Adrien Marcone (EPF Lausanne)
Lipschitz-Killing curvatures and Gauss-Bonnet-Chern theorem for Riemannian manifolds with conical ends
The first part of the talk will be dedicated to the Lipschitz-Killing curvatures of a Riemannian manifold. These are intrinsic quantities depending on the curvature tensor and they can be defined in terms of contractions of double-forms. We will present several of their properties which can be of interest when dealing with the Riemannian setting. The Lipschitz-Killing curvatures can also be constructed by pulling back onthe manifold some differential forms defined on its frame bundle. This construction is particularly useful in view of the Gauss-Bonnet-Chern theorem. In the second part, we will present a Gauss-Bonnet-Chern theorem for Riemannian manifolds with conical ends. Those are non-compact manifolds but with a suffciently strong control of the geometry at infinity allowing the total curvature to converge. The Lipschitz-Killing curvatures of some compact submanifolds appear naturally in this formula, providing an explicit quantification of the Gauss-Bonnet defect i.e. the difference of the Euler-characteristic and the total curvature.
10:20 • Université de Fribourg, Chemin du Musée 23, Lonza (Math II) seminar room
Alexia Yavicoli (University of Buenos Aires & CONICET, Argentina)
Fractals and patterns
I will talk about the relationship between the size of a set and the presence of geometric patterns, such as arithmetic progressions. In particular, there exist large sets in the sense of Hausdorff measure that avoid countably many given linear patterns, but on the other hand large sets in the sense of thickness contain many long linear patterns.
14:00 • Universität Bern, Sidlerstrasse 5, 3012 Bern, Room 228
Dr. Longting Wu (ETH Zürich)
Structures on genus 0 absolute and relative GW theory I
In the series of talks, we first review the moduli of stable maps, WDVV equation, and the construction of quantum cohomology. Next, we change our focus to relative Gromov--Witten (GW) theory and present some recent calculations on surfaces using an analog of WDVV equation in relative theory. Finally, we generalize the definition of relative GW theory to include negative contact orders, and discuss some parallel structures (quantum cohomology, Virasoro operators, etc.) on relative GW theory.
15:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
John Parker (Durham University)
Cusp regions for parabolic ends of hyperbolic manifolds
A hyperbolic manifold or orbifold can be written as the quotient of hyperbolic space by a discrete group of isometries. A cusp end of the orbifold corresponds to parabolic elements in the group. A consequence of discreteness is that these cusp ends contain regions of a certain shape. In dimensions two and three this is classical. More complicated things can happen in higher dimensions. In this talk I will survey the classical results, then I will discuss some more recent results in dimension four which show how continued fractions and Diophantine approximation come into play.
15:45 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Nicole Rohrer (Universität Zürich)
Generalization of Algorithms for Decoding Random Linear Codes
In post-quantum cryptography, the McEliece cryptosystem is one of the main candidates to replace current public key cryptosystems, such as RSA. It has been shown to be NP-complete using Goppa codes. However, information set decoding (ISD) attacks still need to be considered. A major drawback of the McEliece scheme implemented with Goppa codes is its large public key size. An idea of how to reduce it, without lowering its security, is to implement the code over a general finite field. One of the more recent and most efficient ISD attacks is the ball-collision decoding attack published by Bernstein, Lange, and Peters in 2010. In this talk, we look at the binary ball-collision decoding algorithm and present our generalization of it to general finite fields. We analyze its complexity and give some parameter examples.
16:00 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 12
Dr. Philipp Petersen (Oxford University)
Deep Neural Networks and Partial Differential Equations: Approximation Theory and Structural Properties
Novel machine learning techniques based on deep learning, i.e., the data-driven manipulation of neural networks, have reported remarkable results in many areas such as image classification, game intelligence, or speech recognition. Driven by these successes, many scholars have started using them in areas which do not focus on traditional machine learning tasks. For instance, more and more researchers are employing neural networks to develop tools for the discretisation and solution of partial differential equations. Two reasons can be identified to be the driving forces behind the increased interest in neural networks in the area of the numerical analysis of PDEs. On the one hand, powerful approximation theoretical results have been established which demonstrate that neural networks can represent functions from the most relevant function classes with a minimal number of parameters. On the other hand, highly efficient machine learning techniques for the training of these networks are now available and can be used as a black box.
In this talk, we will give an overview of some approaches towards the numerical treatment of PDEs with neural networks and study the two aspects above. We will recall some classical and some novel approximation theoretical results and tie these results to PDE discretisation. Afterwards, providing a counterpoint, we analyse the structure of network spaces and deduce considerable problems for the black box solver. In particular, we will identify a number of structural properties of the set of neural networks that render optimisation over this set especially challenging and sometimes impossible.
16:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room E 1.2