Francis Brown

Francis Brown (born November 5, 1977 ) is a British-French mathematician who deals with number theory and combinatorics of Feynman diagrams .

life and work

Brown studied at the University of Cambridge with a bachelor's degree and at the École normal supérieure and received his doctorate in 2006 from the University of Bordeaux with Pierre Cartier . As a post-doctoral student he was at the Max Planck Institute for Mathematics in Bonn and the Mittag-Leffler Institute . He is Chargé de Recherche of the CNRS at the Institut de Mathématiques de Jussieu in Paris, which is assigned to the universities of Paris VI and VII. He has been a permanent visiting scientist (Visiteur CNRS longue-durée) at IHES since 2012 .

In 2012 he received the Prix ​​Élie Cartan of the Académie des Sciences for the proof of two conjectures about multiple zeta functions . Also in 2012 he received the CNRS bronze medal.

The first conjecture comes from Alexander Goncharov and Yuri Manin and says that all periods in the module space of curves of gender 0 with dots can be represented as linear combinations of values ​​of the multiple zeta function. The second conjecture comes from Michael E. Hoffman and says that all values ​​of the multiple zeta function are linear combinations of those with exponent 2 or 3. The guess is important in the theory of motives . ${\displaystyle {\mathcal {M}}_{0,n}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} \left[{\frac {1}{2\pi i}}\right]}$

In a paper published in Annals of Mathematics in 2012 , he then generally proved that periods of mixed Tate motifs over unbranched can be represented as linear combinations of values ​​of the multiple zeta function. This contains the above first conjecture as the special case of the Tate motif defined by the module space of curves of gender 0 with marked points and the relative cohomology . At the same time in this work he proved a conjecture by Yasutaka Ihara and Pierre Deligne about the connection of mixed Tate motifs over the whole numbers with the motivic fundamental group of the projective straight line minus three points. ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Q} \left[{\frac {1}{2\pi i}}\right]}$ ${\displaystyle {\mathcal {M}}_{0,n}}$ ${\displaystyle n}$ ${\displaystyle H^{l}({\overline {\mathcal {M}}}_{0,n}-A,B-B\cap A)}$

He also deals with zeta functions in quantum field theory , in particular with the question of whether the Feynman integrals occurring in perturbation theory (more precisely, primitive, log-divergent Feynman amplitudes are considered) are expressed by zeta functions and multiple zeta functions in four dimensions in simple model quantum field theories such as theory let and if so by which. In 2009 he proved that these can be expressed for low order by rational combinations of multiple zeta functions and at the same time showed that this, contrary to popular folklore assumption, does not apply in general and that if they can be expressed by multiple zeta functions, the combinations that occur are severely restricted. In 2012, together with Oliver Schnetz, he proved a conjecture by Dirk Kreimer and David Broadhurst from 1995, which expressed a special infinite class of Feynman integrals (Zig-Zag graphs) using values ​​of zeta functions at odd places. The conjecture was made by Connes and Kreimer after extensive numerical calculations from Feynman diagrams. Here he works as part of a program pursued by Cartier, Alain Connes , Maxim Kontsevich , Dirk Kreimer and others to explain the connection between Feynman integrals and zeta functions with the help of a motivic Galois group ( called Cosmic Galois Group by Connes ). ${\displaystyle \phi ^{4}}$

According to Kontsevich, the primitive Feynman amplitudes of the theory are related to the number function of the points of the associated graph function over finite fields (with , p prim) (and are two aspects of a mixed motif behind it ). In 2009 Brown proved a conjecture by Kontsevich (1997) that this function is a polynomial in q in the special case of low-order Feynman graphs (in general, the conjecture according to Prakash Belkale and Patrick Brosnan (2003) is wrong). In 2011 he also gave an effective version of the proof with Schnetz. He found the -invariant for the number-theoretic characterization of the number function of graphs and shows that these are Fourier coefficients of modular forms for special graphs , which resulted in further explicit counterexamples to the conjecture of Kontsevich. ${\displaystyle \phi ^{4}}$ ${\displaystyle \mathbb {F} _{q}}$ ${\displaystyle q=p^{n}}$ ${\displaystyle c_{2}}$

In 2014 he was invited speaker at the ICM in Seoul (Motivic periods and Pn {0; 1; 1}).

literature

• Pierre Deligne Multizetas, d´aprés Francis Brown , Seminaire Bourbaki, No. 1048, January 2012, pdf
• Brown QFT and Arithmetic , Journées Arithmetique, IHES 2013, pdf

Fonts (selection)

• Multiple zeta values ​​and periods of moduli spaces . ${\displaystyle {\mathfrak {M}}_{0,n}}$Ann. Sci. Éc. Standard. Great. (4) 42 (2009), no. 3, 371-489. ArXiv
• Mixed Tate motives over . ${\displaystyle \mathbb {Z} }$Ann. of Math. (2) 175 (2012), no. 2, 949-976. ArXiv
• Dedekind zeta motives for totally real number fields. Invent. Math. 194 (2013), no. 2, 257-311. ArXiv
• Motivic periods and ${\displaystyle P^{1}\setminus \left\{0,1,\infty \right\}}$ . Proceedings of the ICM 2014. online

Web links

• Homepage
• Appreciation on the occasion of the CNRS bronze medal, pdf
• Videos by Francis Brown in the AV portal of the Technical Information Library

Individual evidence

1. ↑ Brown Multiple zeta values ​​and periods of moduli spaces , Annales Scientifiques de l'ENS, Volume 42, 2009, pp. 371-489, abstract
2. ↑ Brown Multiple zeta values ​​and periods: from moduli spaces to Feynman integrals . Combinatorics and Physics, 27-52, Contemp. Math., 539, Amer. Math. Soc., Providence, RI, 2011
3. ↑ Acknowledgment on the occasion of the Elie Cartan Prize, Academie des Sciences, pdf ( Memento of the original from May 25, 2013 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
4. ^ Brown: Mixed Tate motives over Z , Annals of Mathematics, Volume 175, 2012, pp. 949-976, Preprint, pdf
5. ^ Brown, Schnetz Proof of the Zig-Zag conjecture , 2012
6. ↑ Partial results were also obtained by Sternbridge in 1998, who proved them for graphs with 12 or fewer edges
7. ^ Brown, Schnetz Modular Forms and Arithmetic, 2013