George Willis (University of Newcastle, Australia)
Self-replicating groups and scale groups abstract
Abstract:
Self-replicating groups are isometry groups of rooted trees which furnish important examples of groups in geometric and combinatorial group theory. Scale groups are isometry groups of regular trees which arise as subquotients of general totally disconnected, locally compact (t.d.l.c.) groups. It will be explained how these classes of groups are essentially equivalent despite acting on different types of trees. This equivalence allows an exchange of ideas between the study of discrete groups on one hand and t.d.l.c. groups on the other. In particular, the role of scale groups in the structure theory of general t.d.l.c. groups raises general questions about the structure of scale groups.
10:30 • Université de Genève, Conseil Général 7-9, Room 1-05
Dr. Daria Tieplova (ICTP, Trieste)
Fundamental limits of overparametrized shallow neural networks for supervised learning abstract
Abstract:
I will discuss the joint work done with Francesco Camilli and JeanBarbier concerning an information-theoretical analysis of a two-layerneural network trained from input-output pairs generated by a teachernetwork with matching architecture, in overparametrized regimes. Ourresults come in the form of bounds relating i) the mutual informationbetween training data and network weights, or ii) the Bayes-optimalgeneralization error, to the same quantities but for a simpler(generalized) linear model for which explicit expressions are rigorouslyknown. Our bounds, which are expressed in terms of the number of trainingsamples, input dimension and number of hidden units, thus yieldfundamental performance limits for any neural network (and actually anylearning procedure) trained from limited data generated according to ourtwo-layer teacher neural network model. The proof relies on rigorous toolsfrom spin glasses and is guided by ``Gaussian equivalence principles\'\'lying at the core of numerous recent analyses of neural networks. Withrespect to the existing literature, which is either non-rigorous orrestricted to the case of the learning of the readout weights only, ourresults are information-theoretic (i.e. are not specific to any learningalgorithm) and, importantly, cover a setting where all the networkparameters are trained.
14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Dimitriy Rumynin (Warwick)
C_2 graded groups, their real representations and Dyson\'s tenfold way abstract
Abstract:
A C_2 graded group is a pair, a group G and an index two subgroup H. A "real representation" of G is a complex representation of H together with a given (to be chosen) action of the outer coset G\\H. Different choices lead to different theories.Such representations appear independently in three different disciplines : Algebra, Physics and Topology. The goal of the talk is to review the formalism and various choices, including resulting theories. The talk is based upon recent work of the speaker and James Taylor (Oxford) and Matthew B. Young (Utah State).
15:15 • EPF Lausanne
Dr. Mitchell Taylor (ETH Zürich, Switzerland)
Low regularity well-posedness for the general quasilinear Schrödinger equation abstract
Abstract:
We present a new and relatively simple method for proving large data local well-posedness in low regularity Sobolev spaces for general quasilinear Schrödinger equations of the form \\begin{equation*}\\begin{cases}&i\\partial_tu+g^{jk}(u,\\overline{u},\\nabla u,\\nabla\\overline{u})\\partial_j\\partial_k u=F(u,\\overline{u},\\nabla u,\\nabla\\overline{u}),\\hspace{5mm} u:\\mathbb{R}\\times\\mathbb{R}^d\\to\\mathbb{C}^m,\\\\&u(0,x)=u_0(x),\\end{cases}\\end{equation*}assuming only non-degeneracy of the metric, nontrapping and mild regularity/decay of the initial data. As a consequence, we remove the uniform ellipticity assumption from the main result of Marzuola, Metcalfe and Tataru (Arch. Ration. Mech. Anal. 2021) and substantially weaken the regularity/decay assumptions from the pioneering works of Kenig, Ponce, Rolvung and Vega.This is based on joint work with Ben Pineau (UC Berkeley).
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Prof. Dr. Peter Hintz (ETH Zürich, Switzerland)
Perturbations and weak interactions of black holes abstract
Abstract:
Following a brief introduction to general relativity and the Einstein field equations, I will discuss two interrelated recent developments in the mathematical study of black holes and some of the mathematical techniques involved. In the first part, I describe the stability properties of various exact black hole solutions under regular small perturbations, namely the relaxation of an out-of-equilibrium black hole to a stationary state; this has been the subject of intense recent activity by a number of research groups, and is now understood in a fair amount of detail. In the second part, I will discuss work in progress towards the construction of singular perturbations of spacetimes via the insertion of small black holes: this aims to provide the first rigorous examples of spacetimes describing the merger of two black holes with extreme mass ratios.
16:30 • UZH Zentrum, Building KO, Room F 150
Adélie Garin (EPFL)
When TDA meets geometric group theory: A stratification of barcode space using Coxeter complexes abstract
Abstract:
At the intersection of data science and algebraic topology, topological data analysis (TDA) is a recent field of study that provides robust mathematical, statistical and algorithmic methods for analysing the topology and geometry underlying complex data. TDA has proven useful in many applications, including biology, materials science and climate science, and continues to evolve rapidly. Barcodes are frequently used invariants in TDA. They provide topological summaries of the persistent homology of a filtered space. Understanding the structure and geometry of the barcode space is therefore crucial for applications. In this talk, we use Coxeter complexes to define new coordinates on the barcode space. These coordinates define a stratification of the barcode space with n bars, where the highest dimensional strata are indexed by the elements of the symmetric group. This creates a bridge between the fields of TDA, geometric group theory and permutation statistics, which could be exploited by researchers in each field.This presentation is based on joint work with B. Brück. No prerequisites on TDA or Coxeter complexes are required.
17:00 • Université de Neuchâtel, Institut de Mathématiques, B103