I will delve into the notions of nonrational fan and polytope in toric geometry. I will then illustrate some related constructions from the symplectic and complex viewpoint.

Based on joint works with Elisa Prato and Dan Zaffran.

Resource-aware control provides a framework that allows to prescribe, in a principled way, when to use resources efficiently while still guaranteeing a desired quality of service in performing the intended task. What constitutes a resource can take many different shapes, and in fact the literature has seen this paradigm applied to a rich variety of settings pertaining control actuation, data acquisition, computation, sensor updates, communication services, and even access to human operators. This talk describes another area of application to the design of fast optimization solvers in machine learning, where the resource to be aware of is the last sampled state of the system. Recent work in the machine learning and optimization communities seeks to shed light on the behavior of accelerated optimization methods via high-resolution differential equations. These differential equations are continuous-time counterparts of discrete-time optimization algorithms and, remarkably, their convergence properties can be characterized using the powerful tools provided by classical Lyapunov stability analysis. An outstanding open question is how to discretize these continuous flows while maintaining their convergence rates. We provide an answer to this question by employing ideas from resource-aware control. The main idea is to take advantage of the Lyapunov functions employed to characterize the rate of convergence of high-resolution differential equations to design variable-stepsize discretizations that preserve by design the convergence properties of the original dynamics. We illustrate our design approach on the heavy-ball dynamics and compare its performance to other discretization methods.

As for toric symplectic manifolds, wonderful symplectic manifolds can be classified by some convex polytopes. After having introduced these objects, I will outline the classification.

Symplectic integrators tend to be preferred integration scheme for simulating conservative Hamiltonian systems, with important applications in statistical mechanics, Monte Carlo methods, and numerical simulations of gauge field theories. Their success is because, besides exactly preserving the symplectic structure, such methods preserve the Hamiltonian within a bounded error; a result that relies on energy conservation. We will show how this can be extended to dissipative Hamiltonian systems, for which the Hamiltonian is no longer a first integral. In particular, this approach enables the construction of a large class of (accelerated) optimization methods on manifolds and for problems with constraints, which can have interesting applications in machine learning. We will also mention connections with sampling and stochastic processes from this perspective.

Many data sets are in metric format, that is, equipped with distances between the data points. It then is a challenge for geometry to develop concepts and methods that can extract qualitative and quantitative aspects of such data.

We have found that this can also lead to new perspectives on classical concepts of geometry, like curvature. Our approach to curvature quantifies the deviation from hyperconvexity via intersection patterns of balls. This is closely related to the persistent homology employed in topological data analysis. On this basis, we can also develop and explore notions of discrete curvature for graphs, hypergraphs or simplicial complexes.

This talk starts by discussing the notion of ideal in a Lie algebroid and argues that

a foliation on a manifold should be considered an ideal and not only a subalgebroid in the tangent bundle.

After the discussion of some examples and properties of ideals, their Atiyah class is defined and compared with the Atiyah class of a Lie pair. The latter is then interpreted as an obstruction to the existence of an ideal structure on a given subalgebroid.

In this talk, we discuss some recent work on the connections between machine learning and dynamical systems. These come broadly in three categories, namely machine learning via, for and of dynamical systems. In the first direction, we introduce a dynamical approach to deep learning theory with particular emphasis on its connections with approximation and control theory. In the second direction, we discuss the approximation and optimization theory of learning input-output temporal relationships using recurrent neural networks and variants, with the goal of highlighting key new phenomena that arise in learning in dynamic settings. If time allows, in the last direction we discuss some principled methods that learns stable and interpretable dynamical model from data arising in scientific applications.

We will consider modeling shapes and fields via topological and lifted-topological transforms. Specifically, we show how the Euler Characteristic Transform and the Lifted Euler Characteristic Transform can be used in practice for statistical analysis of shape and field data. The Lifted Euler Characteristic is an alternative to the Euler calculus developed by Ghrist and Baryshnikov for real valued functions. We also state a moduli space of shapes for which we can provide a complexity metric for the shapes. We also provide a sheaf theoretic construction of shape space that does not require diffeomorphisms or correspondence. A direct result of this sheaf theoretic construction is that in three dimensions for meshes, 0-dimensional homology is enough to characterize the shape.

After reviewing some basic aspects about finite dimensional completely integrable Hamiltonian systems I will discuss recent work concerning new symplectic invariants and constructions of some classes of integrable systems in low dimensions, which include certain important examples from classical mechanics.

The notion of stress plays important roles in solid mechanics, fluid mechanics, electromagnetism, and relativity theory. Stresses cannot be measured experimentally, and their existence is based on a mathematical theorem that relies on a number of assumptions of physical and mathematical nature. In fact, standard stress cannot explain physical phenomena such as surface tension. Surface tension and other phenomena are modeled using an extension of stress theory to hyper-stresses.

While stress is introduced to the novice as the force divided by the cross-section area, in standard continuum mechanics, formulated in a three-dimensional Euclidean space, the stress tensor is a means to determine the internal forces in a body. Its existence is proved by the so-called Cauchy stress theorem, based on Cauchy postulates. The traditional method of proof of the Cauchy theorem cannot be adapted to hyperstresses, modeled as higher-order tensors.

We present a general stress theory that applies to stresses and hyper-stresses in the setting of general differentiable manifolds. The basic mathematical object is the configuration space---the Banach manifold of k-times continuously differentiable sections of a fiber bundle over a compact base manifold---the material body. The choice of topology is natural so that the set of embeddings of the body manifold in a space manifold is open in the manifold of all mappings. Forces are defined to be elements of the cotangent bundle of the configuration space and their action on virtual velocities---elements of the tangent bundle---is interpreted physically as virtual power.

Stresses and hyper-stresses emerge naturally from a representation theorem as measures, valued in the dual bundles of some jet bundles, which represent forces.