Fundamental group schemes in Arithmetic Geometry
Tuan Chau, Quang Ninh, 21-25, May, 2017

Time:09:00:21/05/2017 to 12:00:25/05/2017

Organisers: Phùng Hồ Hải (Institute of Mathematics, VAST) Ngô Lâm Xuân Châu (Quy Nhơn University)

Scientific Committee: Hélène Esnault (Free University, Berlin) Phùng Hồ Hải (Institute of Mathematics, VAST) Joao Pedro dos Santos (University of Paris VI)

Purpose: To report recent progress in the study of fundamental group schemes and their applications in arithmetic geometry. There will be two mini-courses held by H. Esnault (Berlin) and T. Abe (Tokyo) on Crystals and some invited talks. We shall organize in the afternoon and in the evening some discussion sessions, where the PhD students and the young researchers can propose questions, ideas and discuss them with the senior researchers.

Invited Speakers: Prof. Hélène Esnault (FU Berlin) and Prof. Tomoyuki Abe (Univ. of Tokyo) will give a minicourse on Crystals and Companion (each four one-hour lectures). Other invited speakers: Prof. Xiaotao Sun (Institute of Math., Chinese Academy of Science) Prof. Manuel Blickle (University of Mainz, Germany) Dr. Joao Pedro dos Santos (University of Paris VI) Dr. Pham Hung Quy (FPT University, Hanoi) Dr. Ngô Lâm Xuân Châu (Quy Nhơn University) Hugo Bay-Rousson (University of Paris VI) Marco d’Addezzio (FU Berlin) Efstathia Katsigianni (FU Berlin)


Language: English

Contact: Phùng Hồ Hải, IMH, VAST ( Địa chỉ email này đang được bảo vệ từ spam bots, bạn cần kích hoạt Javascript để xem nó. Bạn cần kích hoạt Javascript để xem nó. ) and Lê Thị Quý ( Địa chỉ email này đang được bảo vệ từ spam bots, bạn cần kích hoạt Javascript để xem nó. Bạn cần kích hoạt Javascript để xem nó. )


8:00-9:00 Abe (Tokyo Univ)
9:30-10:30 Esnault (FU Berlin)
11:00-12:00 dos Santos  (Paris 6)
14h00-15h00 Ngo Lam Xuan Chau (Quy Nhon)
15h30-16h30 Bay-Rousson (Paris 6)
8:00-9:00 Esnault
9:30-10:30 Abe
11:00-12:00 Otabe (Tohoku Univ)
afternoon free
8:00-9:00 Abe
9:3010:30 Esnault
11:00-12:00 Pham Hung Quy (FPT Univ)
14h00-15h00 Katsigianni (FU Berlin)
15h30-16h30 d’Addezzio (FU Berlin)
8:00-9:00 Esnault
9:30-10:30 Abe
Title and Abstract
Ngô Lâm Xuân Châu:
A birational equivalence of algebraic ordinary differential equations of order one and applications
Abstract: In this talk we study a birational equivalence of algebraic ordinary differential equations of order one and its applications to deduce a degree bound for algebraic general solutions of the equivalent classes of autonomous differential equations.
Tomoyuki Abe
Isocrystal and p-adic cohomology
Abstract: A goal of this series of lectures is to overview the p-adic cohomology theory over field of positive characteristic.
In the lecture, I will try to make analogies between l-adic theory or complex theory and p-adic theory as clear as possible.
As a 6 functor formalism for p-adic cohomology theory, I will briefly explain the theory of arithmetic D-modules as well, and will give some applications.
J.P. dos Santos
Differential Galois groups over a DVR: basic properties
Abstract: Differential Galois Theory of equations with a parameter (= integrable connections on schemes over a DVR) produces group schemes which sometimes fail to be algebraic. On the other hand, I’ll explain how, at least in characteristic (0,0), this unpleasant property is a consequence of an automatic process: one picks an algebraic group and one blows a formal subscheme up.
Then, I’ll explain — with the help of standard material — that even in the case of “regularity at infinity” (= connections on proper schemes) group scheme of infinite type appear naturally. To end, I shall argue that, in spite of the preceding peculiarity, “regularity at infinity” does impose some good properties on the associated group schemes. (This is work in collaboration with P. H. Hai.)
Hélène Esnault
Lefschetz Theorems in Arithmetic Geometry
Abstract: We survey classical material around Lefschetz theorems for fundamental groups, and show the relation to parts of Deligne's program in Weil II.
Shusuke Otabe
On a purely inseparable analogue of the Abhyankar conjecture for affine curves
Abstract: Let $U$ be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck's 'etale fundamental group $pi_1^{text{'et}}(U)$. In this talk, I will consider a purely inseparable analogue of this problem, formulated in terms of Nori's profinite fundamental group scheme $pi^N(U)$. I will state a conjectual description of all the infinitesimal quotients of $pi^N(U)$ and give several evidences of it.
Pham Hung Quy

Filter regular sequences and $F$-sinularities

Abstract: In this talk, we give applications of of filter regular sequences in study of some singularities defined by Frobenius action.

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