WELCOME TO ZETAS 2018 : Zeta Functions, Polyzeta Functions, Arithmetical Series : Applications to Motives and Number Theory LAMA Summer School U. Savoie Mont Blanc - CNRS 18-29 June 2018 LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 1 / 20
LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 2 / 20
Why such a Summer School? Origin : every two years a Series of International Meetings on Zeta Functions at the CNRS UMI Poncelet Laboratory in Moscow, traditionally in December : 1st (2006) : ZETA 1 2nd (2008) : ZETA 2. 6th (2016) : ZETA 6 7th (2018) : L-functions and algebraic varieties, in Memory of Alexey Zykin, org. Sergey Gorchinskiy, Marc Hindry, Philippe Lebacque, Sergey Rybakov, Michael Tsfasman. idea : launched by Michel Balazard and Michael Tsfasman to allow communication in the collectivity of the researchers working with zeta functions and L functions, all of them, to gather this collectivity. LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 3 / 20
List of zeta functions In mathematics, a zeta function is (usually) a function analogous to the original example: the Riemann zeta function Zeta functions include: Airy zeta function, related to the zeros of the Airy function Arakawa Kaneko zeta function Arithmetic zeta function Artin Mazur zeta-function of a dynamical system Barnes zeta function or Double zeta function Beurling zeta function of Beurling generalized primes Dedekind zeta-function of a number field Epstein zeta-function of a quadratic form. Goss zeta function of a function field Hasse Weil zeta-function of a variety Height zeta function of a variety Hurwitz zeta-function A generalization of the Riemann zeta function Ihara zeta-function of a graph Igusa zeta-function L-function, a 'twisted' zeta-function. Lefschetz zeta-function of a morphism Lerch zeta-function A generalization of the Riemann zeta function Local zeta-function of a characteristic p variety Matsumoto zeta function Minakshisundaram Pleijel zeta function of a Laplacian Motivic zeta function of a motive Multiple zeta function or Mordell Tornheim zeta-function of several variables p-adic zeta function of a p-adic number Prime zeta function Like the Riemann zeta function, but only summed over primes. Riemann zeta function The archetypal example. Ruelle zeta function Selberg zeta-function of a Riemann surface Shimizu L-function Shintani zeta function Subgroup zeta function Witten zeta function of a Lie group Zeta function of an incidence algebra, a function that maps every interval of a poset to the constant value 1. Despite not resembling a holomorphic function, the special cas integer divisibility is related as a formal Dirichlet series to the Riemann zeta function. Zeta function of an operator or Spectral zeta function See also Other functions called zeta functions, but not analogous to the Riemann zeta function LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 4 / 20
> associated to several Conjectures. In this Summer School : - Riemann Hypothesis (1859), - Manin Conjecture, - Lehmer Conjecture, - Monodromy Conjecture,... Objectives : the courses will lead you to some transversal aspects of these functions, like zeroes, poles, natural boundaries, combinatorics of coefficients, linked to geometrical problems in Singularity theory of varieties and number theoretical problems. LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 5 / 20
Who? Scientific Committee : Georges Comte (Univ. Savoie Mont Blanc) Emmanuel Kowalski (ETH Zürich) Francois Loeser (Univ. Orsay) Mark Pollicott (Univ. Warwick, UK) Emmanuel Royer (Univ. Clermont-Ferrand) Jean-Louis Verger-Gaugry (CNRS, LAMA, USMB) Organizing Committee : adm. : (Sylvie David) now Laure Bastide (LAMA)/coord. Janine Brison (USMB)/coord. Emilie Morales, Juliette Guély (CNRS) Georges Comte, Michel Raibaut, Jean-Louis Verger-Gaugry LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 6 / 20
Our sponsors : Project of this Summer School : submitted in 2017 to the CNRS, accepted and classified as CNRS 2018 Thematic School (Laurence Caillat, Laurent Guillopé, Laurent Serlet) University Savoie Mont Blanc - AAP Relations Internationales /CORI/DRED (Françoise Limousin, Isabelle Couchoud) Two GDRs support the Summer School : - GDRSTN Structuration de la Théorie des Nombres (G. Ricotta), - GDR Singularités et Applications (E. Mann). + Labex Persyval-Lab (Marie-Christine Rousset), Région Auvergne Rhônes Alpes, LAMA (Philippe Briand), Fédération de Recherche en Mathématiques MARA (Auvergne Rhônes Alpes)(Arnaud Guillin), LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 7 / 20
About the Courses (June 18-29, 2 weeks) : each Course is XY 1, 2, 3, or 4, 1h30 each : Siegfried Boecherer : L p-adic and Complex Functions on Classical Groups : Admissible Measures, Special Values I Georges Comte : Real Motivic Milnor Fiber of Semi-algebraic Formulas, o-minimality. Driss Essouabri : Dirichlet Series and Zeta Functions of One or Several Complex Variables. Alexei Pantchichkine : L p-adic and Complex Functions on Classical Groups : Admissible Measures, Special Values II. Michel Raibaut : Igusa Zeta Functions, Motivic Zeta Functions, Monodromy Conjecture. Jean-Louis Verger-Gaugry : Limit Conjectures in Number Theory, Lehmer s Conjecture, Schinzel-Zassenhaus s Conjecture, Dynamical Zeta Function of the β-shift. LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 8 / 20
Summer School ZETAS 2018 : Zeta Functions, Polyzeta Functions Arithmetical Series : Applications to Motives and Number Theory. 9.00-10.30 Welcome 10.15: Presentation of the Summer School Week 1 : June 18-22 2018 Monday Tuesday Wednesday Thursday Friday AP 1 Contributed talks, short talks, clips, 1 DE 2 DE 4 10.30-11.00 Coffee break Coffee break Coffee break Coffee break Coffee break 11.00-12.30 VG 1 GC 1 Mini-Workshop : Voronin Universality Theorem 1 GC 2 VG 2 12.30-14.00 Lunch Lunch Lunch Lunch Lunch 14.00-15.30 participants MR 1 Excursion 1 DE 3 Introduction to PARI/GP with zeta functions 15.30-16.00 Coffee break Coffee break '' Coffee break Coffee break 16.00-17.30 DE 1 SB 1 '' AP 2 PARI/GP (continuation) 20.00 - Aix-Les-Bains Week 2 : June 25-29 2018 LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 9 / 20
Week 2 : June 25-29 2018 9.00-10.30 Monday Tuesday Wednesday Thursday Friday Contributed talks, short talks, clips, 2 MR 2 Contributed talks, short talks, clips, 3 Voronin 3 SB 4 10.30-11.00 Coffee break Coffee break Coffee break Coffee break Coffee break 11.00-12.30 VG 3 GC 3 Voronin 2 GC 4 VG 4 12.30-14.00 Lunch Lunch Lunch Lunch Lunch 14.00-15.30 SB 2 MR 3 Excursion 3 invited talk MR 4 15.30-16.00 Coffee break Coffee break '' Coffee break Coffee break 16.00-17.30 AP 3 SB 3 '' AP 4 20.00 - Le Bourget-du- Lac LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 10 / 20
Two Mini-Workshops : - Voronin s Universality Theorem (probabilistic attack of the Riemann Hypothesis for the Riemann Zeta Function) : 3 sessions : 1st (week 1) with an Introduction by J-L Verger-Gaugry, following a probabilistic approach of Laurincikas, 2nd and 3rd (interactive sessions with other zeta functions and L-functions, week 2) : personalization to your own zeta or L-function, following Voronin s ideas. - Mini-workshop Introduction to the PARI/GP software (for number theory initially) with Bill Allombert, from the Institut Mathématique de Bordeaux LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 11 / 20
3 Sessions of Contributed talks (encouraged) : on Wednesday 20th, Monday 25th, and Wednesday 27th. Possibility of video recording and diffusion on the web - if interested, please contact Laure Bastide for giving the authorization) - Marc Munsch (Monday 25th) : Large Values of L-Functions -... coming soon today : Scientific Program of the contributed talks. Title and abstract to be sent by email to JLVG (etzetas2018@sciencesconf.org), choose the day you prefer for your talk. LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 12 / 20
Video Recording of the Courses : for a new series of the Mathematics Department LAMA (on YouTube,...) Merci de ne pas oublier de rallumer votre portable après chaque Cours. Thank you for not forgetting to switch on again your mobile phone after each Course. ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo The doctoral activities of the LAMA are part of the PhD Program of the Ecole Doctorale of Univ. Grenoble Alpes. LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 13 / 20
Working areas (with WiFi) : BU Bourget (University Library), main Lecture Rooms Nivollet, Revard, with blackboards, Room TLR (first floor at LAMA), with blackboards. WiFi everywhere except at the Résidence Les Arves (a wire is necessary - disconnect your WiFi before connexion - Eduspot and select Université de Savoie ). Eduroam is available. LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 14 / 20
LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 15 / 20
Excursions : on Wednesday 20th June afternoon : Relais du Mont du Chat/ Molard Noir on Wednesday 27th June afternoon : Croix du Nivolet on Saturday 23rd June the whole day : Chamonix if you do not wish to participate in the excursions, please contact urgently Laure Bastide. LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 16 / 20
LAMACampus Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 17 / 20
Evenings on Thursday 21st (proposed), after dinner : Fair of Music at Aix-Les-Bains (by yourself), on Thursday 28th, after dinner : Le Bourget-du-Lac/harbour, beach/ desserts and beverages. LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 18 / 20
T-shirts with logo (one is free for every participant) : if interested, ask Laure Bastide. LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 19 / 20
next : Int. Conf. on Zeta Functions (in Dec 2019) : at CIRM, Marseilles, org. F. Jouve,... ZETA 8, ZETA 9,..., in Moscow. LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 20 / 20