Talks
Here you can find my conference and workshop talks and posters in reversed chronological order. Hovering over the information icon "i" gives details regarding the type of presentation. Generated by jekyll-scholar.
2026
- Ph.D. Defense, Trondheim, NorwayAlgorithms For Nonsmooth Optimization On Riemannian Manifolds and ApplicationsJan 2026
The main focus of this work was the development, convergence analysis, and implementation of methods for optimizing nonsmooth functions that leverage the geometric structure of the problem space. This was done by extending two classical methods for nonsmooth optimization in Euclidean spaces (the bundle method and the proximal gradient method) to the Riemannian setting, and by analyzing and showcasing their performance with several numerical experiments under different assumptions. The algorithms are implemented in the Julia package Manopt.jl, and the numerical experiments in this work are available as notebooks in the package ManoptExamples.jl. The second focus point of this work was the application of the machinery of nonsmooth Riemannian optimization to the Procrustes problem with varying norms. This study justifies the use of a closed form minimizer as a proxy for the more theoretically rigorous minimizers obtained via nonsmooth Riemannian optimization. While this may seem in contrast with the main focus of the present thesis, these conclusions were enabled precisely by the use of nonsmooth Riemannian optimization. Additionally, this work also serves as a bridge between the fields of Riemannian optimization and statistics, by showcasing an application of methods from the former to a problem from the latter.
- Ph.D. Trial Lecture, Trondheim, NorwayGeneralized Derivatives For Nonsmooth OptimizationJan 2026
This lecture covers introductory material on some generalized notions of derivatives for nonsmooth functions. Several types of subdifferential are introduced, and necessary and sufficient optimality conditions are mentioned for (non)convex functions. The goal is to show how these can be used to state algorithms for optimizing nonsmooth functions.
2025
- Math Meets Industry, Trondheim, NorwayOptimization On Riemannian ManifoldsAug 2025
Measured data usually comes with underlying structures that can be exploited to improve the performance of optimization algorithms. These often naturally exhibit a Riemannian manifold structure, allowing the definition of optimization tasks in a way that respects the geometry of the data. For example, when dealing with unit-norm vectors, these can be seen as points on the unit sphere, which is a Riemannian manifold. More abstract examples include the space of rotations or positive definite matrices. This talk explores optimization problems that may be nonsmooth, nonconvex and posed on Riemannian manifolds, and strategies to solve them.
- ICCOPT 2025, Los Angeles, USAThe Convex Riemannian Proximal Gradient MethodJul 2025
We consider a class of (possibly strongly) geodesically convex optimization problems on Hadamard manifolds, where the objective function splits into the sum of a smooth and a possibly nonsmooth function. We introduce an intrinsic convex Riemannian proximal gradient (CRPG) method that employs the manifold proximal map for the nonsmooth step, without operating in the embedding or tangent space. A sublinear convergence rate for convex problems and a linear convergence rate for strongly convex problems is established , and we derive fundamental proximal gradient inequalities that generalize the Euclidean case. Our numerical experiments on hyperbolic spaces and manifolds of symmetric positive definite matrices demonstrate substantial computational advantages over existing methods.
2024
- Colloquium on Applied Mathematics, Göttingen, GermanyNonsmooth Optimization On Riemannian ManifoldsDec 2024
We discuss optimization techniques for real-valued, lower-semicontinuous, and geodesically convex functions defined on Riemannian manifolds. Many real-world applications such as image and signal restoration, denoising, and inpainting can be studied as minimization tasks where the objective function has this type of structure. We showcase a novel algorithm as well as numerical examples.
- EURO 2024, Copenhagen, DenmarkThe Riemannian Convex Bundle MethodJul 2024
Within the context of optimization on manifolds, a research direction of particular interest is the investigation of algorithms fit to optimize non-smooth objectives. This research area is relevant since the need for optimizing non-smooth objective functions arises in many real-world problems and applications such as image and signal restoration, denoising, inpainting, etc. In this talk, we introduce the convex bundle method to solve convex, nonsmooth optimization problems on Riemannian manifolds. Each step of our method is based on a model that involves the convex hull of previously collected subgradients, parallely transported into the current serious iterate. This approach generalizes the dual form of classical bundle subproblems in Euclidean space. Several numerical examples implemented using the Julia package Manopt.jl illustrate the performance of the proposed method and compare it to other non-smooth optimization algorithms.
- 2024 SIAM Conference on Applied Linear Algebra, Paris, FranceNonsmooth Optimization On ManifoldsMay 2024
Within the context of optimization on manifolds, a research direction of particular interest is the investigation of algorithms fit to optimize non-smooth objectives. This research area is relevant since the need for optimizing non-smooth objective functions arises in many real-world problems and applications such as image and signal restoration, denoising, inpainting, etc. In this talk, we discuss optimization techniques for real-valued, lower-semicontinuous, and geodesically convex functions defined on Riemannian manifolds, as many of the applications mentioned previously can be studied as minimization tasks where the objective function has this type of structure. We showcase a novel algorithm as well as numerical examples.
- DNA Seminar, Trondheim, NorwayThe Riemannian Convex Bundle MethodApr 2024
Within the context of optimization on manifolds, a research direction of particular interest is the investigation of algorithms fit to optimize non-smooth objectives. This research area is relevant since the need for optimizing non-smooth objective functions arises in many real-world problems and applications such as image and signal restoration, denoising, inpainting, etc. In this talk, we introduce the convex bundle method to solve convex, nonsmooth optimization problems on Riemannian manifolds. Each step of our method is based on a model that involves the convex hull of previously collected subgradients, parallely transported into the current serious iterate. This approach generalizes the dual form of classical bundle subproblems in Euclidean space. Several numerical examples implemented using the Julia package Manopt.jl illustrate the performance of the proposed method and compare it to other non-smooth optimization algorithms.
2023
-
- MaGIC 2023, Øyer, NorwayThe Convex Bundle Method On Hadamard ManifoldsFeb 2023
The study of optimization problems on manifolds has recently seen increasing interest. We are interested in the case where the objective functions are non-smooth and defined on a manifold. Bundle methods in the Euclidean setting have shown to be very effective in optimizing non-smooth functions. We present our work on extending this algorithm to the case of geodesically convex, lower-semicontinuous functions defined on Hadamard manifolds. Finally, we showcase our preliminary results with a numerical example.
2021
- M.Sc. Dissertation, Milan, ItalyThe Schoen-Yau Positive Mass TheoremMar 2021
Differential geometry provides a natural language for the development of several physical theories. Among these, the general theory of relativity of Albert Einstein holds a prominent place. As is known, the spacetime of such theory generalizes Minkowski spacetime and it is modeled after a pseudo-Riemannian manifold of dimension four that generalizes a Lorentzian space the same way a Riemannian manifold generalizes Euclidean space. The physical concept that is responsible for the transition from Minkowski spacetime to that of general relativity is mass, which forces the former to curve, in accordance to Einstein’s theory. For instance, a body with a certain mass curves spacetime in such a way that it is rendered approximately flat far away from this mass, in absence of any other body. Even at a great distance from an isolated body, information about its mass can be gathered from the distortion that causes spacetime to differ from the flat one. More precisely, if N is a complete spacelike hypersurface of the spacetime (M, \tilde g), the Riemannian metric g induced on it will be asymptotically flat. This means that, by removing a suitable compact subset K from N, each connected component of N ∖K, known as an end of N, is diffeomorphic to the complement of a ball in R^3 and that the metric approximates the Euclidean one at infinity accordingly. The ADM mass of an end, introduced by R. Arnowitt, S. Deser e C. Misner in the late ’50s, turns out to be a certain coefficient in the asymptotic expansion of the metric at spatial infinity. The present thesis will focus on the discussion of the proof of the “positive mass theorem” as done by R. Schoen and S.-T. Yau in their 1979 paper “On the Proof of the Positive Mass Conjecture in General Relativity”. According to this theorem, assuming the hypersurface N to be maximal in (M , \tilde g) and its scalar curvature to be non-negative, the ADM mass of each end is non-negative; furthermore, if an end has zero mass, then (N,g) is isometric to the Euclidean space of dimension three. The first statement is due to an argument from minimal surface theory: after reducing the situation to the case of strictly positive scalar curvature on N, assuming the mass of a fixed end to be negative, the second variation formula together with the local Gauss-Bonnet theorem enable the deduction of a contradiction concerning the Gauss curvature of a particular minimal surface embedded in the manifold in question. The second statement is proven by means of a more analytic approach: if an end has zero mass and g is not Ricci-flat, then by solving a certain elliptic problem, it is possible to prove the existence of a metric that is conformally equivalent to g, asymptotically flat, scalar-flat and with negative mass, in contrast to the first statement. Since in three dimensions the Ricci tensor is zero if and only if the curvature is zero, the conclusion is that (N,g) is isometric to Euclidean space if an end has null mass.
2018
- B.Sc. Dissertation, Milan, ItalyGromov’s Non-squeezing TheoremNov 2018
As is known, symplectic geometry provides a natural environment for the study of classical mechanics in its Hamiltonian formulation. Fundamental ideas in physics, such as mechanical motions and energy, admit a formal representation in symplectic geometry that enables us to study their nature in depth, in terms of geometric objects of the likes of diffeomorphisms, vector fields and differential forms. For instance, motions which we are used to in physics are just phase fluxes generated by a vector field corresponding to a function called Hamiltonian of the system, that in physics is regarded as the energy of such system. The fundamental object studied in symplectic geometry is a class of manifolds, called symplectic manifolds, are nothing but regular even-dimensional differential manifolds endowed with a closed and non-degenerate differential 2-form, usually referred to as ω. By Liouville’s Theorem, as is well understood, mechanical motions preserve phase space volume. More generally, Hamiltonian motions preserve the symplectic volume form of the manifold they exist in, which is the top exterior power of the symplectic form ω. The goal of this paper is, in fact, to explain what specifies mechanical motions inside the class of all volume preserving transformations. More generally, what sets a specific class of diffeomorphisms defined on a symplectic manifold, called Hamiltonian diffeomorphisms, apart from generic volume-preserving ones. One can already grasp the duality of these phenomena: from a geometric point of view, at least under some assumptions on the manifold, the group of Hamiltonian diffeomorphisms is just the path connected component of the identity in the group of all symplectic diffeomorphisms, which in turn are diffeomorphisms that preserve the symplectic form ω, while from the point of view of classical mechanics it is the group of all possible mechanical motions. The pivotal theorem of this essay, that in some way characterizes symplectic transformations, is the so-called Non-squeezing Theorem, also known as the Symplectic camel Theorem, first proven by Mikhail L. Gromov in 1985 in his paper Pseudoholomorphic curves in symplectic manifolds. Said theorem states that one cannot embed a ball into a cylinder via a symplectic map if the radius of the ball is greater than the radius of the cylinder. It can be easily seen that, in turn, one can embed a ball of any radius into a cylinder of any other radius via a generic volume-preserving map. This tells us that symplectic transformations are less flexible than simple volume-preserving diffeomorphisms. From the viewpoint of physics, in one of the countless application of Gromov’s Theorem, this kind of rigidity can be seen in the generalized uncertainty relation, as shown by Maurice A. de Gosson in his article The symplectic egg in calssical and quantum mechanics. The discussion of this point, however, is not the goal of this presentation. The vast majority of what is discussed in this thesis is due to the study of Leonid Polterovich’s ETH lectures, grouped together in his book “The geometry of the group of symplectic diffeomorphisms". As a consequence of this fact, all ideas presented here, together with almost all of the proofs, are due to Polterovich. A handful of the most basic proofs have been worked out independently, while other ones were done with the help of third party books.