. 206, doi:10.1007/978-1-4613-0039-7, ISBN 978-0-387-95373-1, MR 1899299
63. ^ Brass, Peter; Moser, William; Pach, János (2005), "5.1 The Maximum Number of Unit Distances in the Plane", Research problems in discrete geometry, Springer, New York, pp. 183–190, ISBN 978-0-387-23815-9, MR 2163782
64. ^ Dey, Tamal K. (1998), "Improved bounds for planar k-sets and related problems", Discrete & Computational Geometry, 19 (3): 373–382, doi:10.1007/PL00009354, MR 1608878; Tóth, Gábor (2001), "Point sets with many k-sets", Discrete & Computational Geometry, 26 (2): 187–194, doi:10.1007/s004540010022, MR 1843435.
65. ^ Aronov, Boris; Dujmović, Vida; Morin, Pat; Ooms, Aurélien; Schultz Xavier da Silveira, Luís Fernando (2019), "More Turán-type theorems for triangles in convex point sets", Electronic Journal of Combinatorics, 26 (1): P1.8, arXiv:1706.10193, Bibcode:2017arXiv170610193A, doi:10.37236/7224, archived from the original on 2019-02-18, retrieved 2019-02-18
66. ^ Atiyah, Michael (2001), "Configurations of points", Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 359 (1784): 1375–1387, Bibcode:2001RSPTA.359.1375A, doi:10.1098/rsta.2001.0840, ISSN 1364-503X, MR 1853626, S2CID 55833332
67. ^ Finch, S. R.; Wetzel, J. E. (2004), "Lost in a forest", American Mathematical Monthly, 11 (8): 645–654, doi:10.2307/4145038, JSTOR 4145038, MR 2091541
68. ^ Howards, Hugh Nelson (2013), "Forming the Borromean rings out of arbitrary polygonal unknots", Journal of Knot Theory and Its Ramifications, 22 (14): 1350083, 15, arXiv:1406.3370, doi:10.1142/S0218216513500831, MR 3190121, S2CID 119674622
69. ^ Solomon, Yaar; Weiss, Barak (2016), "Dense forests and Danzer sets", Annales Scientifiques de l'École Normale Supérieure, 49 (5): 1053–1074, arXiv:1406.3807, doi:10.24033/asens.2303, MR 3581810, S2CID 672315; Conway, John H., Five $1,000 Problems (Update 2017) (PDF), On-Line Encyclopedia of Integer Sequences, archived (PDF) from the original on 2019-02-13, retrieved 2019-02-12 70. ^ Brandts, Jan; Korotov, Sergey; Křížek, Michal; Šolc, Jakub (2009), "On nonobtuse simplicial partitions" (PDF), SIAM Review, 51 (2): 317–335, Bibcode:2009SIAMR..51..317B, doi:10.1137/060669073, MR 2505583, S2CID 216078793, archived (PDF) from the original on 2018-11-04, retrieved 2018-11-22. See in particular Conjecture 23, p. 327. 71. ^ Socolar, Joshua E. S.; Taylor, Joan M. (2012), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer, 34 (1): 18–28, arXiv:1009.1419, doi:10.1007/s00283-011-9255-y, MR 2902144, S2CID 10747746 72. ^ Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (May 28, 2023). "A chiral aperiodic monotile". arXiv:2305.17743 [math.CO]. 73. ^ Arutyunyants, G.; Iosevich, A. (2004), "Falconer conjecture, spherical averages and discrete analogs", in Pach, János (ed.), Towards a Theory of Geometric Graphs, Contemp. Math., vol. 342, Amer. Math. Soc., Providence, RI, pp. 15–24, doi:10.1090/conm/342/06127, ISBN 978-0-8218-3484-8, MR 2065249 74. ^ Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society, 61 (4): 346–352, doi:10.1090/noti1100 75. ^ Katz, Nets; Tao, Terence (2002), "Recent progress on the Kakeya conjecture", Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), Publicacions Matemàtiques, pp. 161–179, CiteSeerX 10.1.1.241.5335, doi:10.5565/PUBLMAT_Esco02_07, MR 1964819, S2CID 77088 76. ^ Weaire, Denis, ed. (1997), The Kelvin Problem, CRC Press, p. 1, ISBN 978-0-7484-0632-6 77. ^ Brass, Peter; Moser, William; Pach, János (2005), Research problems in discrete geometry, New York: Springer, p. 457, ISBN 978-0-387-29929-7, MR 2163782 78. ^ Mahler, Kurt (1939). "Ein Minimalproblem für konvexe Polygone". Mathematica (Zutphen) B: 118–127. 79. ^ Norwood, Rick; Poole, George; Laidacker, Michael (1992), "The worm problem of Leo Moser", Discrete & Computational Geometry, 7 (2): 153–162, doi:10.1007/BF02187832, MR 1139077 80. ^ Wagner, Neal R. (1976), "The Sofa Problem" (PDF), The American Mathematical Monthly, 83 (3): 188–189, doi:10.2307/2977022, JSTOR 2977022, archived (PDF) from the original on 2015-04-20, retrieved 2014-05-14 81. ^ Chai, Ying; Yuan, Liping; Zamfirescu, Tudor (June–July 2018), "Rupert Property of Archimedean Solids", The American Mathematical Monthly, 125 (6): 497–504, doi:10.1080/00029890.2018.1449505, S2CID 125508192 82. ^ Steininger, Jakob; Yurkevich, Sergey (December 27, 2021), An algorithmic approach to Rupert's problem, arXiv:2112.13754 83. ^ Demaine, Erik D.; O'Rourke, Joseph (2007), "Chapter 22. Edge Unfolding of Polyhedra", Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, pp. 306–338 84. ^ Ghomi, Mohammad (2018-01-01). "Dürer's Unfolding Problem for Convex Polyhedra". Notices of the American Mathematical Society. 65 (1): 25–27. doi:10.1090/noti1609. ISSN 0002-9920. 85. ^ Whyte, L. L. (1952), "Unique arrangements of points on a sphere", The American Mathematical Monthly, 59 (9): 606–611, doi:10.2307/2306764, JSTOR 2306764, MR 0050303 86. ^ ACW (May 24, 2012), "Convex uniform 5-polytopes", Open Problem Garden, archived from the original on October 5, 2016, retrieved 2016-10-04. 87. ^ Pleanmani, Nopparat (2019), "Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph", Discrete Mathematics, Algorithms and Applications, 11 (6): 1950068, 7, doi:10.1142/s179383091950068x, MR 4044549, S2CID 204207428 88. ^ Baird, William; Bonato, Anthony (2012), "Meyniel's conjecture on the cop number: a survey", Journal of Combinatorics, 3 (2): 225–238, arXiv:1308.3385, doi:10.4310/JOC.2012.v3.n2.a6, MR 2980752, S2CID 18942362 89. ^ Bousquet, Nicolas; Bartier, Valentin (2019), "Linear Transformations Between Colorings in Chordal Graphs", in Bender, Michael A.; Svensson, Ola; Herman, Grzegorz (eds.), 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, LIPIcs, vol. 144, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 24:1–24:15, doi:10.4230/LIPIcs.ESA.2019.24, ISBN 978-3-95977-124-5, S2CID 195791634 90. ^ Gethner, Ellen (2018), "To the Moon and beyond", in Gera, Ralucca; Haynes, Teresa W.; Hedetniemi, Stephen T. (eds.), Graph Theory: Favorite Conjectures and Open Problems, II, Problem Books in Mathematics, Springer International Publishing, pp. 115–133, doi:10.1007/978-3-319-97686-0_11, ISBN 978-3-319-97684-6, MR 3930641 91. ^ Chung, Fan; Graham, Ron (1998), Erdős on Graphs: His Legacy of Unsolved Problems, A K Peters, pp. 97–99. 92. ^ Chudnovsky, Maria; Seymour, Paul (2014), "Extending the Gyárfás-Sumner conjecture", Journal of Combinatorial Theory, Series B, 105: 11–16, doi:10.1016/j.jctb.2013.11.002, MR 3171779 93. ^ Toft, Bjarne (1996), "A survey of Hadwiger's conjecture", Congressus Numerantium, 115: 249–283, MR 1411244. 94. ^ Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991), Unsolved Problems in Geometry, Springer-Verlag, Problem G10. 95. ^ Hägglund, Jonas; Steffen, Eckhard (2014), "Petersen-colorings and some families of snarks", Ars Mathematica Contemporanea, 7 (1): 161–173, doi:10.26493/1855-3974.288.11a, MR 3047618, archived from the original on 2016-10-03, retrieved 2016-09-30. 96. ^ Jensen, Tommy R.; Toft, Bjarne (1995), "12.20 List-Edge-Chromatic Numbers", Graph Coloring Problems, New York: Wiley-Interscience, pp. 201–202, ISBN 978-0-471-02865-9. 97. ^ Molloy, Michael; Reed, Bruce (1998), "A bound on the total chromatic number", Combinatorica, 18 (2): 241–280, CiteSeerX 10.1.1.24.6514, doi:10.1007/PL00009820, MR 1656544, S2CID 9600550. 98. ^ Barát, János; Tóth, Géza (2010), "Towards the Albertson Conjecture", Electronic Journal of Combinatorics, 17 (1): R73, arXiv:0909.0413, Bibcode:2009arXiv0909.0413B, doi:10.37236/345. 99. ^ Fulek, Radoslav; Pach, János (2011), "A computational approach to Conway's thrackle conjecture", Computational Geometry, 44 (6–7): 345–355, arXiv:1002.3904, doi:10.1016/j.comgeo.2011.02.001, MR 2785903. 100. ^ Gupta, Anupam; Newman, Ilan; Rabinovich, Yuri; Sinclair, Alistair (2004), "Cuts, trees and ${\displaystyle \ell _{1))$-embeddings of graphs", Combinatorica, 24 (2): 233–269, CiteSeerX 10.1.1.698.8978, doi:10.1007/s00493-004-0015-x, MR 2071334, S2CID 46133408 101. ^ Hartsfield, Nora; Ringel, Gerhard (2013), Pearls in Graph Theory: A Comprehensive Introduction, Dover Books on Mathematics, Courier Dover Publications, p. 247, ISBN 978-0-486-31552-2, MR 2047103. 102. ^ Hliněný, Petr (2010), "20 years of Negami's planar cover conjecture" (PDF), Graphs and Combinatorics, 26 (4): 525–536, CiteSeerX 10.1.1.605.4932, doi:10.1007/s00373-010-0934-9, MR 2669457, S2CID 121645, archived (PDF) from the original on 2016-03-04, retrieved 2016-10-04. 103. ^ Nöllenburg, Martin; Prutkin, Roman; Rutter, Ignaz (2016), "On self-approaching and increasing-chord drawings of 3-connected planar graphs", Journal of Computational Geometry, 7 (1): 47–69, arXiv:1409.0315, doi:10.20382/jocg.v7i1a3, MR 3463906, S2CID 1500695 104. ^ Pach, János; Sharir, Micha (2009), "5.1 Crossings—the Brick Factory Problem", Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures, Mathematical Surveys and Monographs, vol. 152, American Mathematical Society, pp. 126–127. 105. ^ Demaine, E.; O'Rourke, J. (2002–2012), "Problem 45: Smallest Universal Set of Points for Planar Graphs", The Open Problems Project, archived from the original on 2012-08-14, retrieved 2013-03-19. 106. ^ Conway, John H., Five$1,000 Problems (Update 2017) (PDF), Online Encyclopedia of Integer Sequences, archived (PDF) from the original on 2019-02-13, retrieved 2019-02-12
107. ^ mdevos; Wood, David (December 7, 2019), "Jorgensen's Conjecture", Open Problem Garden, archived from the original on 2016-11-14, retrieved 2016-11-13.
108. ^ Ducey, Joshua E. (2017), "On the critical group of the missing Moore graph", Discrete Mathematics, 340 (5): 1104–1109, arXiv:1509.00327, doi:10.1016/j.disc.2016.10.001, MR 3612450, S2CID 28297244
109. ^ Blokhuis, A.; Brouwer, A. E. (1988), "Geodetic graphs of diameter two", Geometriae Dedicata, 25 (1–3): 527–533, doi:10.1007/BF00191941, MR 0925851, S2CID 189890651
110. ^ Florek, Jan (2010), "On Barnette's conjecture", Discrete Mathematics, 310 (10–11): 1531–1535, doi:10.1016/j.disc.2010.01.018, MR 2601261.
111. ^ Broersma, Hajo; Patel, Viresh; Pyatkin, Artem (2014), "On toughness and Hamiltonicity of $2K_2$-free graphs" (PDF), Journal of Graph Theory, 75 (3): 244–255, doi:10.1002/jgt.21734, MR 3153119, S2CID 1377980
112. ^ Jaeger, F. (1985), "A survey of the cycle double cover conjecture", Annals of Discrete Mathematics 27 – Cycles in Graphs, North-Holland Mathematics Studies, vol. 27, pp. 1–12, doi:10.1016/S0304-0208(08)72993-1, ISBN 978-0-444-87803-8.
113. ^ Heckman, Christopher Carl; Krakovski, Roi (2013), "Erdös-Gyárfás conjecture for cubic planar graphs", Electronic Journal of Combinatorics, 20 (2), P7, doi:10.37236/3252.
114. ^ Chudnovsky, Maria (2014), "The Erdös–Hajnal conjecture—a survey" (PDF), Journal of Graph Theory, 75 (2): 178–190, arXiv:1606.08827, doi:10.1002/jgt.21730, MR 3150572, S2CID 985458, Zbl 1280.05086, archived (PDF) from the original on 2016-03-04, retrieved 2016-09-22.
115. ^ Akiyama, Jin; Exoo, Geoffrey; Harary, Frank (1981), "Covering and packing in graphs. IV. Linear arboricity", Networks, 11 (1): 69–72, doi:10.1002/net.3230110108, MR 0608921.
116. ^ Babai, László (June 9, 1994). "Automorphism groups, isomorphism, reconstruction". Handbook of Combinatorics. Archived from the original (PostScript) on 13 June 2007.
117. ^ Lenz, Hanfried; Ringel, Gerhard (1991), "A brief review on Egmont Köhler's mathematical work", Discrete Mathematics, 97 (1–3): 3–16, doi:10.1016/0012-365X(91)90416-Y, MR 1140782
118. ^ Fomin, Fedor V.; Høie, Kjartan (2006), "Pathwidth of cubic graphs and exact algorithms", Information Processing Letters, 97 (5): 191–196, doi:10.1016/j.ipl.2005.10.012, MR 2195217
119. ^ Schwenk, Allen (2012). Some History on the Reconstruction Conjecture (PDF). Joint Mathematics Meetings. Archived from the original (PDF) on 2015-04-09. Retrieved 2018-11-26.
120. ^ Ramachandran, S. (1981), "On a new digraph reconstruction conjecture", Journal of Combinatorial Theory, Series B, 31 (2): 143–149, doi:10.1016/S0095-8956(81)80019-6, MR 0630977
121. ^ Kühn, Daniela; Mycroft, Richard; Osthus, Deryk (2011), "A proof of Sumner's universal tournament conjecture for large tournaments", Proceedings of the London Mathematical Society, Third Series, 102 (4): 731–766, arXiv:1010.4430, doi:10.1112/plms/pdq035, MR 2793448, S2CID 119169562, Zbl 1218.05034.
122. ^ Tuza, Zsolt (1990). "A conjecture on triangles of graphs". Graphs and Combinatorics. 6 (4): 373–380. doi:10.1007/BF01787705. MR 1092587. S2CID 38821128.
123. ^ Brešar, Boštjan; Dorbec, Paul; Goddard, Wayne; Hartnell, Bert L.; Henning, Michael A.; Klavžar, Sandi; Rall, Douglas F. (2012), "Vizing's conjecture: a survey and recent results", Journal of Graph Theory, 69 (1): 46–76, CiteSeerX 10.1.1.159.7029, doi:10.1002/jgt.20565, MR 2864622, S2CID 9120720.
124. Kitaev, Sergey; Lozin, Vadim (2015). Words and Graphs. Monographs in Theoretical Computer Science. An EATCS Series. doi:10.1007/978-3-319-25859-1. ISBN 978-3-319-25857-7. S2CID 7727433 – via link.springer.com.
125. Kitaev, Sergey (2017-05-16). A Comprehensive Introduction to the Theory of Word-Representable Graphs. International Conference on Developments in Language Theory. arXiv:1705.05924v1. doi:10.1007/978-3-319-62809-7_2.
126. Kitaev, S. V.; Pyatkin, A. V. (April 1, 2018). "Word-Representable Graphs: a Survey". Journal of Applied and Industrial Mathematics. 12 (2): 278–296. doi:10.1134/S1990478918020084. S2CID 125814097 – via Springer Link.
127. Kitaev, Sergey V.; Pyatkin, Artem V. (2018). "Графы, представимые в виде слов. Обзор результатов" [Word-representable graphs: A survey]. Дискретн. анализ и исслед. опер. (in Russian). 25 (2): 19–53. doi:10.17377/daio.2018.25.588.
128. ^ Marc Elliot Glen (2016). "Colourability and word-representability of near-triangulations". arXiv:1605.01688 [math.CO].
129. ^ Kitaev, Sergey (2014-03-06). "On graphs with representation number 3". arXiv:1403.1616v1 [math.CO].
130. ^ Glen, Marc; Kitaev, Sergey; Pyatkin, Artem (2018). "On the representation number of a crown graph". Discrete Applied Mathematics. 244: 89–93. arXiv:1609.00674. doi:10.1016/j.dam.2018.03.013. S2CID 46925617.
131. ^ Spinrad, Jeremy P. (2003), "2. Implicit graph representation", Efficient Graph Representations, American Mathematical Soc., pp. 17–30, ISBN 978-0-8218-2815-1.
132. ^ "Seymour's 2nd Neighborhood Conjecture". faculty.math.illinois.edu. Archived from the original on 11 January 2019. Retrieved 17 August 2022.
133. ^ mdevos (May 4, 2007). "5-flow conjecture". Open Problem Garden. Archived from the original on November 26, 2018.
134. ^ mdevos (March 31, 2010). "4-flow conjecture". Open Problem Garden. Archived from the original on November 26, 2018.
135. ^ Hrushovski, Ehud (1989). "Kueker's conjecture for stable theories". Journal of Symbolic Logic. 54 (1): 207–220. doi:10.2307/2275025. JSTOR 2275025. S2CID 41940041.
136. ^ a b c Shelah S (1990). Classification Theory. North-Holland.
137. ^ Shelah, Saharon (2009). Classification theory for abstract elementary classes. College Publications. ISBN 978-1-904987-71-0.
138. ^ Peretz, Assaf (2006). "Geometry of forking in simple theories". Journal of Symbolic Logic. 71 (1): 347–359. arXiv:math/0412356. doi:10.2178/jsl/1140641179. S2CID 9380215.
139. ^ Cherlin, Gregory; Shelah, Saharon (May 2007). "Universal graphs with a forbidden subtree". Journal of Combinatorial Theory. Series B. 97 (3): 293–333. arXiv:math/0512218. doi:10.1016/j.jctb.2006.05.008. S2CID 10425739.
140. ^ Džamonja, Mirna, "Club guessing and the universal models." On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
141. ^ Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae. 159 (1): 1–50. arXiv:math/9802134. Bibcode:1998math......2134S. doi:10.4064/fm-159-1-1-50. S2CID 8846429.
142. ^ Baldwin, John T. (July 24, 2009). Categoricity (PDF). American Mathematical Society. ISBN 978-0-8218-4893-7. Archived (PDF) from the original on July 29, 2010. Retrieved February 20, 2014.
143. ^ Shelah, Saharon (2009). "Introduction to classification theory for abstract elementary classes". arXiv:0903.3428 [math.LO].
144. ^ Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
145. ^ Makowsky J, "Compactness, embeddings and definability," in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
146. ^ Keisler, HJ (1967). "Ultraproducts which are not saturated". J. Symb. Log. 32 (1): 23–46. doi:10.2307/2271240. JSTOR 2271240. S2CID 250345806.
147. ^ Malliaris, Maryanthe; Shelah, Saharon (10 August 2012). "A Dividing Line Within Simple Unstable Theories". arXiv:1208.2140 [math.LO]. Malliaris, M.; Shelah, S. (2012). "A Dividing Line within Simple Unstable Theories". arXiv:1208.2140 [math.LO].
148. ^ Conrey, Brian (2016), "Lectures on the Riemann zeta function (book review)", Bulletin of the American Mathematical Society, 53 (3): 507–512, doi:10.1090/bull/1525
149. ^ Singmaster, David (1971), "Research Problems: How often does an integer occur as a binomial coefficient?", American Mathematical Monthly, 78 (4): 385–386, doi:10.2307/2316907, JSTOR 2316907, MR 1536288.
150. ^ Guo, Song; Sun, Zhi-Wei (2005), "On odd covering systems with distinct moduli", Advances in Applied Mathematics, 35 (2): 182–187, arXiv:math/0412217, doi:10.1016/j.aam.2005.01.004, MR 2152886, S2CID 835158
151. ^ "Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key". Archived from the original on 2016-03-27. Retrieved 2016-03-18.
152. ^ Robertson, John P. (1996-10-01). "Magic Squares of Squares". Mathematics Magazine. 69 (4): 289–293. doi:10.1080/0025570X.1996.11996457. ISSN 0025-570X.
153. ^ Aigner, Martin (2013), Markov's theorem and 100 years of the uniqueness conjecture, Cham: Springer, doi:10.1007/978-3-319-00888-2, ISBN 978-3-319-00887-5, MR 3098784
154. ^ Huisman, Sander G. (2016). "Newer sums of three cubes". arXiv:1604.07746 [math.NT].
155. ^ Dobson, J. B. (1 April 2017), "On Lerch's formula for the Fermat quotient", p. 23, arXiv:1103.3907v6 [math.NT]
156. ^ Ribenboim, P. (2006). Die Welt der Primzahlen. Springer-Lehrbuch (in German) (2nd ed.). Springer. pp. 242–243. doi:10.1007/978-3-642-18079-8. ISBN 978-3-642-18078-1.
157. ^ Mazur, Barry (1992), "The topology of rational points", Experimental Mathematics, 1 (1): 35–45, doi:10.1080/10586458.1992.10504244, S2CID 17372107, archived from the original on 2019-04-07, retrieved 2019-04-07
158. ^ Kuperberg, Greg (1994), "Quadrisecants of knots and links", Journal of Knot Theory and Its Ramifications, 3: 41–50, arXiv:math/9712205, doi:10.1142/S021821659400006X, MR 1265452, S2CID 6103528
159. ^ Burklund, Robert; Hahn, Jeremy; Levy, Ishan; Schlank, Tomer (2023). "K-theoretic counterexamples to Ravenel's telescope conjecture". arXiv:2310.17459 [math.AT].
160. ^ Dimitrov, Vessilin; Gao, Ziyang; Habegger, Philipp (2021). "Uniformity in Mordell–Lang for curves" (PDF). Annals of Mathematics. 194: 237–298. arXiv:2001.10276. doi:10.4007/annals.2021.194.1.4. S2CID 210932420.
161. ^ Guan, Qi'an; Zhou, Xiangyu (2015). "A solution of an ${\displaystyle L^{2))$ extension problem with optimal estimate and applications". Annals of Mathematics. 181 (3): 1139–1208. arXiv:1310.7169. doi:10.4007/annals.2015.181.3.6. JSTOR 24523356. S2CID 56205818.
162. ^ 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. Bibcode:1996InMat.124..437M. doi:10.1007/s002220050059. MR 1369424. S2CID 3590991.
163. ^ Cohen, Stephen D.; Fried, Michael D. (1995), "Lenstra's proof of the Carlitz–Wan conjecture on exceptional polynomials: an elementary version", Finite Fields and Their Applications, 1 (3): 372–375, doi:10.1006/ffta.1995.1027, MR 1341953
164. ^ Casazza, Peter G.; Fickus, Matthew; Tremain, Janet C.; Weber, Eric (2006). "The Kadison-Singer problem in mathematics and engineering: A detailed account". In Han, Deguang; Jorgensen, Palle E. T.; Larson, David Royal (eds.). Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7–12, 2005, University of Central Florida, Florida. Contemporary Mathematics. Vol. 414. American Mathematical Society. pp. 299–355. doi:10.1090/conm/414/07820. ISBN 978-0-8218-3923-2. Retrieved 24 April 2015.
165. ^ Mackenzie, Dana. "Kadison–Singer Problem Solved" (PDF). SIAM News. No. January/February 2014. Society for Industrial and Applied Mathematics. Archived (PDF) from the original on 23 October 2014. Retrieved 24 April 2015.
166. ^ a b Agol, Ian (2004). "Tameness of hyperbolic 3-manifolds". arXiv:math/0405568.
167. ^ Kurdyka, Krzysztof; Mostowski, Tadeusz; Parusiński, Adam (2000). "Proof of the gradient conjecture of R. Thom". Annals of Mathematics. 152 (3): 763–792. arXiv:math/9906212. doi:10.2307/2661354. JSTOR 2661354. S2CID 119137528.
168. ^ Moreira, Joel; Richter, Florian K.; Robertson, Donald (2019). "A proof of a sumset conjecture of Erdős". Annals of Mathematics. 189 (2): 605–652. arXiv:1803.00498. doi:10.4007/annals.2019.189.2.4. S2CID 119158401.
169. ^ Stanley, Richard P. (1994), "A survey of Eulerian posets", in Bisztriczky, T.; McMullen, P.; Schneider, R.; Weiss, A. IviÄ‡ (eds.), Polytopes: abstract, convex and computational (Scarborough, ON, 1993), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 440, Dordrecht: Kluwer Academic Publishers, pp. 301–333, MR 1322068. See in particular p. 316.
170. ^ Kalai, Gil (2018-12-25). "Amazing: Karim Adiprasito proved the g-conjecture for spheres!". Archived from the original on 2019-02-16. Retrieved 2019-02-15.
171. ^ Santos, Franciscos (2012). "A counterexample to the Hirsch conjecture". Annals of Mathematics. 176 (1): 383–412. arXiv:1006.2814. doi:10.4007/annals.2012.176.1.7. S2CID 15325169.
172. ^ Ziegler, Günter M. (2012). "Who solved the Hirsch conjecture?". Documenta Mathematica (Extra Volume "Optimization Stories"): 75–85.
173. ^ Kauers, Manuel; Koutschan, Christoph; Zeilberger, Doron (2009-07-14). "Proof of Ira Gessel's lattice path conjecture". Proceedings of the National Academy of Sciences. 106 (28): 11502–11505. arXiv:0806.4300. Bibcode:2009PNAS..10611502K. doi:10.1073/pnas.0901678106. ISSN 0027-8424. PMC 2710637.
174. ^ Chung, Fan; Greene, Curtis; Hutchinson, Joan (April 2015). "Herbert S. Wilf (1931–2012)". Notices of the AMS. 62 (4): 358. doi:10.1090/noti1247. ISSN 1088-9477. OCLC 34550461. The conjecture was finally given an exceptionally elegant proof by A. Marcus and G. Tardos in 2004.
175. ^ Savchev, Svetoslav (2005). "Kemnitz' conjecture revisited". Discrete Mathematics. 297 (1–3): 196–201. doi:10.1016/j.disc.2005.02.018.
176. ^ Green, Ben (2004). "The Cameron–Erdős conjecture". The Bulletin of the London Mathematical Society. 36 (6): 769–778. arXiv:math.NT/0304058. doi:10.1112/S0024609304003650. MR 2083752. S2CID 119615076.
177. ^ "News from 2007". American Mathematical Society. AMS. 31 December 2007. Archived from the original on 17 November 2015. Retrieved 2015-11-13. The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the Cameron-Erdős conjecture..."
178. ^ Brown, Aaron; Fisher, David; Hurtado, Sebastian (2017-10-07). "Zimmer's conjecture for actions of SL(𝑚,ℤ)". arXiv:1710.02735 [math.DS].
179. ^ Xue, Jinxin (2014). "Noncollision Singularities in a Planar Four-body Problem". arXiv:1409.0048 [math.DS].
180. ^ Xue, Jinxin (2020). "Non-collision singularities in a planar 4-body problem". Acta Mathematica. 224 (2): 253–388. doi:10.4310/ACTA.2020.v224.n2.a2. S2CID 226420221.
181. ^ Richard P Mann. "Known Historical Beggar-My-Neighbour Records". Retrieved 2024-02-10.
182. ^ Bowditch, Brian H. (2006). "The angel game in the plane" (PDF). School of Mathematics, University of Southampton: warwick.ac.uk Warwick University. Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18.
183. ^ Kloster, Oddvar. "A Solution to the Angel Problem" (PDF). Oslo, Norway: SINTEF ICT. Archived from the original (PDF) on 2016-01-07. Retrieved 2016-03-18.
184. ^ Mathe, Andras (2007). "The Angel of power 2 wins" (PDF). Combinatorics, Probability and Computing. 16 (3): 363–374. doi:10.1017/S0963548306008303. S2CID 16892955. Archived (PDF) from the original on 2016-10-13. Retrieved 2016-03-18.
185. ^ Gacs, Peter (June 19, 2007). "THE ANGEL WINS" (PDF). Archived from the original (PDF) on 2016-03-04. Retrieved 2016-03-18.
186. ^ Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (2023). "An aperiodic monotile". arXiv:2303.10798v2 [math.CO].
187. ^ Larson, Eric (2017). "The Maximal Rank Conjecture". arXiv:1711.04906 [math.AG].
188. ^ Kerz, Moritz; Strunk, Florian; Tamme, Georg (2018), "Algebraic K-theory and descent for blow-ups", Inventiones Mathematicae, 211 (2): 523–577, arXiv:1611.08466, Bibcode:2018InMat.211..523K, doi:10.1007/s00222-017-0752-2, MR 3748313, S2CID 253741858
189. ^ Song, Antoine. "Existence of infinitely many minimal hypersurfaces in closed manifolds" (PDF). www.ams.org. Retrieved 19 June 2021. ..I will present a solution of the conjecture, which builds on min-max methods developed by F. C. Marques and A. Neves..
190. ^ "Antoine Song | Clay Mathematics Institute". ...Building on work of Codá Marques and Neves, in 2018 Song proved Yau's conjecture in complete generality
191. ^ Wolchover, Natalie (July 11, 2017), "Pentagon Tiling Proof Solves Century-Old Math Problem", Quanta Magazine, archived from the original on August 6, 2017, retrieved July 18, 2017
192. ^ Marques, Fernando C.; Neves, André (2013). "Min-max theory and the Willmore conjecture". Annals of Mathematics. 179 (2): 683–782. arXiv:1202.6036. doi:10.4007/annals.2014.179.2.6. S2CID 50742102.
193. ^ Guth, Larry; Katz, Nets Hawk (2015). "On the Erdos distinct distance problem in the plane". Annals of Mathematics. 181 (1): 155–190. arXiv:1011.4105. doi:10.4007/annals.2015.181.1.2.
194. ^ Henle, Frederick V.; Henle, James M. "Squaring the Plane" (PDF). www.maa.org Mathematics Association of America. Archived (PDF) from the original on 2016-03-24. Retrieved 2016-03-18.
195. ^ Brock, Jeffrey F.; Canary, Richard D.; Minsky, Yair N. (2012). "The classification of Kleinian surface groups, II: The Ending Lamination Conjecture". Annals of Mathematics. 176 (1): 1–149. arXiv:math/0412006. doi:10.4007/annals.2012.176.1.1.
196. ^ Connelly, Robert; Demaine, Erik D.; Rote, Günter (2003), "Straightening polygonal arcs and convexifying polygonal cycles" (PDF), Discrete & Computational Geometry, 30 (2): 205–239, doi:10.1007/s00454-003-0006-7, MR 1931840, S2CID 40382145
197. ^ Faber, C.; Pandharipande, R. (2003), "Hodge integrals, partition matrices, and the ${\displaystyle \lambda _{g))$ conjecture", Ann. of Math., 2, 157 (1): 97–124, arXiv:math.AG/9908052, doi:10.4007/annals.2003.157.97
198. ^ Shestakov, Ivan P.; Umirbaev, Ualbai U. (2004). "The tame and the wild automorphisms of polynomial rings in three variables". Journal of the American Mathematical Society. 17 (1): 197–227. doi:10.1090/S0894-0347-03-00440-5. MR 2015334.
199. ^ Hutchings, Michael; Morgan, Frank; Ritoré, Manuel; Ros, Antonio (2002). "Proof of the double bubble conjecture". Annals of Mathematics. Second Series. 155 (2): 459–489. arXiv:math/0406017. doi:10.2307/3062123. hdl:10481/32449. JSTOR 3062123. MR 1906593.
200. ^
201. ^ Teixidor i Bigas, Montserrat; Russo, Barbara (1999). "On a conjecture of Lange". Journal of Algebraic Geometry. 8 (3): 483–496. arXiv:alg-geom/9710019. Bibcode:1997alg.geom.10019R. ISSN 1056-3911. MR 1689352.
202. ^ Ullmo, E (1998). "Positivité et Discrétion des Points Algébriques des Courbes". Annals of Mathematics. 147 (1): 167–179. arXiv:alg-geom/9606017. doi:10.2307/120987. JSTOR 120987. S2CID 119717506. Zbl 0934.14013.
203. ^ Zhang, S.-W. (1998). "Equidistribution of small points on abelian varieties". Annals of Mathematics. 147 (1): 159–165. doi:10.2307/120986. JSTOR 120986.
204. ^ 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 (2017). "A formal proof of the Kepler conjecture". Forum of Mathematics, Pi. 5: e2. arXiv:1501.02155. doi:10.1017/fmp.2017.1.
205. ^ Hales, Thomas C.; McLaughlin, Sean (2010). "The dodecahedral conjecture". Journal of the American Mathematical Society. 23 (2): 299–344. arXiv:math/9811079. Bibcode:2010JAMS...23..299H. doi:10.1090/S0894-0347-09-00647-X.
206. ^ Park, Jinyoung; Pham, Huy Tuan (2022-03-31). "A Proof of the Kahn-Kalai Conjecture". arXiv:2203.17207 [math.CO].
207. ^ Dujmović, Vida; Eppstein, David; Hickingbotham, Robert; Morin, Pat; Wood, David R. (August 2021). "Stack-number is not bounded by queue-number". Combinatorica. 42 (2): 151–164. arXiv:2011.04195. doi:10.1007/s00493-021-4585-7. S2CID 226281691.
208. ^ Huang, C.; Kotzig, A.; Rosa, A. (1982). "Further results on tree labellings". Utilitas Mathematica. 21: 31–48. MR 0668845..
209. ^ Hartnett, Kevin (19 February 2020). "Rainbow Proof Shows Graphs Have Uniform Parts". Quanta Magazine. Retrieved 2020-02-29.
210. ^ Shitov, Yaroslav (1 September 2019). "Counterexamples to Hedetniemi's conjecture". Annals of Mathematics. 190 (2): 663–667. arXiv:1905.02167. doi:10.4007/annals.2019.190.2.6. JSTOR 10.4007/annals.2019.190.2.6. MR 3997132. S2CID 146120733. Zbl 1451.05087. Retrieved 19 July 2021.
211. ^ 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. arXiv:1511.05020. doi:10.1016/j.jctb.2019.11.008. ISSN 0095-8956. S2CID 29791394.
212. ^ 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. arXiv:1602.07557. doi:10.1016/j.jctb.2019.11.007. ISSN 0095-8956. S2CID 220369443.
213. ^ 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. arXiv:1609.05747. doi:10.1016/j.jctb.2019.11.006. ISSN 0095-8956. S2CID 119625722.
214. ^ 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. arXiv:1612.07189. doi:10.1016/j.jctb.2019.12.002. ISSN 0095-8956. S2CID 119175309.
215. ^ Zang, Wenan; Jing, Guangming; Chen, Guantao (2019-01-29). "Proof of the Goldberg–Seymour Conjecture on Edge-Colorings of Multigraphs". arXiv:1901.10316v1 [math.CO].
216. ^ Abdollahi A., Zallaghi M. (2015). "Character sums for Cayley graphs". Communications in Algebra. 43 (12): 5159–5167. doi:10.1080/00927872.2014.967398. S2CID 117651702.
217. ^ Huh, June (2012). "Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs". Journal of the American Mathematical Society. 25 (3): 907–927. arXiv:1008.4749. doi:10.1090/S0894-0347-2012-00731-0.
218. ^ Chalopin, Jérémie; Gonçalves, Daniel (2009). "Every planar graph is the intersection graph of segments in the plane: extended abstract". In Mitzenmacher, Michael (ed.). Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009. ACM. pp. 631–638. doi:10.1145/1536414.1536500.
219. ^ Aharoni, Ron; Berger, Eli (2009). "Menger's theorem for infinite graphs". Inventiones Mathematicae. 176 (1): 1–62. arXiv:math/0509397. Bibcode:2009InMat.176....1A. doi:10.1007/s00222-008-0157-3.
220. ^ Seigel-Itzkovich, Judy (2008-02-08). "Russian immigrant solves math puzzle". The Jerusalem Post. Retrieved 2015-11-12.
221. ^ Diestel, Reinhard (2005). "Minors, Trees, and WQO" (PDF). Graph Theory (Electronic Edition 2005 ed.). Springer. pp. 326–367.
222. ^ Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2002). "The strong perfect graph theorem". Annals of Mathematics. 164: 51–229. arXiv:math/0212070. Bibcode:2002math.....12070C. doi:10.4007/annals.2006.164.51. S2CID 119151552.
223. ^ 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.
224. ^ Chen, Zhibo (1996). "Harary's conjectures on integral sum graphs". Discrete Mathematics. 160 (1–3): 241–244. doi:10.1016/0012-365X(95)00163-Q.
225. ^ Friedman, Joel (January 2015). "Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: with an Appendix by Warren Dicks" (PDF). Memoirs of the American Mathematical Society. 233 (1100): 0. doi:10.1090/memo/1100. ISSN 0065-9266. S2CID 117941803.
226. ^ Mineyev, Igor (2012). "Submultiplicativity and the Hanna Neumann conjecture". Annals of Mathematics. Second Series. 175 (1): 393–414. doi:10.4007/annals.2012.175.1.11. MR 2874647.
227. ^ Namazi, Hossein; Souto, Juan (2012). "Non-realizability and ending laminations: Proof of the density conjecture". Acta Mathematica. 209 (2): 323–395. doi:10.1007/s11511-012-0088-0.
228. ^ Pila, Jonathan; Shankar, Ananth; Tsimerman, Jacob; Esnault, Hélène; Groechenig, Michael (2021-09-17). "Canonical Heights on Shimura Varieties and the André-Oort Conjecture". arXiv:2109.08788 [math.NT].
229. ^ Bourgain, Jean; Ciprian, Demeter; Larry, Guth (2015). "Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three". Annals of Mathematics. 184 (2): 633–682. arXiv:1512.01565. Bibcode:2015arXiv151201565B. doi:10.4007/annals.2016.184.2.7. hdl:1721.1/115568. S2CID 43929329.
230. ^ Helfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT].
231. ^ Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT].
232. ^ Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].
233. ^ Zhang, Yitang (2014-05-01). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. ISSN 0003-486X.
234. ^ "Bounded gaps between primes - Polymath Wiki". asone.ai. Archived from the original on 2020-12-08. Retrieved 2021-08-27.
235. ^ Maynard, James (2015-01-01). "Small gaps between primes". Annals of Mathematics: 383–413. arXiv:1311.4600. doi:10.4007/annals.2015.181.1.7. ISSN 0003-486X. S2CID 55175056.
236. ^ Cilleruelo, Javier (2010). "Generalized Sidon sets". Advances in Mathematics. 225 (5): 2786–2807. doi:10.1016/j.aim.2010.05.010. hdl:10261/31032. S2CID 7385280.
237. ^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (I)", Inventiones Mathematicae, 178 (3): 485–504, Bibcode:2009InMat.178..485K, CiteSeerX 10.1.1.518.4611, doi:10.1007/s00222-009-0205-7, S2CID 14846347
238. ^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (II)", Inventiones Mathematicae, 178 (3): 505–586, Bibcode:2009InMat.178..505K, CiteSeerX 10.1.1.228.8022, doi:10.1007/s00222-009-0206-6, S2CID 189820189
239. ^ "2011 Cole Prize in Number Theory" (PDF). Notices of the AMS. 58 (4): 610–611. ISSN 1088-9477. OCLC 34550461. Archived (PDF) from the original on 2015-11-06. Retrieved 2015-11-12.
240. ^ "Bombieri and Tao Receive King Faisal Prize" (PDF). Notices of the AMS. 57 (5): 642–643. May 2010. ISSN 1088-9477. OCLC 34550461. Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18. Working with Ben Green, he proved there are arbitrarily long arithmetic progressions of prime numbers—a result now known as the Green–Tao theorem.
241. ^ Metsänkylä, Tauno (5 September 2003). "Catalan's conjecture: another old diophantine problem solved" (PDF). Bulletin of the American Mathematical Society. 41 (1): 43–57. doi:10.1090/s0273-0979-03-00993-5. ISSN 0273-0979. Archived (PDF) from the original on 4 March 2016. Retrieved 13 November 2015. The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu.
242. ^ Croot, Ernest S. III (2000). Unit Fractions. Ph.D. thesis. University of Georgia, Athens. Croot, Ernest S. III (2003). "On a coloring conjecture about unit fractions". Annals of Mathematics. 157 (2): 545–556. arXiv:math.NT/0311421. Bibcode:2003math.....11421C. doi:10.4007/annals.2003.157.545. S2CID 13514070.
243. ^ Lafforgue, Laurent (1998), "Chtoucas de Drinfeld et applications" [Drinfelʹd shtukas and applications], Documenta Mathematica (in French), II: 563–570, ISSN 1431-0635, MR 1648105, archived from the original on 2018-04-27, retrieved 2016-03-18
244. ^ Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255. Archived (PDF) from the original on 2011-05-10. Retrieved 2016-03-06.
245. ^ Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics. 141 (3): 553–572. CiteSeerX 10.1.1.128.531. doi:10.2307/2118560. JSTOR 2118560. OCLC 37032255. Archived from the original on 16 September 2000.
246. ^ Lee, Choongbum (2017). "Ramsey numbers of degenerate graphs". Annals of Mathematics. 185 (3): 791–829. arXiv:1505.04773. doi:10.4007/annals.2017.185.3.2. S2CID 7974973.
247. ^ Lamb, Evelyn (26 May 2016). "Two-hundred-terabyte maths proof is largest ever". Nature. 534 (7605): 17–18. Bibcode:2016Natur.534...17L. doi:10.1038/nature.2016.19990. PMID 27251254.
248. ^ Heule, Marijn J. H.; Kullmann, Oliver; Marek, Victor W. (2016). "Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer". In Creignou, N.; Le Berre, D. (eds.). Theory and Applications of Satisfiability Testing – SAT 2016. Lecture Notes in Computer Science. Vol. 9710. Springer, [Cham]. pp. 228–245. arXiv:1605.00723. doi:10.1007/978-3-319-40970-2_15. ISBN 978-3-319-40969-6. MR 3534782. S2CID 7912943.
249. ^ Linkletter, David (27 December 2019). "The 10 Biggest Math Breakthroughs of 2019". Popular Mechanics. Retrieved 20 June 2021.
250. ^ Piccirillo, Lisa (2020). "The Conway knot is not slice". Annals of Mathematics. 191 (2): 581–591. doi:10.4007/annals.2020.191.2.5. S2CID 52398890.
251. ^ Klarreich, Erica (2020-05-19). "Graduate Student Solves Decades-Old Conway Knot Problem". Quanta Magazine. Retrieved 2022-08-17.
252. ^ Agol, Ian (2013). "The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves, and Jason Manning)" (PDF). Documenta Mathematica. 18: 1045–1087. arXiv:1204.2810v1. doi:10.4171/dm/421. S2CID 255586740.
253. ^ Brendle, Simon (2013). "Embedded minimal tori in ${\displaystyle S^{3))$ and the Lawson conjecture". Acta Mathematica. 211 (2): 177–190. arXiv:1203.6597. doi:10.1007/s11511-013-0101-2.
254. ^ Kahn, Jeremy; Markovic, Vladimir (2015). "The good pants homology and the Ehrenpreis conjecture". Annals of Mathematics. 182 (1): 1–72. arXiv:1101.1330. doi:10.4007/annals.2015.182.1.1.
255. ^ Austin, Tim (December 2013). "Rational group ring elements with kernels having irrational dimension". Proceedings of the London Mathematical Society. 107 (6): 1424–1448. arXiv:0909.2360. Bibcode:2009arXiv0909.2360A. doi:10.1112/plms/pdt029. S2CID 115160094.
256. ^ Lurie, Jacob (2009). "On the classification of topological field theories". Current Developments in Mathematics. 2008: 129–280. arXiv:0905.0465. Bibcode:2009arXiv0905.0465L. doi:10.4310/cdm.2008.v2008.n1.a3. S2CID 115162503.
257. ^ a b "Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" (PDF) (Press release). Clay Mathematics Institute. March 18, 2010. Archived from the original on March 22, 2010. Retrieved November 13, 2015. The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman.
258. ^ Morgan, John; Tian, Gang (2008). "Completion of the Proof of the Geometrization Conjecture". arXiv:0809.4040 [math.DG].
259. ^ Rudin, M.E. (2001). "Nikiel's Conjecture". Topology and Its Applications. 116 (3): 305–331. doi:10.1016/S0166-8641(01)00218-8.
260. ^ Norio Iwase (1 November 1998). "Ganea's Conjecture on Lusternik-Schnirelmann Category". ResearchGate.
261. ^ Tao, Terence (2015). "The Erdős discrepancy problem". arXiv:1509.05363v5 [math.CO].
262. ^ Duncan, John F. R.; Griffin, Michael J.; Ono, Ken (1 December 2015). "Proof of the umbral moonshine conjecture". Research in the Mathematical Sciences. 2 (1): 26. arXiv:1503.01472. Bibcode:2015arXiv150301472D. doi:10.1186/s40687-015-0044-7. S2CID 43589605.
263. ^ Cheeger, Jeff; Naber, Aaron (2015). "Regularity of Einstein Manifolds and the Codimension 4 Conjecture". Annals of Mathematics. 182 (3): 1093–1165. arXiv:1406.6534. doi:10.4007/annals.2015.182.3.5.
264. ^ Wolchover, Natalie (March 28, 2017). "A Long-Sought Proof, Found and Almost Lost". Quanta Magazine. Archived from the original on April 24, 2017. Retrieved May 2, 2017.
265. ^ Newman, Alantha; Nikolov, Aleksandar (2011). "A counterexample to Beck's conjecture on the discrepancy of three permutations". arXiv:1104.2922 [cs.DM].
266. ^ Voevodsky, Vladimir (1 July 2011). "On motivic cohomology with Z/l-coefficients" (PDF). annals.math.princeton.edu. Princeton, NJ: Princeton University. pp. 401–438. Archived (PDF) from the original on 2016-03-27. Retrieved 2016-03-18.
267. ^ Geisser, Thomas; Levine, Marc (2001). "The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky". Journal für die Reine und Angewandte Mathematik. 2001 (530): 55–103. doi:10.1515/crll.2001.006. MR 1807268.
268. ^ Kahn, Bruno. "Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry" (PDF). webusers.imj-prg.fr. Archived (PDF) from the original on 2016-03-27. Retrieved 2016-03-18.
269. ^
270. ^ Mattman, Thomas W.; Solis, Pablo (2009). "A proof of the Kauffman-Harary Conjecture". Algebraic & Geometric Topology. 9 (4): 2027–2039. arXiv:0906.1612. Bibcode:2009arXiv0906.1612M. doi:10.2140/agt.2009.9.2027. S2CID 8447495.
271. ^ Kahn, Jeremy; Markovic, Vladimir (2012). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". Annals of Mathematics. 175 (3): 1127–1190. arXiv:0910.5501. doi:10.4007/annals.2012.175.3.4.
272. ^ Lu, Zhiqin (September 2011) [2007]. "Normal Scalar Curvature Conjecture and its applications". Journal of Functional Analysis. 261 (5): 1284–1308. arXiv:0711.3510. doi:10.1016/j.jfa.2011.05.002.
273. ^ Dencker, Nils (2006), "The resolution of the Nirenberg–Treves conjecture" (PDF), Annals of Mathematics, 163 (2): 405–444, doi:10.4007/annals.2006.163.405, S2CID 16630732, archived (PDF) from the original on 2018-07-20, retrieved 2019-04-07
274. ^