Techniques of machine learning (ML) find a rapidly increasing range of applications touching upon social, economic, and technological aspects of everyday life. They are also being used with great enthusiasm to fill in gaps in our scientific knowledge by data-based modelling approaches. I have followed these developments for a while with interest, concern, and mounting disappointment. When these technologies are employed to take over decisive functionality in safety-critical applications, we would like to exactly know how to guarantee their compliance with pre-defined guardrails and limitations. Moreover, when they are utilized as building blocks in scientific research, it would violate scientific standards -in my opinion- if these building blocks were used without a throrough understanding of their functionality, including inaccuracies, uncertainties, and other pitfalls. In this context, I will juxtapose (a subset of) deep neural network methods with the family of entropy-optimal Sparse Probabilistic Approximation (sSPA) techniques developed recently by Illia Horenko (RPTU Kaiserslautern-Landau) and colleagues.
For a group or semigroup or ring G, solving equations where the coefficients are elements in G and the solutions take values in G can be seen as akin to solving systems of linear equations in linear algebra, Diophantine equations in number theory, or more generally, polynomial systems in algebraic geometry.
In this talk I will give a survey about solving equations in infinite non-abelian groups, with emphasis on free groups, and show how imposing certain constraints on the solutions can tilt the balance between decidability and undecidability.
A notion of symmetry has been fundamental in mathematics and physics for many years. A new type of "quantum" symmetry has emerged in theory of operator algebras, quantum groups, quantum invariants in low dimensional topology, integrable systems, vertex operator algebras, quantum field theory and condensed matter physics since 1980's. I will present development in these areas from an operator algebraic viewpoint. Emphasis is given to the Jones theory of subfactors, chiral conformal field theory and two-dimensional topological order. No knowledge on these topics is assumed as a prerequisite.
In their seminal work “Complexity of Bezout’s Theorem,” Mike Shub and Steve Smale developed a geometric framework to study the complexity of solving polynomial systems. This program introduced ideas that connect algebraic geometry, probability, and complexity theory, with key concepts including homotopy methods, condition numbers, and intersection theory. A major milestone in this field was the solution of Smale’s 17th problem—finding a zero of a polynomial, on average, in polynomial time—through the contributions of Beltrán and Pardo, Bürgisser and Cucker, and Lairez.
In this talk, I will outline the geometric aspects of this program, tracing its origins and broader developments. I will revisit Shub and Smale’s original proposal to study geodesics in the "condition metric" and demonstrate how optimal transport provides a new perspective, reconnecting with the foundational ideas of the story.
Given a smooth manifold M, geometers and topologists ask the question "which geometries, i.e. which kind of Riemannian metrics" can one put on M? In recent years, this problem has been studied intensively for metrics with positive scalar curvature. The main questions are:is there such a metric at all?
if so: can one "classify" these metrics?It turns out that both questions have interesting answers. Surprisingly, much of this turns out to be rather topological: there are well studied topological constructions ("surgery") of such metrics. There are obstructions, which are based on characterisitc classes and numbers. This is happening in particular through the use of the spectral theory of the Dirac operator.
We will give a brief survey of what the theory and then focus on some recent developments, using higher index theory.
Surprisingly there is a simple and beautiful description of the geometry of conjugation within any split subgroup H of the full isometry group G of Euclidean space. We prove (yes, also in the talk) with elementary Linear Algebra methods that for any h∈H, the conjugacy class can geometrically be described by a subset of Euclidean space, namely the move-set of its linearisation. Examples of groups H considered include affine Coxeter groups, certain other crystallographic groups, and the group G itself. This is joint work with Elizabeth Mili\'cevi\'c and Anne Thomas.
We give a survey over aspherical closed manifolds. Aspherical is the purely homotopy theoretic condition that the universal covering is contractible. There are many well-known examples such as closed Riemannian manifolds with non-positive sectional curvature, but also very exotic examples such as closed aspherical manifolds that do not admit a triangulation. We discuss some prominent conjectures, e.g., the Borel Conjecture about the topological rigidity, and the Singer Conjecture about the concentration of L^2-Betti numbers in the middle dimension. The main point is to get a better understanding why the condition aspherical has so many consequences about the topology, geometry, algebra, and analysis about such manifolds.
Consider the motion of an ideal (i.e. inviscid) incompressible fluid inside a 2-dimensional bounded domain. It is described by the Euler equations for the velocity field, and, equivalently, by the Lagrange equations for the fluid particle trajectories. A surprising property of the flows was found recently: for any sufficiently regular flow, the particle trajectories are real analytic curves, while the velocity field may be of finite regularity (Uriel Frisch has expressed this fact as a "very smooth ride in the stormy sea"). In particular, the flow lines of any stationary (time independent) flow are real-analytic curves. The further development of this result is the theorem stating that the set of stationary flows can be endowed with a structure of an analytic Banach manifold. The general hypothesis says that the phase space of the fluid contains a (very strange) attractor whose every component is an analytic Banach manifold, and the set of stationary flows is one of its components. In this talk I'll tell about the nontrivial history of this discovery, explain the ideas of the proof, and tell about the conjectures on the global structure of the fluid as a dynamical system. No preliminary knowledge of the fluid dynamics is assumed.
I will explain the strong interplay between geometric structures on manifolds and actions of discrete groups and give an insight in new developments of the past twenty years, focussing in particular on groups acting non-positively curved symmetric spaces. In the end I might touch on how the mathematical developments also led to applications in machine learning.
In this talk I will present some recent results concerning the global regularity of the three-dimensional Primitive Equations of oceanic and atmospheric dynamics with various anisotropic viscosity and turbulence mixing diffusion. However, in the non-viscous (inviscid) case it can be shown that there is a one-parameter family of initial data for which the corresponding smooth solutions of the inviscid Primitive Equations develop finite-time singularities (blowup).
Capitalizing on the above results, one is able to provide rigorous justification for the derivation of the Primitive Equations of planetary scale oceanic dynamics from the three-dimensional Navier-Stokes equations, for vanishing small values of the aspect ratio of the depth to horizontal width.
In addition, I will also show the global well-posedeness of the coupled three-dimensional viscous Primitive Equations with a micro-physics phase change moisture model for cloud formation.
Furthermore, I will also consider the singular limit behavior of a tropical atmospheric model with moisture, as ε → 0, where ε > 0 is a moisture phase transition small convective adjustment relaxation time parameter.
In this talk we discuss several topics from theory and practice of climate modelling. Results from recent high resolution computations with the climate model ICON are shown and physical and mathematical implications are described. This comprises the role of submesoscale turbulence in the dynamics of the ocean and the current practice in subgridscale modelling in light of recent analytical results of the ocean primitive equations. Finally we introduce a stochastic approach to climate modelling related to Hasselmann’s paradigm and study its mathematical properties.
The Rokhlin lemma is a finite approximation property that underpins a great many constructions in classical ergodic theory, including most spectacularly those at the basis of the Ornstein isomorphism theory for Bernoulli shifts. In the 1970s Ornstein and Weiss showed amenability to be the natural setting for finite approximation in dynamics by establishing a general form of the Rokhlin lemma in this setting, and this led, among other things, to a much broader recasting of the Ornstein isomorphism theory. Over the last couple of decades a growing interest in the interplay between dynamics and the geometric and analytic structure of groups has set the stage for a resurgence of applications of the Ornstein-Weiss Rokhlin lemma, once again confirming its utility as a powerful and versatile tool. I will sketch some of these recent applications, ranging in flavour from orbit equivalence to operator algebras.
The theory of combinatorial limits provides analytic tools to represent and study large discrete structures and has led to new views on a wide range of topics in mathematics and computer science. After introducing this rapidly developing area of combinatorics, we focus on several problems from extremal combinatorics, which we will view through lenses of combinatorial limits.In particular, we will present a counterexample to a conjecture of Lovász concerning finitely forcible optima, which was one of the two most cited conjectures in the theory of combinatorial limits. We will also demonstrate how analytic tools have been successfully used to resolve long standing open problems in extremal combinatorics, such as a 30-year-old conjecture of Erdős and Sós on uniform Turán densities of 3-uniform hypergraphs and a 30-year-old problem on the existence of high chromatic graphs with small Ramsey multiplicity.
The asymmetric simple exclusion process (ASEP) is a model for translation in protein synthesis and traffic flow; it can be defined as a Markov chain describing particles hopping on a one-dimensional lattice. I will give an overview of some of the connections of the stationary distribution of the ASEP to combinatorics and special functions. I will also mention some open problems and observations about positivity in Markov chains.
Algebraic curves (Riemann surfaces) are among the most studied objects in mathematics due to the fact that they can be approached from the point of view of algebraic geometry, complex analysis or Galois theory. In 1984, Mark Green put forward a deceptively simple conjecture concerning the structure of the equations of an algebraic curve in its canonical embedding, amounting to the statement that the complexity of each curve of genus g can be recovered in a precise way from the equations among its canonical forms. I will present an introduction to this circle of ideas, then explain how ideas coming from topology and geometric group theory led to a recent solution of Green's Conjecture for generic curves in arbitrary characteristics.
The Norwegian mathematician Sophus Lie (1842-1899) was founder of a whole new branch of mathematics („Lie theory“) by creating the groups and algebras which carry his name. Indeed, Lie groups are the main tool for describing continuous symmetries! From 1886 to 1898, Sophus Lie was a professor of mathematics in Leipzig as the successor of Felix Klein. Wilhelm Killing's first paper on the classification of simple Lie algebras from 1888 was the first hint that the new area would produce wast amounts of research data - which it does up to this day. We will retrace the role and different stages as well as formats of research data in Lie theory, and we will explain what the NY Times meant in 2007 by "A Calculation The Size Of Manhattan“ (spoiler alert: it has to do with the exceptional Lie group E_8).
We consider the classical problem of prescribing the scalar curvature of a manifold, dating back to works by Kazdan and Warner. On the sphere, when one uses conformal metrics, this is also known as Nirenberg's problem. This problem is not always solvable: one of the difficulties arising when studying it is the loss of compactness, mainly caused by the action of the Möbius group and by the criticality of Sobolev's embeddings. This phenomenon, referred to as "bubbling" is mainly understood in low dimensions, where bubbles must be either single or at finite distance from each other. We will review some history on the problem, discussing the main ideas involved to tackle the above issues. We will also present some recent work with M.Mayer concerning existence and non-existence results of new type for the higher-dimensional case.
In this talk we consider interacting particle systems, their description via graphical respresentations (stochastic flows) and their dual processes. We then focus on systems where the underlying lattice is given by the complete graph and consider the mean-field limit for which the number of vertices tends to infinity. We are not only interested in the mean-field limit of a single process, but also in how several coupled processes behave in the limit. These turn out to be closely related (dual in some sense) to recursive tree processes (RTP), which are generalizations of Markov chains with a tree-like time parameter, that were studied by Aldous and Bandyopadyay in discrete time (alongside corresponding recursive distributional equations (RDE)). We illustrate our theory with a particle system with cooperative branching and deaths.
This is joint work with Tibor Mach and Jan Swart (Prague).
What does a random surface look like? And what does it mean anyway to pick a surface uniformly at random?
These questions and related others arise naturally in the study of Liouville quantum gravity, and have been the subject of intense research on the mathematical side in the last few years. I will try to give a panoramic view of a canonical notion of random surface, including counter-intuitive aspects of its geometry. I will also explain some of the outstanding conjectures in this field, as well as some progress that has been made in particular concerning Liouville Brownian motion, a natural notion of diffusion in this geometry.
We give a quick review of Parisi and Wu's stochastic quantisation procedure and apply it to the non-linear sigma model as well as the Yang-Mills model. We then review a number of recent results on the resulting equations.
We show how several parabolic equations and systems can be derived from hyperbolic systems of conservation laws just by performing a suitable quadratic change of time. The simplest example is the heat equation which follows from the isothermal Euler equations. This idea allows us to transfer well known techniques from the hyperbolic world to the parabolic setting, for instance for some mean-curvature flows.
We estimate, in a number field, the maximal number of linearly independent elements with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices. Our arguments combine group theory, ramification theory, and the geometry of numbers. This is joint work with Mikolaj Fraczyk and Péter Maga.
Locally symmetric spaces are immensely interesting objects that lie at the crossroads of number theory, representation theory and homogeneous dynamics. Since the work of DeGeorge and Wallach in '70s and Gromov in '80s it has been known that many topological invariants of locally symmetric spaces are controlled only in terms of volume. One of the most ambitious conjectures in that direction was proposed by Gelander. He conjectured that any arithmetic locally symmetric space M is homotopy equivalent to a simplicial complex where the degree of vertices are bounded in terms of dimension of M and the total number of cells is linear in the volume of M. I will present some of the recent results on the growth of topological invariants of locally symmetric spaces and explain the main ideas behind the proof of Gelander's conjecture for hyperbolic 3-manifolds.
Buildings were introduced by Tits and Bruhat-Tits as abstract analogs of symmetric spaces associated to algebraic groups over non-archimedian local fields with or without valuations.
Since then their theory has been enriched by a variety of methods as well as numerous applications: Buildings may be descibed as subgroup structures, as metric spaces or combinatorial objects with a close interplay between these different viewpoints.
In this talk I would like to introduce you to the theory of buildings and highlight some of their applications.
Algebraic geometry starts with cubic polynomials and the hypersurfaces they define form one of the most important classes of algebraic varieties (including elliptic curves and the famous cubic surfaces). Another and a priori totally unrelated class of geometric objects is provided by K3 surfaces. However, it has turned out by work of Hassett, Kuznetsov and many others that these two geometries are mysteriously linked. This talk will explain what we know about the relation and how it brings together transcendental and algebraic aspects.
I will present recent mathematical advances on the global evolution of matter fields (governed by the Euler equations) and the nonlinear stability of spacetimes (governed by the Einstein equations), including results on spherically symmetric compressible fluid flows around a Schwarzschild black hole, on the dispersion of massive scalar fields in Minkowski spacetime, and on the validation of the f(R)-theory of gravity. Blog: philippelefloch.org
We give a survey of some recent regularity and uniqueness results for extremals and minimizers of non-parametric variational integrals satisfying natural ellipticity conditions.
Regular model sets (a special class of cut and project sets), which go back to Yves Meyer (1972) in mathematics and to Peter Kramer (1982) in physics, form a versatile class of structures with amazing harmonic properties. These sets are also known as mathematical quasicrystals, and include the famous Penrose tiling with fivefold symmetry as well as its various generalisations to other non-crystallographic symmetries. They are widely used to model the structures discovered in 1982 by Dan Shechtman (2011 Nobel Laureate in Chemistry). More recently, also systems such as the square-free integers or the visible lattice points have been studied in this context, leading to the theory of weak model sets. This is an extension of the class of regular model sets that was also briefly considered by Meyer and by Schreiber in the 1970s, but has not seen any systematic investigation. Due to the connection with B-free integers and lattice systems, which are of renewed interest in the light of Sarnak's research program around M"obius orthogonality, weak model sets are now being studied in more detail by several groups. This talk will review some of the developments, and introduce important concepts from the field of aperiodic order, with focus on spectral aspects.
Starting from just a sheet of paper, by folding, stacking, crumpling, sometimes tearing, we will explore a variety of phenomena, from magic tricks and geometry to nonlinear elasticity and the traditional Japanese art of origami. Much of the show consists of table-top demos, which you can try later with friends and family.
So, take a sheet of paper . . .
Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalization. The spectral theory of tensors was pioneered by Lim and Qi over a decade ago, and it has found numerous applications. This lecture offers a first introduction, with emphasis on algebraic aspects.
In dem Vortrag geben wir eine kurze historische Übersicht über die Spektraltheorie von zufälligen Feldern mit stationären Zuwächsen. Behandelt werden auch intrinsisch stationäre Felder und ihre Beziehung zu Pontryagin-Räumen.
Prototypical models for collective dynamics are found in opinion dynamics, flocking, self-organization of biological organisms, and rendezvous in mobile networks. They are driven by the “social engagement” of agents with their local neighbors.
We discuss the emergent behavior of such systems. Their large-time behavior leads to the emergence of different patterns, depending on the propagation of connectivity of the underlying graphs. In particular, global interactions lead to the emergence of consensus, leaders etc. The collective dynamics of large crowds of agents lend itself to “social hydrodynamics”, driven by the corresponding engagement through local means. We discuss the global regularity of social hydrodynamics for sub-critical initial configurations.
Felix Klein and Sophus Lie, both professors in Leipzig in the 1880-90's, were among the most influential mathematicians during the last decades of the 19th century, and their ideas have had a profound impact on the development of modern mathematics through the 20th century. They happened to meet for the first time in Berlin in the fall 1869, at the very beginning of their careers. Despite their different personality, age, and educational background, they developed a close friendship and shared a deep interest in geometry, which resulted in an intense collaboration and exchange of ideas during the following three years 1870-72. Moreover, their friendship and vivid letter correspondence continued for two more decades. Unfortunately, from their correspondence prior to 1878 only the letters from Klein to Lie have survived, but they will be published in the nearest future. In this talk we shall primarily focus on the circumstances leading to their close friendship and collaboration through 1869-72, seen in the light of these letters.
Many natural flows are driven by buoyancy forces, perhaps the most familiar being those resulting from density variations due to temperature differences in the presence of a gravitational field. The simplest setting to study this sort of system is so-called Rayleigh-Bénard convection, the buoyancy driven flow in a horizontal layer of fluid heated from below and cooled from above. This seminal problem has received tremendous theoretical and experimental attention over the last century but many riddles remain, especially for buoyancy-driven turbulent transport. Following an introduction to the phenomena and its applications along with a review of the current state of theory and experiments on turbulent Rayleigh-Bénard convection, I will describe some of the light that mathematical analysis has shed on the subject in recent years.
