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Thomas Royen | |
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Born | Thomas Royen 6 July 1947 |

Residence | Schwalbach am Taunus, Germany |

Citizenship | Germany |

Alma mater | Goethe University Frankfurt University of Freiburg Technical University Dortmund (PhD) |

Known for | Proof of Gaussian correlation inequality |

Scientific career | |

Fields | Mathematics, Statistics |

Thesis | On Convergence Against Stable Laws (1975) |

**Thomas Royen** (born July 6, 1947 in Frankfurt am Main) is a retired German professor of statistics who has been affiliated with the University of Applied Sciences Bingen. Royen came to prominence in the spring of 2017 for a relatively simple proof for the Gaussian Correlation Inequality (GCI), a conjecture that originated in the 1950s, which he had published three years earlier without much recognition.^{ [1] } A proof of this conjecture, which lies at the intersection of geometry, probability theory and statistics, had eluded top experts for decades.^{ [2] }

**Statistics** is a branch of mathematics working with data collection, organization, analysis, interpretation and presentation. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

The **University of Applied Sciences Bingen** is a university located in Bingen am Rhein, Germany. It was founded in 1897. The University of Applied Sciences Bingen consists of two faculties: the faculty of life sciences and engineering and the faculty of technology, informatics and business.

A **mathematical proof** is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is always true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture.

Royen was born in 1947 to Paul Royen, a professor with the institute for inorganic chemistry at the Goethe University Frankfurt and Elisabeth Royen, also a chemist. From 1966 to 1971, he studied mathematics and physics at his father's university and the University of Freiburg. After graduating, he worked as a tutor at the University of Freiburg, before transferring to the Technical University of Dortmund for his doctoral thesis. After attaining his PhD in 1975 with a thesis called *Über die Konvergenz gegen stabile Gesetze* (*On Convergence Against Stable Laws*), he worked as a wissenschaftlicher Assistent at Dortmund University's institute for statistics. Married with children, Royen lives in Schwalbach am Taunus.

**Inorganic chemistry** deals with the synthesis and behavior of inorganic and organometallic compounds. This field covers all chemical compounds except the myriad organic compounds, which are the subjects of organic chemistry. The distinction between the two disciplines is far from absolute, as there is much overlap in the subdiscipline of organometallic chemistry. It has applications in every aspect of the chemical industry, including catalysis, materials science, pigments, surfactants, coatings, medications, fuels, and agriculture.

**Goethe University Frankfurt** is a university located in Frankfurt, Germany. It was founded in 1914 as a citizens' university, which means it was founded and funded by the wealthy and active liberal citizenry of Frankfurt. The original name was **Universität Frankfurt am Main**. In 1932, the university's name was extended in honour of one of the most famous native sons of Frankfurt, the poet, philosopher and writer/dramatist Johann Wolfgang von Goethe. The university currently has around 45,000 students, distributed across four major campuses within the city.

**Mathematics** includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

In 1977, Royen started working as a statistician for the pharmaceutical company Hoechst AG. From 1979 to 1985, he worked at the company's own educational facility teaching mathematics and statistics. Starting in 1985 until becoming an emeritus in 2010, he taught statistics and mathematics at the University of Applied Sciences Bingen in Rhineland-Palatinate.^{ [3] }

**Hoechst AG** was a German chemicals then life-sciences company that became Aventis Deutschland after its merger with France's Rhône-Poulenc S.A. in 1999. With the new company's 2004 merger with Sanofi-Synthélabo, it became a subsidiary of the resulting Sanofi-Aventis pharmaceuticals group.

* Emeritus*, in its current usage, is an adjective used to designate a retired chairperson, professor, pastor, bishop, pope, director, president, prime minister, rabbi, emperor, or other person.

**Rhineland-Palatinate** is a state of Germany.

Royen worked mainly on probability distributions, in particular multivariate chi-squares and gamma distributions, to improve some frequently used statistical test procedures. Nearly half of his circa 30 publications were written when he was aged over sixty. Because he was annoyed over some contradictory reviews and in a few cases also over the incompetence of a referee, he decided in his later years, when his actions had no influence anymore on his further career, to publish his papers on the online platform arXiv.org and sometimes in a less renowned Indian journal to fulfill, at least formally, the condition of a peer review.^{ [4] }

In probability theory and statistics, a **probability distribution** is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable X is used to denote the outcome of a coin toss, then the probability distribution of X would take the value 0.5 for *X* = heads, and 0.5 for *X* = tails. Examples of random phenomena can include the results of an experiment or survey.

In probability theory and statistics, the **gamma distribution** is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are three different parametrizations in common use:

- With a shape parameter
*k*and a scale parameter*θ*. - With a shape parameter
*α*=*k*and an inverse scale parameter*β*= 1/*θ*, called a rate parameter. - With a shape parameter
*k*and a mean parameter*μ*=*kθ*=*α*/*β*.

**arXiv** is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, physics, astronomy, electrical engineering, computer science, quantitative biology, statistics, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, and had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month.

On July 17, 2014, a few years after his retirement, when brushing his teeth, Royen had a flash of insight: how to use the Laplace transform of the multivariate gamma distribution to achieve a relatively simple proof for the Gaussian correlation inequality, a conjecture on the intersection of geometry, probability theory and statistics, formulated after work by Dunnett and Sobel (1955) and the American statistician Olive Jean Dunn (1958),^{ [5] } that had remained unsolved since then. He sent a copy of his proof to Donald Richards, an acquainted American mathematician, who worked on a proof of the GCI for 30 years. Richards immediately saw the validity of Royen's proof and subsequently helped him to transform the mathematical formulas into LaTeX. When Royen contacted other reputed mathematicians, though, they didn't bother to investigate his proof, because Royen was relatively unknown, and these mathematicians therefore estimated the chance that Royen's proof would be false as very high.^{ [2] }

In mathematics, the **Laplace transform** is an integral transform named after its inventor Pierre-Simon Laplace. It transforms a function of a real variable *t* to a function of a complex variable s. The transform has many applications in science and engineering.

The **Gaussian correlation inequality** (**GCI**), formerly known as the **Gaussian correlation conjecture** (**GCC**), is a mathematical theorem in the fields of mathematical statistics and convex geometry. A special case of the inequality was published as a conjecture in a paper from 1955; further development was given by Olive Jean Dunn in 1958; and the general case was stated in 1972, also as a conjecture.

In mathematics, a **conjecture** is a conclusion or proposition based on incomplete information, for which no proof or disproof has yet been found. Conjectures such as the Riemann hypothesis or Fermat's Last Theorem have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.

Royen published this proof in an article with the title *A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions* on arXiv ^{ [6] } and subsequently in the Far East Journal of Theoretical Statistics,^{ [7] } a relatively unknown periodical based in Allahabad, India, for which Royen was at the time voluntarily working as a referee himself. Due to this, his proof went at first largely unnoticed by the scientific community,^{ [8] } until in late 2015 two Polish mathematicians, Rafał Latała and Dariusz Matlak, wrote a paper in which they reorganized Royen's proof in a way that was intended to be easier to follow.^{ [1] } In July 2015, Royen supplemented his proof with a further paper in arXiv *Some probability inequalities for multivariate gamma and normal distributions*.^{ [9] }

**Allahabad**, officially known as **Prayagraj**, and also known as **Illahabad** and **Prayag**, is a city in the Indian state of Uttar Pradesh. It is the administrative headquarters of Allahabad district—the most populous district in the state and 13th most populous district in India—and the Allahabad division.

On March 28, 2017 Natalie Wolchover of * Quanta Magazine * published a story about Royen's proof, after which he gained more academic and public recognition for his achievement, which is not only remarkable because of its mathematical significance but also of Royen's (from a mathematical perspective) advanced age (67) when he found the proof of the GCI. In addition to this, it's also noteworthy that the world of mathematics and statistics has apparently become so vast that experts in a field can be unaware two years after publication of a proof, that an important conjecture in their own field has been proved.

**Louis de Branges de Bourcia** is a French-American mathematician. He is the Edward C. Elliott Distinguished Professor of Mathematics at Purdue University in West Lafayette, Indiana. He is best known for proving the long-standing Bieberbach conjecture in 1984, now called de Branges's theorem. He claims to have proved several important conjectures in mathematics, including the generalized Riemann hypothesis.

The **Kepler conjecture**, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing and hexagonal close packing arrangements. The density of these arrangements is around 74.05%.

**Probability** is a measure of the likeliness that an event will occur. Probability is used to quantify an attitude of mind towards some proposition of whose truth we are not certain. The proposition of interest is usually of the form "A specific event will occur." The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1, we call probability. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.

In convex analysis, a non-negative function *f* : **R**^{n} → **R**_{+} is **logarithmically concave** if its domain is a convex set, and if it satisfies the inequality

**Thomas Callister Hales** is an American mathematician working in the areas of representation theory, discrete geometry, and formal verification. In representation theory he is known for his work on the Langlands program and the proof of the fundamental lemma over the group Sp(4). In discrete geometry, he settled the Kepler conjecture on the density of sphere packings and the honeycomb conjecture. In 2014, he announced the completion of the Flyspeck Project, which formally verified the correctness of his proof of the Kepler conjecture.

In statistics, the **multivariate t-distribution** is a multivariate probability distribution. It is a generalization to random vectors of the Student's

In mathematics, the **Gaussian isoperimetric inequality**, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states that among all sets of given Gaussian measure in the *n*-dimensional Euclidean space, half-spaces have the minimal Gaussian boundary measure.

Inequalities are very important in the study of information theory. There are a number of different contexts in which these inequalities appear.

In mathematics, the **Fortuin–Kasteleyn–Ginibre (FKG) inequality** is a correlation inequality, a fundamental tool in statistical mechanics and probabilistic combinatorics, due to Cees M. Fortuin, Pieter W. Kasteleyn, and Jean Ginibre (1971). Informally, it says that in many random systems, increasing events are positively correlated, while an increasing and a decreasing event are negatively correlated. It was obtained by studying the random cluster model.

The **Tracy–Widom distribution**, introduced by Craig Tracy and Harold Widom, is the probability distribution of the normalized largest eigenvalue of a random Hermitian matrix.

**Caucher Birkar** is a UK-based Iranian Kurdish mathematician and a professor at the University of Cambridge.

**Inter-universal Teichmüller theory** is the name given by mathematician Shinichi Mochizuki to a theory he developed in the 2000s, following his earlier work in arithmetic geometry. According to Mochizuki, it is "an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve". The theory was made public in a series of four preprints posted in 2012 to his website. The most striking claimed application of the theory is to provide a proof for various outstanding conjectures in number theory, in particular the abc conjecture. Mochizuki and a few other mathematicians claim that the theory indeed yields such a proof but this has so far not been accepted by the mathematical community.

**Boson sampling** constitutes a restricted model of non-universal quantum computation introduced by S. Aaronson and A. Arkhipov. It consists of sampling from the probability distribution of identical bosons scattered by a linear interferometer. Although the problem is well defined for any bosonic particles, its photonic version is currently considered as the most promising platform for a scalable implementation of a boson sampling device, which makes it a non-universal approach to linear optical quantum computing. Moreover, while not universal, the boson sampling scheme is strongly believed to implement a classically hard task using far fewer physical resources than a full linear-optical quantum computing setup. This makes it an outstanding candidate for demonstrating the power of quantum computation in the near term.

**Olive Jean Dunn** was an American mathematician and statistician, and professor of biostatistics at the University of California Los Angeles (UCLA). She described methods for computing confidence intervals and also codified the Bonferroni correction's application to confidence intervals. She authored the textbook, *Basic Statistics: A Primer for the Biomedical Sciences* in 1977.

**Maryna Sergiivna Viazovska** is a Ukrainian mathematician who, in 2016, solved the sphere-packing problem in dimension 8 and, in collaboration with others, in dimension 24. Previously, the problem had been solved only for three or fewer dimensions, and the proof of the three-dimensional version involved long computer calculations. In contrast, Viazovska's proof for 8 and 24 dimensions is "stunningly simple".

- 1 2 Rafal Latala and Dariusz Matlak, Royen's proof of the Gaussian correlation inequality, arXiv:1512.08776
- 1 2 "A Long-Sought Proof, Found and Almost Lost".
*Quanta Magazine*. Natalie Wolchover. March 28, 2017. Retrieved May 1, 2017. - ↑ "Curriculum Vitae Thomas Royen".
*Forschungsplattform des Landes Rheinland-Pfalz*. Landesregierung Rheinland-Pfalz. - ↑ "Der Beweis" (in German). Sibylle Anderl. April 7, 2017.
- ↑ Dunn, Olive Jean (March 1959). "Estimation of the medians for dependent variables".
*The Annals of Mathematical Statistics*.**30**(1): 192–197. JSTOR 2237135. - ↑ Thomas Royen:
*A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions*, in:*arXiv.org*, 13. August 2014, to Download, 7. April 2017. supplemented by Thomas Royen:*Some probability inequalities for multivariate gamma and normal distributions*, in:*arXiv.org*, 2. Juli 2015, to download, 7. April 2017 - ↑ Thomas Royen:
*A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions,*in:*Far East Journal of Theoretical Statistics*, Part 48 Nr. 2, Pushpa Publishing House, Allahabad 2014, p.139–145 - ↑ In the Quanta magazine article for instance Tilmann Gneiting, a statistician at the Heidelberg Institute for Theoretical Studies, just 65 miles from Bingen, said he was shocked to learn in July 2016, two years after the fact, that the GCI had been proved.
- ↑ Royen, Thomas (2015-07-02). "Some probability inequalities for multivariate gamma and normal distributions". arXiv: 1507.00528 [math.PR].

- Profile of Thomas Royen on Science Portal Rheinland-Pfalz (RLP)
- Thomas Royen, "A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions", arXiv : 1408.1028
- Thomas Royen, "Some probability inequalities for multivariate gamma and normal distributions", arXiv : 1507.00528
- Thomas Royen, "A note on the existence of the multivariate gamma distribution", arXiv : 1606.04747
- ‹See Tfd› (in French) Bourbaki, Frank Barthe on YouTube. January 14, 2017
- Informal interview with Thomas Royen at University of Applied Sciences Bingen on YouTube (in German)

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