1. Prof. Phùng Hồ Hải - Institute of Mathematics, VAST
Title: On the geometry of parametric linear differential equations over the punctured affine line.
Abstract:
Linear differential equations over the complex plane, punctured at a finite set of points, are primarily classified by their singularity at the punctured points. The 21-th problem in Hilbert's list asks about geometric features of such systems. Namely, given a finite dimensional complex representation of the topological fundamental group of the (punctured) plane, does there exist an equation with regular singularity (also called Fuchsian), the monodromy of which is the given representation. The answer to this question is known as the Riemann-Hilbert correspondence. In the simplest case of the complex plane punctured at a single point, this correspondence tells us that a regular singular system of differential equations is of Euler form: $dF=z^{-1}AFdz$, where $A$ is a square matrix of complex numbers. This reflects the fact that the fundamental group of the complex plane, punctured at one point, is the additive group of integers. The similar claim for an arbitrary system is known as Levelt's decomposition theorem. From the point of view of algebraic geometry, one might want to replace the complex number field by an arbitrary field, and even by a ring. In this talk, after reviewing classical theory, we discuss some recent progress in this research direction. This is a joint work with J.P. dos Santos (Sorbonne, Paris) and Pham Thanh Tam (HPU2, Hanoi).
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2. Prof. Phan Thành Nam - LMU Munich, Germany
Title: The mathematical theory of interacting Bose gases at high temperature
Abstract: We will discuss a rigorous derivation of the macroscopic behavior of an interacting Bose gas at above the critical temperature of the Bose-Einstein condensation. Starting from the many-body Schrödinger equation, we prove that the equilibrium quantum state is characterized by a nonlinear Gibbs measure in the mean-field semiclassical limit. Since the Gibbs measure is very singular, the derivation requires a Wick renormalization and several new correlation inequalities. This is joint work with Mathieu Lewin and Nicolas Rougerie.
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3. Prof. Đàm Thanh Sơn - University of Chicago, USA
Title: Schrödinger symmetry and applications in quantum mechanics
Abstract: Nonrelativistic quantum systems are mathematically described by the Schrödinger equation. In general, the group of symmetry of the Schrödinger equation is the Galilean group. In some cases the symmetry group is enlarged to the so-called Schrödinger group. We will discuss some physical systems realizing this Schrödinger symmetry and some consequences of the symmetry.
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4. Prof. Ngô Việt Trung - Institute of Mathematics, VAST
Title: Depth functions and symbolic depth functions of homogeneous ideals
Abstract: Depth is an important invariant of an ideal in a local ring. Over a polynomial ring, it is strongly related to the minimal free resolution, which describes the relations between the elements of the ideal. For many reasons, we want to know the behavior of the function of the depth of ordinary or symbolic powers of a homogeneous ideal. It was conjectured that these functions may behave wildly. These conjectures have been solved recently by a group of people partly during their research stays at VIASM. This talk will give a survey on this topic.
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5. Prof. Vũ Hà Văn
Title: Reaching a consensus on random networks: The power of few
Abstract:
A community of n individuals is split into two camps: Red and Blue. The individuals are connected by a network (graph), which has an influence on their opinions. Everyday, each individual changes his/her color according to the majority color in his/her neighborhood (the majority rule). A color wins if at some day, everyone has this color. This process has been investigated heavily in probability, game theory, and physics.
We study this problem when the underlying graph (network) is random. It is clear that at the beginning, if the two camps have the same size (n/2 each), then the probability that a particular color wins is 1/2. The most natural question here is:
--How much advantage one camp should have at the beginning so it can win with high probability, say 90% ?
The answer is 6 (six). It is surprising in the sense that it does NOT depend on the size (n) of the community. In general, for probability 1-c, one needs an advantage of size C, where C depends on c, but not on n.
In the talk I am going to give a brief introduction to the subject and some ideas that lead to the phenomenon, and discuss various open questions.
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