Time and venue
The sessions take place every Thursday from 09:15 to 10:45
6B of the Maison du Nombre on Campus Belval
Dedicated to PhD students, the PhD Seminar is an important part of the research training offered at the
Department of Mathematics
University of Luxembourg
. The PhD Seminar has a twofold
- To give PhD students the opportunity to present their fields of research
and some topics and problems studied in those fields.
The usual audience of the PhD Seminar is composed by PhD students and
young researchers of the Department of Mathematics. The atmosphere is informal, so that all participants
can ask the questions they want.
Comments and suggestions on the presentations are also highly encouraged.
- To provide a good general mathematical background, in relation with the
research interests of groups of PhD students
and other participants of the PhD Seminar.
In this perspective, the PhD Seminar proposes all over the academic year
series of introductory minicourses, held by
the young researchers of the Department of Mathematics.
Who can be a speaker?
All contributions for talks or minicourses are welcome, including from people external to Luxembourg's
Department of Mathematics!
If you want to make a presentation, please provide the organizers with a title and an abstract of your talk.
Lecture notes in any form are welcome as well.
Summer semester: Talks
10 March 2017
30 March 2017
Optimal transport and spaces with Ricci curvature bounded below
Abstract Given a Riemannian manifold (that is a particular metric space on which the notion volume can be defined) a very natural question is to understand how the volume of balls varies depending on its radius and/or its center. Formalizing this idea of the interplay between distances and volumes led to a rich theory known as "theory of Riemannian manifolds with Ricci curvature bounded below". Rather than focusing on technical details, I will illustrate the power of this theory with numerous examples. On the other hand one can see a volume element as a distribution of mass that one wants to transport (reorganize) to an other volume element (distribution of mass). Given a cost of moving volume elements, one might wonder what is the most cost-effective way of transporting the mass. This problem is formally known as "optimal transport". When the cost depends explicitly on the distance, optimal transport strongly intertwines volumes and distances. The volume-distance intrication appearing both in optimal transport and the theory of Riemannian manifolds with Ricci curvature bounded below suggests a strong link between the two theories. In the last 15 years, this link has been discovered and explicated : any statement of the theory of Ricci curvature bounded below can be rephrased in an optimal transport fashion. The optimal transport point of view does not require any smoothness or regularity assumptions and can therefore be used to study rougher spaces.
06 April 2017
About the mathematical legacy of S. Ramanujan: Congruences between modular forms
Abstract In his article “On certain arithmetical function” in 1916, S. Ramanujan defines the arithmetic function Tau. First, we will present his results concerning the arithmetic properties of this function, and later we will restate them in terms of the modern language of modular forms. Secondly, we will discuss open questions that arose at the end of his work . Ramanujan’s achievements inspired the work of two Fields medalist J.P. Serre and P. Deligne who were the first ones to clarify the deep nature of his results. Their work gave birth to the theory of Galois representations attached to modular forms which is now a central topic in number theory and for instance, it is the main tool in A. Wiles proof of Fermat’s Last Theorem.
13 April 2017
Operads of compactified configuration spaces
Abstract In this talk we are going to discuss how different types of algebras can be encoded by an algebraic object called operad. Then we are going to show how a certain compactification of configuration spaces can give us a geometric interpretation of homotopy (Lie and associative)-algebras. This interpretation can also serve as an independent description of what a homotopy Lie-algebra is (so the audience does not have to know that beforehand). We will also outline how we can use this interpretation to construct interesting algebraic structures.
27 April 2017
A parametrization of complex projective structures on surfaces
Abstract The aim of this talk will be to give an introduction of the concept of complex projective structures on surfaces and to describe a parametrization result of them. We will firstly give the necessary definitions and we will describe how to build the developing map and the holonomy representation of such a structure. Finally, we will discuss the connection between these objects and the data of a Riemann surface structure and a holomorphic quadratic differential on it.
04 May 2017
Stochastic Flow Processes and Brownian Motion on Manifolds
Abstract We study stochastic flow processes and diffusions on smooth Riemannian manifolds starting from the well-known notion of a flow to a vector field. As an application we sketch how these concepts can be used to give a very simple proof for existence and uniqueness of a solution to the Dirichlet problem. We give a brief overview how to define Brownian motion on (smooth Riemannian) manifolds: the extrinsic approach as solution to the usual martingale problem using a Whitney embedding and the Eells-Elworthy-Malliavin approach using the projection from the orthonormal frame bundle. All notions will be briefly introduced during the talk as needed concerning the broad audience.
Abstract In generality, homotopy theory is the study of mathematical contexts in which functions are equipped with a concept of homotopy between them. A key aspect of the theory is that the concept of isomorphism is relaxed to that of homotopy equivalence: Where a function is regarded as invertible if there is a reverse function such that both composites are equal to the identity, for a homotopy equivalence one only requires the composites to be homotopic to the identity. Model categories give rise to a large class of homotpy theories, and provide a setting that is suitable for concrete calculations. My aim is to explain the basics of category theory, define model categories and show how to work in this setting. I will state the Quillen transfer theorem and illustrate it in the theory of homological algebra. Given the time, I will say something about the derived critical locus in the derived algebraic geometry.
Abstract Aggregation functions defined on linguistic scales (i.e., finite chains) have been intensively investigated for about two decades. Among these functions, fuzzy connectives (such as uninorms) are binary operations that play in an important role in fuzzy logic. In this talk we focus on the characterization of the class of idempotent uninorms on finite chains. Indeed, we provide an axiomatic characterization of the idempotent uninorms in terms of three conditions only : quasitriviality, symmetry and non-decreasing monotonicity. Moreover, we provide a graphical characterization of these operations in terms of their contour plots. Finally, we present an algebraic translation of the previous graphical characterization in terms of single-peaked linear orderings.
The PhD seminar is replaced once per year by the « PhD Away Days », i.e. a weekend
during which the PhD candidates present their research area or related topics to their fellow PhD students
hence strengthening their presentation skills and socialise with the other PhD candidates. How to organise
the mathematical content (GMS style talks, mini courses etc.) and the journey and corresponding activities
varies depending on the organising PhD representative(s).
Dates and Location
The PhD Away Days 2017 took place in Durbuy, Belgium from September 29 to
September 30, 2017
The presentations were assessed by Christian DÖBLER and James THOMPSON.
The titles and abstracts of the talks can be downloaded here.