Roger Heath-Brown | |
---|---|

Born | 12 October 1952 |

Citizenship | United Kingdom |

Alma mater | University of Cambridge |

Known for | Analytic number theory, Heath-Brown–Moroz constant |

Awards | Smith's Prize (1976) Berwick Prize (1981) Fellow of the Royal Society (1993) Senior Berwick Prize (1996) Pólya Prize (2009) |

Scientific career | |

Fields | Pure mathematics |

Institutions | University of Oxford |

Thesis | Topics in Analytic Number Theory (1979) |

Doctoral advisor | Alan Baker |

Doctoral students | Timothy Browning James Maynard |

Website | www |

**David Rodney "Roger" Heath-Brown** FRS (born 12 October 1952),^{ [1] } is a British mathematician working in the field of analytic number theory.^{ [2] }

He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervisor was Alan Baker.^{ [3] }^{ [4] }

In 1979 he moved to the University of Oxford, where from 1999 he held a professorship in pure mathematics.^{ [1] } He retired in 2016.^{ [5] }

Heath-Brown is known for many striking results. He proved that there are infinitely many prime numbers of the form *x*^{3} + 2*y*^{3}.^{ [6] } In collaboration with S. J. Patterson in 1978 he proved the Kummer conjecture on cubic Gauss sums in its equidistribution form. He has applied Burgess's method on character sums to the ranks of elliptic curves in families. He proved that every non-singular cubic form over the rational numbers in at least ten variables represents 0.^{ [7] } Heath-Brown also showed that Linnik's constant is less than or equal to 5.5.^{ [8] } More recently, Heath-Brown is known for his pioneering work on the so-called determinant method. Using this method he was able to prove a conjecture of Serre ^{[ citation needed ]} in the four variable case in 2002.^{ [9] } This particular conjecture of Serre was later dubbed the "dimension growth conjecture" and this was almost completely solved by various works of Browning, Heath-Brown, and Salberger by 2009.^{ [10] }

The London Mathematical Society has awarded Heath-Brown the Junior Berwick Prize (1981), the Senior Berwick Prize (1996),^{ [11] } and the Pólya Prize (2009). He was made a Fellow of the Royal Society in 1993,^{ [1] } and a corresponding member of the Göttingen Academy of Sciences in 1999.^{ [12] }

He was an invited speaker at International Congress of Mathematicians in 1983 in Warsaw and in 2010 in Hyderabad on the topic of "*Number Theory.*"^{ [13] }

In 2012 he became a fellow of the American Mathematical Society.^{ [14] }

In September of 2007, he co-authored (along with Joseph H. Silverman) the preface to the Oxford University Sixth Edition of * An Introduction to the Theory of Numbers * by G.H. Hardy and E.M. Wright.

**Sir Andrew John Wiles** is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal by the Royal Society. He was appointed Knight Commander of the Order of the British Empire in 2000, and in 2018 was appointed as the first Regius Professor of Mathematics at Oxford. Wiles is also a 1997 MacArthur Fellow.

**Number theory** is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers or defined as generalizations of the integers.

**Jean-Pierre Serre** is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003.

**Linnik's theorem** in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that there exist positive *c* and *L* such that, if we denote p(*a*,*d*) the least prime in the arithmetic progression

In mathematics, the **Birch and Swinnerton-Dyer conjecture** describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. As of 2021, only special cases of the conjecture have been proven.

In mathematics, Helmut Hasse's **local–global principle**, also known as the **Hasse principle**, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the *p*-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers *and* in the *p*-adic numbers for each prime *p*.

**Klaus Friedrich Roth** was a German-born British mathematician who won the Fields Medal for proving Roth's theorem on the Diophantine approximation of algebraic numbers. He was also a winner of the De Morgan Medal and the Sylvester Medal, and a Fellow of the Royal Society.

**Nigel James Hitchin** FRS is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University of Oxford.

**Terence Chi-Shen Tao** is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory.

**Philip Hall** FRS, was an English mathematician. His major work was on group theory, notably on finite groups and solvable groups.

**Ben Joseph Green** FRS is a British mathematician, specialising in combinatorics and number theory. He is the Waynflete Professor of Pure Mathematics at the University of Oxford.

In mathematics, **arithmetic geometry** is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.

In number theory and algebraic geometry, a **rational point** of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point.

This is a glossary of **arithmetic and diophantine geometry** in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.

The **Ax–Kochen theorem**, named for James Ax and Simon B. Kochen, states that for each positive integer *d* there is a finite set *Y _{d}* of prime numbers, such that if

In mathematics, **Kummer sum** is the name given to certain cubic Gauss sums for a prime modulus *p*, with *p* congruent to 1 modulo 3. They are named after Ernst Kummer, who made a conjecture about the statistical properties of their arguments, as complex numbers. These sums were known and used before Kummer, in the theory of cyclotomy.

**Barry Charles Mazur** is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology.

**Richard Paul Winsley Thomas** FRS is a British mathematician working in several areas of geometry. He is a professor at Imperial College London. He studies moduli problems in algebraic geometry, and ‘mirror symmetry’—a phenomenon in pure mathematics predicted by string theory in theoretical physics.

**George Leo Watson** was a British mathematician, who specialized in number theory.

**Andrew Victor Sutherland** is an American mathematician and Principal Research Scientist at the Massachusetts Institute of Technology. His research focuses on computational aspects of number theory and arithmetic geometry. He is known for his contributions to several projects involving large scale computations, including the Polymath project on bounded gaps between primes, the L-functions and Modular Forms Database, the sums of three cubes project, and the computation and classification of Sato-Tate distributions.

- 1 2 3 "Prof Roger Heath-Brown, FRS".
*Debrett's People of Today*. Archived from the original on 6 September 2012. Retrieved 28 December 2010. - ↑ "Prof. Roger Heath-Brown, University of Oxford, FRS" . Retrieved 4 May 2017.
- ↑ Official home page Archived 5 August 2004 at the Wayback Machine
- ↑ Roger Heath-Brown at the Mathematics Genealogy Project
- ↑ "Vice chancellor's oration". Gazette.web.ox.ac.uk. Retrieved on 2018-08-29.
- ↑ Heath-Brown, D.R. (2001). "Primes represented by
*x*^{3}+ 2*y*^{3}".*Acta Mathematica*.**186**: 1–84. doi: 10.1007/BF02392715 . - ↑ D. R. Heath-Brown,
*Cubic forms in ten variables*, Proceedings of the London Mathematical Society,**47**(3), pages 225–257 (1983) doi : 10.1112/plms/s3-47.2.225 - ↑ D. R. Heath-Brown,
*Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression*, Proceedings of the London Mathematical Society,**64**(3), pages 265–338 (1992) doi : 10.1112/plms/s3-64.2.265 - ↑ D.R. Heath-Brown,
*The density of rational points on curves and surfaces*, Annals of Mathematics,**155**(2), pages 553–598 (2002) - ↑ T. D. Browning,
*Quantitative Arithmetic of Projective Varieties*, Progress in Mathematics,**277**, Birkhauser - ↑ Berwick prizes page at The MacTutor History of Mathematics archive
- ↑ "Professor Roger Heath-Brown". The Mathematical Institute, University of Oxford . Retrieved 28 December 2010.
- ↑ "ICM Plenary and Invited Speakers since 1897". International Congress of Mathematicians.
- ↑ List of Fellows of the American Mathematical Society. Retrieved 19 January 2013.

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