Umut Çetin (London School of Economics)
Mathematics of Market Microstructure
10:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Alejandra Garrido (Universidad Complutense de Madrid)
Simple locally compact groups from full groups and branch groups abstract
Abstract:
Since the start of the century there has been increased interest in totally disconnected locally compact (tdlc) groups and a structure theory is starting to emerge. A special role is played by the class S of tdlc groups that are compactly generated, topologically simple and not discrete. The theory is in need of more examples with a view to some sort of classification. Full groups of homeomorphisms of Cantor space are a rich source of simple groups. In joint work with Colin Reid, it is shown that, given the right conditions on the "seed" group, the full group can be given a totally disconnected locally compact topology and it contains a big simple compactly generated subgroup. An example of this is Neretin\'s group of almost automorphisms of a tree. Further examples include commensurators of profinite regular branch groups and full groups of Burger-Mozes universal groups.It is also shown that this construction accounts for all groups in one of the five types of groups in S, up to local isomorphism (the totally disconnected analogue of having the same Lie algebra in the connected case).I will focus more on the examples than on the theory, since we are looking for more examples than those above.
10:30 • Université de Genève, Conseil Général 7-9, Room 1-05
Dr. Johannes Krah (University of Bielefeld)
On proper splinters in positive characteristic abstract
Abstract:
A scheme X is a splinter if for any finite surjective morphism f: Y \\to X the pullback map O_X \\to f_* O_Y splits as O_X-modules. By the direct summand conjecture, now a theorem due to André, every regular Noetherian ring is a splinter. Whilst for affine schemes the splinter property can be viewed as a local measure of singularity, the splinter property imposes strong constraints on the global geometry of proper schemes over a field of positive characteristic. For instance, the structure sheaf of a proper splinter in positive characteristic has vanishing positive-degree cohomology. I will report on joint work with Charles Vial concerning further obstructions on the global geometry of proper splinters in positive characteristic.
13:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 25
The Subspace Flatness Conjecture and Faster Integer Programming abstract
Abstract:
In a seminal paper, Kannan and Lov\\’asz (1988) considered a quantity $\\mu_{KL}(\\Lambda,K)$which denotes the best volume-based lower bound on the \\emph{covering radius} $\\mu(\\Lambda,K)$ of a convexbody $K$ with respect to a lattice $\\Lambda$. Kannan and Lov\\’asz proved that $\\mu(\\Lambda,K) \\leq n \\cdot \\mu_{KL}(\\Lambda,K)$ and the Subspace Flatness Conjecture by Dadush (2012) claims a $O(\\log n)$ factor suffices, which would matchthe lower bound from the work of Kannan and Lov\\’asz.We settle this conjecture up to a constant in the exponent by proving that $\\mu(\\Lambda,K) \\leq O(\\log^{3}(n)) \\cdot \\mu_{KL} (\\Lambda,K)$. Our proof isbased on the Reverse Minkowski Theorem due to Regev and Stephens-Davidowitz (2017).Following the work of Dadush (2012, 2019), we obtain a $(\\log n)^{O(n)}$-time randomized algorithm tosolve integer programs in $n$ variables.Another implication of our main result is a near-optimal \\emph{flatness constant} of $O(n \\log^{3}(n))$.This is joint work with Victor Reis.
14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
Prof. Dr. Antoine Gloria (Sorbonne Université)
What is large-scale regularity? abstract
Abstract:
The aim of this talk is to investigate what survives of the standard estimates valid for operators with constant coefficients in the case of variable coefficients. The general strategy is based on quantifying how far the (inverse) operator with variable coefficients is from an (inverse) operator with constant coefficients, and obtain the desired estimates by perturbation. Whereas this is classically done at the level of the coefficients themselves (Meyers’ estimates, Schauder theory e.g.), in this talk I will use closeness in the sense of homogenization. As an illustration, I will discuss large-scale Meyers’ estimates, large-scale Lipschitz estimates, and conclude with large-scale dispersive estimates.
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Rodrigo Navarro Betancourt (Dublin)
Kashiwara-Vergne from string topology abstract
Abstract:
The group KRV acts transitively and freely on solutions to the Kashiwara-Vergne conjecture, a problem in Lie theory alluding to the Baker-Campbell-Hausdorff series. Alekseev-Kawazumi-Kuno-Naef showed that, surprisingly, KRV can be recovered from the group of automorphisms of the Goldman-Turaev Lie bialgebra of the thrice-punctured sphere. The Goldman bracket and the Turaev cobracket are defined on homotopy classes of free loops, and they are both instances of intersection products stemming from string topology. In this talk, we will give a new characterisation of KRV exploiting this tie to string topology.
15:30 • Université de Genève, Conseil Général 7-9, Room 1-07
Dr. Samir Canning (ETHZ)
What is... the cohomology of moduli spaces of curves? abstract
Abstract:
\'\'The moduli space of curves was first studied by Riemann. I will explain what it is, how to compactify it, and how to attempt to compute the cohomology of its compactifications. Ideas from a broad range of mathematics are necessary, including low dimensional topology, algebraic geometry, and number theory.
16:30 • UZH Zentrum, Building KO2, Room F 150
Harmonic maps and the vectorial obstacle problem: singularities vs free boundaries abstract
Abstract:
I will discuss some recent results obtained in collaboration with A. Figalli, S. Kim and H. Shahgholian. We consider minimizers of the Dirichlet energy among maps constrained to take values outside a smooth domain O in R^m. These minimizers can be thought of as solutions of a vectorial obstacle problem, or as harmonic maps into the manifold-with-boundary given by the complement of O. I will discuss results concerning the regularity of the minimizers, the location of their singularities, and the structure of the free boundary.
16:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43