Tóm tắt các bài giảng:

1. GS. Henri Berestycki – CNRS / EHESS, Pháp

Tiêu đề: Propagation in non homogeneous media and applications

Bản tóm tắt:

I will start by introducing reaction-diffusion equations and reviewing some of the classical theory concerning the spreading properties for homogeneous Fisher and Kolmogorov-Petrovsky-Piskunov (KPP) equations. A well known invasion speed governs the asymptotic speed of propagation. This equation plays an important role in a variety of contexts in ecology, biology and physics. A fundamental question in this theory is to understand the generalization to heterogeneous versions of such equations. In this lecture, after presenting some motivation, I will describe some recent results in this direction. I will describe the effect of inclusion of a line (a “road”) with fast diffusion on biological invasions in the plane (the “field”), otherwise homogeneous. The results shed light on oriented diffusion in an excitable medium. I report here on results from a series of joint works with Jean-Michel Roquejoffre and Luca Rossi.

2. GS. Hwang, Jun-Muk – Viện Nghiên cứu cao cấp Hàn Quốc (KIAS)

Tiêu đề: Cartan-Fubini type extension theorems

Bản tóm tắt:

Cartan-Fubini type extension theorems give various settings where local structure-preserving holomorphic maps (transcendental objects) can be extended to global holomorphic maps (algebraic objects). They can be viewed as
holomorphic generalizations of Liouville’s theorem in conformal geometry. We will give an introductory survey of recent progress on this topic.

3. GS. Marc Levine – Đại học Duisburg-Essen, Đức

Tiêu đề: An overview of motivic homotopy theory

Bản tóm tắt:

Motivic homotopy theory was constructed by Morel and Voevodosky in the 1990s. It’s first applications were to a solution of the Milnor conjecture and Bloch-Kato conjecture on the Galois symbol. Since then the theory has be opened to a systematic study of its basic invariants and structures, as well as providing numerous new applications. Morel, Cisinski-Deglise, and Röndigs-Ostvar have described the relation of the motivic stable homtoopy category to Voevodsky’s triangulated category of motives. Morel’s computation of the endomorphism of the sphere spectrum points out a close relationship to quadratic forms, and has given rise to the construction of interesting new “oriented” cycle theories. Asok and Fasel have applied computations of unstable motivic homotopy groups to stability problems for algebraic vector bundles. Isaksen and others have computed motivic versions of Adams-Novikov and Adams spectral sequences, and used this information to improve known computations of these spectral sequences in classical homotopy theory. We will discuss the basic ideas going into the construction of motivic homotopy theory and some of these results and applications.

4. GS. François Loeser – Đại học Pierre and Marie Curie, Pháp

Tiêu đề: Trace formulas for motivic volumes

Bản tóm tắt:

The aim of this lecture is to present some recent trace formulas obtained in [11],[28],[27],[17] for varieties over valued fields using motivic integration. We start by recalling in §1 the trace formula of Grothendieck that provides a cohomological expression for the number of points of varieties over finite fields. We also review Grothendieck’s function-sheaf dictionary and briefly present some applications. Section 2 is devoted to introducing the motivic Serre invariant from Loeser and Sebag [22], which can be considered as a right substitute for counting point over finite fields; indeed, for smooth varieties over a valued field with perfect residue field, it provides a measure of the set of unramified points. The construction proceeds in analogy with an invariant that Serre defined for locally analytic p-adic varieties, which we recall first.

We are then in position to state in §3 the trace formula of Nicaise and Sebag [28] that provides a cohomological interpretation for the Euler characteristic of the motivic Serre invariant. In the remaining of the section we explain the connection with monodromy and the Milnor fiber as expressed in the original trace formula of Denef and Loeser [11]. We then explain the strategy of proof, relying on explicit computations on resolutions and motivic integration.

In section 4 we shall present another approach due to Hrushovski and Loeser [17]. It is based on non-archimedean geometry and avoids explicit computations on resolutions. It uses the version of motivic integration of Hrushovski and Kazhdan [16], that we review, and ultimately relies on a classical form of the Lefschetz fixed point formula.

5. GS. Cédric Villani – Viện Henri Poincaré (UPMC/CNRS), Pháp

 Tiêu đề: Synthetic theory of Ricci curvature bounds

Bản tóm tắt:

Synthetic theory of Ricci curvature bounds is reviewed, from the conditions which led to its birth, up to some of its latest developments.