Hugo Parlier (University of Luxembourg)
Abstract:
The crossing lemma for simple graphs gives a lower bound on the necessary number of crossings of any planar drawing of a graph in terms of its number of edges and vertices. Viewed through the lens of topology, this leads to questions about arcs and curves on surfaces such as how many crossings must a collection of m homotopically distinct curves on a surface of genus g induce.The talk will be about joint work with Alfredo Hubard where we explore some of these, using tools from the hyperbolic geometry of surfaces in the process.
Prof. Dr. Daren Wei (National University of Singapore)
Time Change Rigidity of Unipotent Flows abstract
Abstract:
Two non-isomorphic ergodic measure preserving flows can become isomorphic if one of the systems undergoes an appropriate time change. In this case we will say that these flows are Kakutani equivalent to each other. We say that an ergodic flow is loosely Kronecker if it is Kakutani equivalent to the straight line flow on (say) a two torus in an irrational direction (the exact direction is immaterial as these are all equivalent to each other). Landmark work of Ratner from the late 70s (that paved the way to her even more famous results on orbit closures and equidistribution of unipotent flows) establishes that 1) the horocycle flow on any finite area surface of constant negative curvature is loosely Kronecker. 2) the product of two such flows is not loosely Kronecker. It remained an open problem whether e.g. products of two horocycle flows are Kakutani equivalent to each other. We show unipotent flows are very rigid under time changes, and indeed unless the flows are loosely Kronecker, two unipotent flows are Kakutani equivalent if and only if they are isomorphic as measure preserving flows. This is a joint work with Elon Lindenstrauss
13:30 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 28
Aloïs Demory (Genève)
Primitive real algebraic surfaces in 3-dimensional toric varieties abstract
Abstract:
The study of topology of real algebraic hypersurfaces is classically divided into two complementary directions : on one hand, finding restrictions on the topology of the real part of real algebraic hypersurfaces with given Newton polytope, and on the other hand, constructing real algebraic varieties with interesting topological properties of their real part. Primitive patchworking is a very fruitful combinatorial construction tool introduced by O. Viro that allowed to construct many maximal (with respect to the Smith-Thom inequality) real algebraic hypersurfaces in various smooth ambient spaces. The very specific topological properties of the hypersurfaces produced using this method are quite well studied in the case of hypersurfaces in smooth toric varieties. We present an ongoing attempt to extend some of these properties to primitive surfaces in arbitrary 3-dimensional toric varieties. As a consequence, new maximal surfaces in certain singular and non-singular toric 3-folds are constructed.
14:00 • Université de Genève, Conseil Général 7-9, Room 6-13
Samuel Lavenir (EPFL)
Abstract:
In this talk, I will discuss the problem of extending functors to fiberwise constructions in the setting of higher topoi. I shall explain the relevance of such extensions both from a practical point of view, and through the lens of the internal logic of the topos. After discussing some known results about fiberwise extensions of localization functors, we will focus on the dual situation of cellularizations. We will see that extending such colocalizations is almost never possible, and relate our result to classical statements by Chachólski and Hernández. Along the way, emphasis will be put on the pertinence of higher categorical language to formalize the problem of fiberwise constructions.
Tommaso Cortopassi (Scuola Normale Superiore di Pisa)
Abstract:
We introduce a new stability estimate for comparing the regular Lagrangian flow of a Sobolev vector field to a piecewise affine approximation generated by an explicit Euler-like method, in the spirit of Crippa-De Lellis\'s estimates. We use this estimate to prove approximation results for solutions of the continuity equation, which can be represented as the push forward of the initial datum via the regular Lagrangian flow. We give two examples: a probabilistic one using Dirac deltas to approximate the initial datum and a deterministic one using a diffuse approximation instead. In both cases, we assume no regularity on the mesh partitioning the spatial domain.
14:15 • Universität Basel, Spiegelgasse 5, Seminarraum 05.002
Niklas Gassner (Universität Zürich)
Codes over Rings, their Generic Decoding, and their use in Code-Based Cryptography abstract
Abstract:
Code-based cryptography is an area of post-quantum cryptography. Several code-based candidates compete in the standardization process of the National Institute of Standards and Technology. A common drawback of code-based cryptosystems is their often large public key size. An approach to decrease this size is the usage of convolutional codes. The idea behind this is that the public key sizes scales linearly in the maximum degree of its polynomial entries. However, thus far the complexity of generic attacks has been exponential in the size of the sliding generator or parity-check matrix. We give a framework which allows us to iteratively decode convolutional codes with information-set decoding. This method relies on the choice of several parameters whose choice affects the success probability of the algorithm and the running time, so we give tools to determine whether a choice of parameters is suitable. We also discuss reasons why the algorithm sometimes doesn\'t terminate and how to circumvent said issues. Finally, we use an implementation of our algorithm to attack two cryptosystem based on convolutional codes. For the first cryptosystem, we managed to recover about 74% of messages in our experiment, each in less than 10 hours. For the second proposal, we used our algorithm to give estimates on the running time and success probability. In two cases where the estimate was low we also verified the result by running the full algorithm. 80% of the experiments were successful and they indicate that the algorithm has a complexity significantly below the security estimates provided by the authors.
15:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 28
Ben Lowe (University of Chicago)
Finiteness of Totally Geodesic Hypersurfaces in Negative Curvature abstract
Abstract:
There is a broad body of work devoted to proving theorems of the following form: spaces with infinitely many special sub-spaces are either nonexistent or rare. Such finiteness statements are important in algebraic geometry, number theory, and the theory of moduli space and locally symmetric spaces. I will talk about joint work with Simion Filip and David Fisher proving a finiteness statement of this kind in a differential geometry setting. Our main theorem is that a closed negatively curved analytic Riemannian manifold with infinitely many closed totally geodesic hypersurfaces must be isometric to an arithmetic hyperbolic manifold. The talk will be more focused on providing background and context than details of proofs and should be accessible to a general audience.
15:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Greg Pavliotis (Imperial College London)
Interacting particle systems and their mean field limit: phase transitions, control, and inference abstract
Abstract:
In this talk, we will present recent results on the quantitative study of stochastic interacting particle systems and of their mean field limit. We will start by exploring the relationship between the large N limit of the constant in the Logarithmic Sobolev Inequality and the presence or absence of phase transitions for the mean field PDE. We will then present methods for computing both stable and unstable stationary states of the mean field PDE and for steering the dynamics towards a chosen steady state using optimal control methodologies. Finally, we will study the problem of inferring the interaction potential from discrete space-time observations of the McKean-Vlasov PDE.
16:00 • EPF Lausanne, MA B2 485
Prof. Dr. Carlos Jerez-Hanckes (Universidad Adolfo Ibañez, Santiago, Chile)
New Insights on Wave Scattering by Multiple Open Arcs: Lightning-Fast Methods and Shape Holomorphy abstract
Abstract:
In this talk, we will focus on solving time-harmonic, acoustic, elastic and polarized electromagnetic waves scattered by multiple finite-length open arcs in unbounded two-dimensional domain. We will first recast the corresponding boundary value problems with Dirichlet or Neumann boundary conditions, as weakly- and hyper-singular boundary integral equations (BIEs), respectively. Then, we will introduce a family of fast spectral Galerkin methods for solving the associated BIEs. Discretization bases of the resulting BIEs employ weighted Chebyshev polynomials that capture the solutions\' edge behavior. We will show that these bases guarantee exponential convergence in the polynomial degree when assuming analyticity of sources and arc geometries. Numerical examples will demonstrate the accuracy and robustness of the proposed methods with respect to number of arcs and wavenumber. Moreover, we will show that for general weakly- and hyper-singular boundary integral equations their solutions depend holomorphically upon perturbations of the arcs\' parametrizations. These results are key to prove the shape holomorphy of domain-to-solution maps associated to BIEs appearing in uncertainty quantification, inverse problems and deep learning, to name a few applications. Also, they pose new questions you may have the answer to!
16:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
Dr. Antoine Jego (EPFL, Lausanne (From Nov 2024: Ceremade, Paris Dauphine))
Crossing exponent of the Brownian loop soup abstract
Abstract:
We study the clusters of loops in a Brownian loop soup in some bounded two-dimensional domain with subcritical intensity θ ∈ (0, 1/2]. We obtain an exact expression for the asymptotic probability of the existence of a cluster crossing a given annulus of radii r and rs as r → 0 (s > 1 fixed). Relying on this result, we then show that the probability for a macroscopic cluster to hit a given disc of radius r decays like | log r|−1+θ+o(1) as r → 0. Finally, we characterise the polar sets of clusters, i.e. sets that are not hit by the closure of any cluster, in terms of logα-capacity.
17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43