General Information
Time and venue
The sessions take place every
Thursday from 11:00 to 12:00 in room
MSA 3.230 (Maison Du Savoir) on Campus Belval.
Aim
Dedicated to PhD students, the PhD Seminar is an important part of the research training offered at the
Department of Mathematics of the
University of Luxembourg. The PhD Seminar has a twofold
objective:
 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.
Organisers
Summer semester: Talks

21 February 2019
Organisational meeting

01 March 2019
Robert BAUMGARTH,
An Introduction to Stochastic Differential Geometry
Abstract The most wellstudied example of a stochastic process is Brownian motion. The deep connection between the Laplacian as being the generator of Brownian motion (up to a constant) naturally extends to the setting of (smooth Riemannian) manifolds. This has led to a completely new research area called Stochastic Differential Geometry, i.e. the stochastic analysis on manifolds. Stochastic methods open ways to solve analytic and geometric problems in a much more elegant fashion. Hence, we discuss briefly modern notions and key concepts of this relatively young and interdisciplinary subject.
Starting from the wellknown notion of a flow to a vector field, we will show that stochastic flows naturally extend to a flow to a partial differential operator. As an application we sketch how these concepts can be used to give an elegant and short proof for existence and uniqueness of a solution to the Dirichlet problem.
This leads to the definition of Brownian motion on (smooth Riemannian) manifolds: the intrinsic approach as solution to the usual martingale problem using a Whitney embedding and the EellsElworthyMalliavin approach using the projection from the orthonormal frame bundle.
Finally, we briefly discuss applications. In particular, curvature and gradient estimates of the heat semigroup.
Concerning the broad audience, all notions will be briefly introduced during the talk as needed.

Abstract After an intuitive motivation I will present two equivalent definitions of the Grothendieck topology. Then I will introduce a formulation of the sheaf condi
tion on a site i.e. a category with a Grothendieck topology. All definitions will be followed by examples.

Abstract Algebraic geometry has allowed geometry to reach more abstract objects in more general categories. An example of this is noncommutative geometry. In this theory, no notion of point can be defined. A nice middle step towards noncommutative manifolds (and other abstract geometric objects in modern algebraic geometry) is the category of supermanifolds. These manifolds are already out of reach of classical geometry (Riemannian, differential, etc.), and the usual notion of point is not good enough. In the talk, I will start with a fast introduction to supergeometry following two of the standard approaches, the second one widely known and used by the community. My goal is to present a modified version of the socalled functor of points of a supermanifold, which allows to enlarge the category of supermanifolds and allows to give a more intuitive notion of point in such a geometric ambience which, restricted to the classical geometry case, coincides with the usual. For this, it will be necessary to introduce some ideas from category theory, which are quite important for any theory concerning algebraic geometry.

04 April 2019
Alexey KALUGIN,
An introduction to the theory of mixed Hodge structures
Abstract De Rham cohomology of every complex smooth projective variety carries extra structure defined by a decomposition into holomorphic and antiholomorphic parts called Hodge decomposition. In his groundbreaking papers Deligne introduced a notion of a mixed Hodge structure as a generalization of Hodge decomposition when a variety is not projective and smooth. Since then mixed Hodge structures obtained a lot of applications in the number theory, mirror symmetry, quantizations ... I am going to give a gentle introduction to this beautiful work of Deligne and also discuss some classical application as well as the most recent one.

Abstract In 1976, two cryptographers Diffie and Hellman have proposed a way to share a secret key between two people, say Alice and Bob. For this construction, Alice and Bob first need to agree on a group to build on their shared secret. Several generalizations of this protocol have been proposed in the years. In this talk we will discuss how such protocols can be constructed from elliptic curves, widely used in elliptic curve cryptography. Namely, we will focus on two key exchange protocols: first, the classical ECDH (Elliptic Curve DiffieHellman) and the more recent, conjecturally quantumsecure, SIDH (Supersingular Isogeny DiffieHellman) by De Feo, Jao and Plût, from 2011. We will first start by a suitable framework to study such keyexchange protocols by recalling the DiffieHellman protocol. In a second step, we will give the necessary mathematical background on elliptic curves to explain the construction of SIDH.
Herefore, we mainly work with isogenies between elliptic curves, that are special group morphisms. We will also focus on some algorithmic questions related to the problem.

25 April 2019
Valentin GARINO,
Introduction to Malliavin Calculus and application to Stochastic Differential Equation
Abstract A Stochastic Differential Equation is roughly an ordinary Differential Equation with an additional perturbation, that is a random noise. As a consequence, its solutions are timeindexed random variables. A crucial problem is whether or not these random variables admit a density with respect to Lebesgue’s measure. This question was partially solved by Hormander and Malliavin and is still an active research field. Malliavin’s proof was also the birth act of Malliavin Calculus, which is basically a complete IntegroDifferential theory for stochastic processes.
After introducing the notions related to Stochastic Integral Calculus and Stochastic Differential Equations, I will discuss the basic notions of Malliavin Calculus, which are the relevant tools to address this problem. I will then finish by sketching the proof of Hormander’s Theorem.

02 May 2019
Lara DAW,
Sojourn time dimensions of fractional Brownian motion
Abstract Describing the properties of the sample paths of stochastic processes (Xt)t≥0 is one of the leading threads of modern stochastic analysis. A naturally connected question consists in describing the geometric properties of the graph of {(t, Xt) : t ≥ 0}, in terms of box, packing and Hausdorff dimensions. Another description of random trajectories was proposed in terms of sojourn times. The objective is to describe the (asymptotic) proportion of time spent by a stochastic process in a given region. Sojourn times have been studied by many authors and play a key role in understanding various features of the paths of stochastic processes, especially those of Fractional Brownian motion.
After introducing some notions related to Fractional Brownian motion and sojourn sets associated to it, I will introduce the microscopic Hausdorff dimension and two different densities. Computing the macroscopic Hausdorff dimension, logarithmic and macroscopic densities of the sojourn sets will give us some geometric properties of the path of Fractional Brownian motion. I will also give a uniform macroscopic dimension result for the Fractional Brownian level sets.
KEYWORDS: Sojourn time; logarithmic density; pixel density; macroscopic Hausdorff dimension; fractional Brownian motion.
Aim
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 2019 took place in
Bordeaux, France from October 19 to
October 21, 2019.
The presentations were assessed by Mikołaj KASPRZAK and George KERCHEV and James THOMPSON.
Organisers
The titles and abstracts of the talks can be downloaded here.
This year’s PhD Away Days were organised in cooperation with the PhD candidates of the Université de Bordeaux. The organisers would like to address their special thanks to Baptiste Huguet for sacrificing his precious time.