► German mathematician Peter Schultz is considered a popular candidate for in 2018. Image source: Bonn University Center for Theoretical Physics Research BCTP
Written by | Yang Xiaoxiao
Editor | Chen Xiaoxue
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Although this year's Fields Medal will not be officially announced at the International Congress of Mathematicians (ICM) in Rio de Janeiro in early August, Wikipedia listed Peter Scholze as the winner in April to recognize his achievements in the field of algebraic geometry (later removed).
Fields Medal, full name is the International Outstanding Mathematics Discovery Award, and is hailed as the Nobel Prize in the mathematics community. The award was prepared and established by Canadian mathematician John Fields. The award conditions are extremely strict. 2-4 young mathematicians with outstanding contributions are selected every four years, and they cannot be over 40 years old. This should not only recognize the outstanding contributions that the mathematician has made, but also encourage them to make new achievements and development in the future.
The young leader in the field of algebraic geometry in China Xu Chenyang once joked that unlike researchers in the fields of biology, physics, chemistry, etc., who have to wait for the Nobel Prize Committee phone calls every year, lucky mathematicians can continue to do research with confidence after they are over 40 years old.
Perhaps in order to prevent outstanding mathematicians who are about to be 40 from leaving any regrets. Most of the winners of the Fields Medal have reached the last four years before the age of 40, but Schultz is still under 31 years old. This does not prevent him from becoming the biggest favorite to win the award. In 2016, Quanta Magazine and other scientific websites predicted [1] that Schultz is very likely to win the 2018 Fields Medal: "Sultz has become one of the most influential mathematicians in the world, a rare talent that appears every few decades."
Peter Schultz was born in 1987 in Dresden , Germany. His father is a physicist, his mother is a computer scientist, and his sister is a chemist. Schultz once joked that his family occupied various fields of natural sciences [2]. When he was in high school, Schultz won three gold medals in the International Olympic Mathematics Competition [3], which allowed him to discover his math talent and to inspire to become a mathematician. After graduating from high school, he entered , Bonn University, and continued to pursue his ideals. Schultz, immersed in mathematics, completed his bachelor's degree in just three semesters and his master's degree in two semesters.
In 2010, a news circulated in a forum on number theory . Several doctoral students from Bonn University abbreviated the 228-page mathematical proof of "The Geometry and Cohomology of Some Simple Shimura Varieties" co-authored by Michael Harris and Richard Taylor to 37 pages [5]. Among them, Schultz found a way to avoid the most complicated part of the original proof. Two years later, Schultz became a professor at the University of Bonn.
Schultz's method of learning mathematics is different from most scholars. As he himself said, he never learns basic theoretical knowledge like "linear algebra ". He will directly look at the theories he is interested in. Only when he encounters difficulties will he learn the background knowledge needed to solve this theory.
This learning method originated from the proof of the proof of the Female theorem he taught himself at the age of 16. When n is greater than 2, the equation has no integer solution [6], which is Fermat's Grand Theorem. Although this is just a seemingly simple algebra problem, its proof involves many cutting-edge mathematics, which it took three hundred years for the mathematical community to prove. By understanding this proof, Schultz learned related knowledge such as mode form theory and elliptic curves. This also made him have a strong interest in algebraic geometry.
Schultz mainly studies the use of geometric tools to solve the polynomial equation. The most breakthrough work in Fermat's theorem lies in the use of p-in-number theory to prove it. p-entry numbers are a powerful tool in number theory research.
In 2012, Schultz proposed a new concept and related technologies of perfection space [7], which can simplify the arithmetic problem of p-to-number.This technology promotes existing theorem applications, finds examples in the space introduced by other mathematicians, and provides new research perspectives for many mathematical problems to be solved.
Schultz's work falls under the category of the "Langlands Program", the core part of which is a series of interrelated inferences and theorems on the relationship between number theory, geometry and analysis. Schultz expanded the Langlands program to "three-dimensional hyperbolic space" and a wider structure. By constructing the shaped like complete space of three-dimensional hyperbolic space, he discovered a new set of mutual antonyms. His colleague Eugen Hellmann, a mathematician at the University of Bonn, once commented: "Schultz has discovered a supreme and precise way to integrate previous work in the field. This elegant theoretical framework can surpass all known results."
Although only 30 years old, Schultz has won many supreme honors in the field, including the Ramanujin Award, the Fermat Award, the Leibniz Award, and the New Horizons Award, and has also declined the New Horizons Award. The sudden celebrity effect made Schultz a little at a loss. He tried his best to avoid the interference caused by his achievements, but instead focused on his research and life.
"I'm just trying to understand what already exists, maybe re-expressing it in my own language. I don't think I'm really starting to do research." He once said this.
According to convention, the International Conference of Mathematicians will invite the winners of that year to give an hourly speech. On this year's speech list [8], besides Schultz there is another favorite to win the award, Geordie Williamson.
►Jody Williamson, Australian mathematician and professor at the University of Sydney. Image source: media.eurekalert.org
Williamson, 37, is the youngest academician in the history of the Australian Academy of Sciences. He is an expert in the field of geometric representation of group theory . He successfully proved the Kazhdan-Lusztig conjecture with algebraic methods, and proposed a technical means to solve many problems in group theory [9].
In addition, several other mathematicians have also been predicted by foreign media as popular candidates for the Fields Medal.
►Simon Brand, German mathematician and professor at Columbia University in the United States. Image source: http://media.scgp.stonybrook.edu
German mathematician Simon Brendle has made many achievements in the fields of differential geometric and nonlinear partial micro equations. He solved the main problem of Yamabe equation in conformal geometry, proved the basic problem of differential spherical surface theorem in overall differential geometry, also proved the long-standing problem Hsiang–Lawson conjecture in the theory of minimal surfaces, and studied the related problems of Riemann manifold rigidity theorem [10].
►Malina Viazovska, Ukrainian mathematician. Image source: Official website of the Clay Mathematics Institute (http://www.claymath.org/maryna-viazovska)
Another is a female mathematician. Ukrainian mathematician Maryna Viazovska is a professor at of the Federal Institute of Technology of Lausanne, Switzerland [11]. As an expert in discrete geometry, she used a simple and concise method to solve the problem of sphere stacking in 8-dimensional space and achieved some success in the modular form theory.
►Ziprian Manolescu, Romanian American mathematician. Image source: UCLA official website
Another popular candidate is Romanian-American mathematician Ciprian Manolescu. Manolescu is about to be 40 years old. He is now a professor of mathematics at the University of California, Los Angeles. He mainly studies gauge field theory and low-dimensional topology [12]. In 2013, he proved that in 5-dimensional and above flow patterns, the triangle anatomical conjecture is not valid.
Who will be the last Fields Medal? Let’s look forward to the International Mathematician Conference on August 1st! Of course, no matter who wins this award, this math feast of mathematics provides us with an opportunity to better understand the development and contribution of cutting-edge mathematics.
Reference:
[1] https://www.quantamagazine.org/will-peter-scholze-win-the-fields-medal-in-2018-20160628/
[2] https://www.quantamagazine.org/peter-scholze-and-the-future-of-arithmetic-geometry-20160628/
[3] https://www.imo-official.org/search.aspx
[4] Harris M, Taylor R. The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151)[M]. Princeton university press, 2001.
[5] Scholze P. The local Langlands correspondence for GL n over padic fields[J]. Inventions mathematicae, 2013, 192(3): 663-715.
[6] Wiles A. Modular elliptic curves and Fermat's last theorem[J]. Annals of mathematics, 1995, 141(3): 443-551.
[7] Scholze P. Perfectoid spaces[J]. Publications mathématiques de l'IHÉS, 2012, 116(1): 245-313.
[8] http://www.icm2018.org/portal/en/plenary-lectures
[9] http://sydney.edu.au/science/people/g.williamson.php
[10] http://www.columbia.edu/~sab2280/publ.html
[11] https://people.epfl.ch/cgi-bin/people?id=280037op=admindatalang=encvlang=en
[12] http://www.math.ucla.edu/~cm/
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