Shahar Mendelson (The Australian National University)
Probabilistic methods in Analysis
10:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Dr. Sam Hagh Shenas Noshari (Université de Fribourg)
On the equivariant cohomology of certain generalizations of symmetric spaces abstract
Abstract:
Symmetric spaces form an important and much studied class of Riemannian manifolds. From the point of view of transformation groups, they are particular instances of homogeneous spaces, and as such come equipped with a natural Lie group action, the so-called isotropy action. What can be said about the equivariant cohomology ring of such an action? Neglecting the multiplicative structure, O. Goertsches has shown that each such ring is as simple as one could reasonably hope for. Namely, it is free as a module over a certain polynomial ring, the ring of invariants of the acting Lie group.
The purpose of this talk is to introduce equivariant cohomology, to present various known results concerning the equivariant cohomology of isotropy actions, and to indicate how Goertsches result can be extended to a certain class of homogeneous spaces that generalizes the class of symmetric spaces.
10:20 • Université de Fribourg, room Phys 2.52
Prof. Dr. Dhruv Ranganathan (Cambridge University)
Enumerative geometry for curves in an algebraic torus abstract
Abstract:
I will discuss various aspects of the enumerative geometry of curves in an algebraic torus, formulated as the logarithmic enumerative geometry of a toric pair. I will explain how logarithmic double ramification cycles can be used to give a complete, albeit complex, solution to the logarithmic GW theory of all toric varieties, relative to their full toric boundary. I will then explain what happens in various special geometries when this solution can be made explicit. One simplification leads very quickly to traditional tropical correspondence theorems, recovering work of Mikhalkin, Nishinou-Siebert, and others. Another simplification leads to tropical refined curve counting, recovering work of Bousseau via integrable systems techniques. I will then explain how the logarithmic GW/DT conjectures (aka the LMNOP conjectures) come into the story via “triple double” ramification cycles. The talk is based on joint work with A. Urundolil Kumaran and D. Maulik, and touches upon forthcoming work of P. Kennedy-Hunt, Q. Shafi, and A. Urundolil-Kumaran.
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Thérèse Moerschell (EPF Lausanne)
Examples of non-uniqueness for the advection-diffusion equation abstract
Abstract:
The advection-diffusion equation is known to have unique solutions for any vector field that is L^2 in time and in space. But what happens when we have slightly less than square integrability? In this talk we will explore two examples of vector fields in L^p(0,T;L^q(\\T^d)) made of shear flows that prove the non-uniqueness of solutions whenever we have p<2 or q<2. We will first show that they give different solutions to the advection equation and then use the Feynman-Kac formula to show that diffusion has little effect if our parameters are well-tuned.
This is part of my Master\'s thesis, supervised by Massimo Sorella and Maria Colombo.
14:15 • Universität Basel, Spiegelgasse 5, SR 05.002
Diana Rauseo (University of Zurich)
Constructing Cryptographic Accumulators from Key-Exchange Protocols abstract
Abstract:
Cryptographic accumulators allow membership testing in a set without disclosing the members of the set and relying on a trusted authority. This thesis provides a comprehensive exploration of cryptographic accumulators and their applications. We examine cryptographic accumulators based on the strong RSA assumptions and the trapdoor-less accumulator proposed by Nyberg and Sander. We then analyze key exchange protocols based on computational problems in groups. Our own contribution is to construct accumulators from these protocols. Finally, we discuss some of their applications such as membership testing, timestamping and their usage in cryptocurrencies. <BR> <BR> (**This eSeminar will take place over Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact zita.fiquelideabreu@math.uzh.ch **)
15:00 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 25
Marco Moraschini (University of Bologna)
Simplicial volume and aspherical manifolds abstract
Abstract:
Simplicial volume is a homotopy invariant for compact manifolds introduced by Gromov that measures the complexity of a manifold in terms of singular simplices. A celebrated question by Gromov (~’90) asks whether all oriented closed connected aspherical manifolds with zero simplicial volume also have vanishing Euler characteristic. In this talk, we will describe the problem and we will show counterexamples to some variations of the previous question. Moreover, we will describe some new strategies to approach the problem as well as the relation between Gromov’s question and other classical problems in topology.This is part of a joint work with Clara Löh and George Raptis.
15:45 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Prof. Dr. Dan Abramovich (Brown University)
The Chow ring of a weighted blowup abstract
Abstract:
The Chow groups of a blowup of a smooth variety along a smooth subvariety are described in Fulton\'s book using Grothendieck\'s "key formula", involving the Chow groups of the blown up variety, the center of blowup, and the Chern classes of its normal bundle.If interested in weighted blowups, one expects everything to generalize directly. This is in hindsight correct, except that at every turn there is an interesting and delightful surprise, shedding light on the original formulas for usual blowups, especially when one wants to pin down the integral Chow ring of a stack theoretic weighted blowup.As an application, one obtains a quick derivation of a formula, due to Di Lorenzo-Pernice-Vistoli and Inchiostro, of the Chow ring of the moduli space \\bar{M}_{1,2}.
16:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 25
T-coercivity: a practical tool for the study of variational formulations abstract
Abstract:
Variational formulations are a popular tool to analyse linear PDEs (eg. neutron diffusion, Maxwell equations, Stokes equations ...), and it also provides a convenient basis to design numerical methods to solve them. Of paramount importance is the inf-sup condition, designed by Ladyzhenskaya, Necas, Babuska and Brezzi in the 1960s and 1970s. As is well-known, it provides sharp conditions to prove well-posedness of the problem, namely existence and uniqueness of the solution, and continuous dependence with respect to the data. Then, to solve the approximate, or discrete, problems, there is the (uniform) discrete inf-sup condition, to ensure existence of the approximate solutions, and convergence of those solutions to the exact solution. Often, the two sides of this problem (exact and approximate) are handled separately, or at least no explicit connection is made between the two.In this talk, I will focus on an approach that is completely equivalent to the inf-sup condition for problems set in Hilbert spaces, the T-coercivity approach. This approach relies on the design of an explicit operator to realize the inf-sup condition. If the operator is carefully chosen, it can provide useful insight for a straightforward definition of the approximation of the exact problem. As a matter of fact, the derivation of the discrete inf-sup condition often becomes elementary, at least when one considers conforming methods, that is when the discrete spaces are subspaces of the exact Hilbert spaces. In this way, both the exact and the approximate problems are considered, analysed and solved at once.In itself, T-coercivity is not a new theory, however it seems that some of its strengths have been overlooked, and that, if used properly, it can be a simple, yet powerful tool to analyse and solve linear PDEs. In particular, it provides guidelines such as, which abstract tools and which numerical methods are the most “natural” to analyse and solve the problem at hand. In other words, it allows one to select simply appropriate tools in the mathematical, or numerical, toolboxes. This claim will be illustrated on classical linear PDEs, and for some generalizations of those models.
16:30 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 35/36
Constrained Sparse Approximation over the Cube abstract
Abstract:
Due to the recent development in machine learning, data science and signal processing more and more data is generated, but only part of it might be necessary in order to already make predictions in a sufficiently good manner. Therefore, the question arises to best approximate a signal b by linear combinations of no more than \\sigma vectors from a suitable dictionary A. Additionally, many areas of application - as for example portfolio selection theory, sparse linear discriminant analysis, general linear complementarity problems or pattern recognition - require the solution x to satisfy certain polyhedral constraints. This talk presents an analysis of the NP-hard minimization problem min{||b-Ax||: x \\in [0,1]^n, |supp(x)| \\leq \\sigma} and its natural relaxation min{||b-Ax||: x \\in [0,1]^n, \\sum x_i \\leq \\sigma}. Our analysis includes a probabilistic view on when the relaxation is exact. We will also consider the problem from a deterministic point of view and provide a bound on the distance between the images of optimal solutions of the original problem and its relaxation under A. This leads to an algorithm for generic integer matrices A \\in \\mathbb{Z}^{m \\times n} and achieves a polynomial running time provided that m and ||A||_{\\infty} are fixed.
17:00 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2