Dedicated to PhD students, the PhD Seminar is an important part of the research training offered at the
. The PhD Seminar has a twofold
objective:
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.
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22 February 2018
Organisational meeting
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Abstract In this talk, I will begin with Thomas Royen’s simple proof of the long-standing Gaussian correlation conjecture, which lies in the intersection of convex geometry and probability. If time permits, I will mention another conjecture in the Gaussian world, the Gaussian (moment) product conjecture, and some recent progress.
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Abstract In this talk, I will talk about congruent numbers. After giving their definition and a brief historical introduction, I will explain various formulation of the problem. At the end, I will explain the correlation between the congruent numbers problem and the weak Birch and Swinnerton-Dyer Conjecture on elliptic curves.
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22 March 2018
Alexey KALUGIN,
Geometric structures in conformal field theory
Abstract Chiral algebras were invented as geometric counterpart of vertex algebras. Notion of Chiral algebra plays crucial rule in the geometric Langlands correspondence. I am going to discuss basic definitions and constructions from this theory.
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Abstract After discussing the L-functions in the ad hoc fashion we will introduce the Local zeta integrals in which we will state the multiplicity one theorem in the form of distribution of a space of Schwartz-Bruhat functions on a given number field. Finally, the main result of Tate's thesis on the Global zeta integrals will get a treatment.
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19 April 2018
Luca NOTARNICOLA,
Multilinear maps and secure key exchange in cryptography
Abstract Recently, many cryptographers have put a lot of research in multilinear maps for cryptography. These are a generalization of bilinear maps, also known as bilinear pairings, and which have already been proved to have many applications in cryptography, as for instance a secure key exchange between three people, known as the 3-partite one-round Diffie-Hellman key exchange by Antoine Joux, 2000, a direct generalization of the classical Diffie-Hellman key exchange protocol by Diffie and Hellman in 1976. Roughly speaking, by key exchange protocol, we mean that two (or more?) people want to share a common message (a key) by communicating over an insecure channel. The goals of this talk are first to first define such multilinear maps; second to study some elliptic curve arithmetic and bilinear pairings (especially, the Weil pairing) on elliptic curves, in order to, third, understand the key exchange protocol on elliptic curves by Joux. To conclude, a direct application of multilinear maps in order to generalize these results.
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26 April 2018
Andrea GALASSO,
Schur-Horn Problem and Symplectic Geometry
Abstract The Schur-Horn problem is the following: if H is an Hermitian matrix with given eigenvalues, what could the vector of its diagonal entries be? In this talk I would like to explain how this problem is related to the convexity properties of the moment map in symplectic geometry.
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Abstract At the end of the 18th century, Ernst Chladni, a physicist and musician, made an interesting discovery: he observed that when he excited a metal plate with the bow of his violin, he could hear sounds of different frequency. The plate was fixed only at its center, and when Chladni put some sand on it, then for each frequency a curious pattern appeared, today known as Chladni figures. Some time later, Kirchhoff pointed out that these patterns correspond to nodal sets of eigenfunctions of the biharmonic operator. In this talk, we introduce random Laplacian eigenfunctions on the three-dimensional torus, known as arithmetic random waves. More precisely, we prove a limit theorem for the nodal surface of arithmetic random waves as the eigenvalue goes to infinity.
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Abstract Our talk concentrates on summation and regularization methods of divergent series. Particular focus will be put on the famous regularization of the divergent series 1+2+3+... to the value of -1/12, which is of uttermost importance in modern physics. The talk splits into two sessions. The first session will be an introductory part, whereas the technical details will be given in the second session.
(Jill Marie-Anne ECKER, May 17) The introductory part contains an historical overview on divergent series. This includes the treatment of divergent series by mathematicians of the 18 th century, such as Euler, Leibniz or Bernoulli. Moreover, we will show the importance of divergent series in modern physics such as quantum field theory and string theory.
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Abstract The Malgrange-Ehrenpreis Theorem states that every non-zero linear partial differential operator with constant coefficients in R^n has a fundamental solution. This theorem was a first evidence of the impact of distribution theory in its application to linear partial differential equations and, therefore, there were found several different proofs of it in subsequent years. The aim of this talk is to shortly survey the different proofs and to focus on the classical proof of Ehrenpreis and Malgrange by giving some illustra
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Abstract Our talk concentrates on summation and regularization methods of divergent series. Particular focus will be put on the famous regularization of the divergent series 1+2+3+... to the value of -1/12, which is of uttermost importance in modern physics. The talk splits into two sessions. The first session will be an introductory part, whereas the technical details will be given in the second session.
(Thi Hanh VO, May 31) The second part includes several summation methods of divergent series. The definitions of the summation methods are accompanied by a plethora of examples of summations of famous divergent series. Several theorems allow to compare the strength of the various summation methods. Furthermore, we will present regularization via the analytical continuation of the zeta function, with a particular focus on the derivation of the result 1+2+3+...=-1/12. We conclude by giving other values of the zeta-function in terms of the Bernoulli numbers.
