Q-Math Seminar
Gabriel Escrig (PhD Candidate at GICC - UCM)
Quantum Bayesian Inference with Renormalization for Gravitational Waves
Tuesday the 8th of April, 2025, 13:00, Room 2.2.D08
Advancements in gravitational-wave interferometers, particularly the next generation, are poised to enable the detections of orders of magnitude more gravitational waves from compact binary coalescences. While these surges in detections will profoundly advance gravitational wave astronomy and multimessenger astrophysics, they also pose significant computational challenges in parameter estimation.We introduce a hybrid quantum algorithm qBIRD, which performs quantum Bayesian Inference with Renormalization and Downsampling to infer gravitational wave parameters. Testing the algorithm with both simulated and observed gravitational waves from binary black hole mergers using quantum simulators, we show that its accuracy is comparable to that of classical Markov Chain Monte Carlo methods. Our current inference runs focus on a subset of parameters, such as chirp mass and mass ratio, due to constraints from classical hardware limitations in simulating quantum algorithms. However, qBIRD is completely scalable, and its full potential will be unlocked when these constraints are eliminated through a small-scale quantum computer with sufficient logical qubits.Link for online session (Active on request): https://eu.bbcollab.com/guest/22f7877a774148a3aee3e398a4a86380
Wend Werner (Universität Münster - Mathematisches Institut)
Enveloping algebras as algebras of bounded Hilbert space operators
Tuesday the 1st of April, 2025, 13:00, Room 2.2.D08
The transition from macro- to microscopic Physics can be carried out in a number of different ways. One of them is based on the deformation of algebras, a topic that is also of great interest within pure mathematics itself. Other so called quantization procedures yield Hilbert space operators very early on; deformation quantization, however, while being quite successful from a purely algebraic point of view, has often difficulties in finding some. We have a look at the universal enveloping algebra $U(g)$ of a Lie algebra $g$. When set up properly, such an algebra can be thought of as being algebraically generated by some physical relevant observables, and it also comes equipped with a very natural deformation into a commutative algebra. In joint work with Elizabeth Gillaspy we show that U(g) can be represented as an algebra of bounded Hilbert space operators iff the underlying Lie algebra is nilpotent. We additionally try to find a pattern inside the class of all such realizations of $U(g)$ for a fixed Lie algebra $g$.Link for online session (Active on request): https://eu.bbcollab.com/guest/22f7877a774148a3aee3e398a4a86380
Francisco Javier González Doña (UC3M)
An insight into commutants, idempotents and structure of operators.
Wednesday the 26th of March, 2025, 13:00, Room 2.2.D08
The structure of normal operators has been well understood since the 1970s, thanks to the spectral theorem. As a consequence of this result, strong connections between the (bi)commutant, the structure of invariant subspaces and the Borelian functional calculus arise. However, for non-normal operators on Banach spaces, this connection remains unclear and far from fully understood.In this talk, we will sketch the aforementioned notions for normal operators, and will introduce some recent ideas to study the structure of general operators from this point of view. If time permits, we will also present some results in this direction for composition operators acting on the Hardy space of the unit disc, where techniques from geometric function theory play a prominent role.Link for online session (Active on request): https://eu.bbcollab.com/guest/22f7877a774148a3aee3e398a4a86380
Pere Ara (UAB)
Separated graphs and dynamics
Tuesday the 4th of March, 2025, 13:00, Room 2.2.D08
A separated graph is a pair $(E,C)$, where $E$ is a directed graph, $C=\bigsqcup _{v\in E^ 0} C_v$, and $C_v$ is a partition of $r^{-1}(v)$ (into pairwise disjoint nonempty subsets) for every vertex $v$. In recent years, separated graphs have been used to provide combinatorial models of several structures, often related to dynamical systems. This can be understood as a generalization of the common use of usual directed graphs in symbolic dynamics. I will survey some of these developments, including the failure of Tarski's dichotomy in the setting of topological actions, the construction of a family of ample groupoids with prescribed type semigroup, and the modeling of actions on the Cantor set.Link for online session (Active on request): https://eu.bbcollab.com/guest/22f7877a774148a3aee3e398a4a86380
Jordi Gaset (CUNEF)
Disformal transformations: bringing order to modified gravity
Tuesday the 18th of February, 2025, 13:00, Room 2.2.D08
Recently it has been discovered that a generalization of conformal transformations, called disformal, relate large sections of the modified gravity landscape. We will present a novel approach to disformal transformation using Lie groups. Even if two models are related by a disformal transformation, the transfer of properties is not obvious because these transformations mix positions and velocities. We will show how to prove equivalence between two models in general situations.Link for online session (Active on request): https://eu.bbcollab.com/guest/22f7877a774148a3aee3e398a4a86380
Rubén Izquierdo (UNIR - ICMAT)
Graded Poisson brackets in classical field theories
Tuesday the 4th of February, 2025, 13:00, Room 2.2.D08
In this talk I will present some recent theory regarding the graded structure on brackets in classical field theory. More particularly, I will focus on presenting the tensorial version of the brackets, generalizing the classical bivector field - Poisson bracket correspondence of classical mechanics to the multisymplectic formalism of classical fields. If time allows it, I will present how to define dynamics with this structures and present some possible applications to almost regular Lagrangians.Link for online session (Active on request): https://eu.bbcollab.com/guest/22f7877a774148a3aee3e398a4a86380
Tomáš Gonda (University of Innsbruck)
Empirical distributions and strong laws of large numbers in categorical probability
Tuesday the 21st of January, 2025, 13:00, Room 2.2.D08
The Glivenko–Cantelli theorem shows that relative frequencies of events in an IID sequence converge to the original distribution one takes samples from. In this work, we provide the tools to study limits of relative frequencies, i.e. empirical distributions, in categorical probability. We identify two axioms, permutation-invariance and empirical adequacy, that a morphism should satisfy so that it can be interpreted as a map that takes an infinite sequence and produces a sample from its empirical distribution. Since not all sequences have an empirical distribution, it must be a partial map. To model these, we develop the theory of quasi-Markov categories, a variant of Markov categories in which morphisms need not have a well-defined output for each input. Our main contribution there is a construction that adds partial morphisms to a Markov category in a way that preserves many of the key properties of the original Markov category.Assuming that idempotents split and the two axioms, we derive representability (correspondence between Markov kernels and distribution-valued measurable functions) and prove abstract forms of the (strong) law of large numbers and the de Finetti theorem.We also provide several concrete constructions of such empirical distribution morphisms for measure-theoretic probability. In this way, our abstract LLN can specialize to the Glivenko-Cantelli theorem and the strong LLN respectively.Link for online session (Active on request): https://eu.bbcollab.com/guest/22f7877a774148a3aee3e398a4a86380
Marco Pacelli (Scuola Superiore Meridionale di Napoli)
Information geometry and statistical potentials: Classical and Quantum
Tuesday the 10th of December, 2024, 13:00, Room 2.2.D08
Statistical manifolds provide the abstract realisation of parametric models in statistics and information theory. They consist of a Riemannian manifold equipped with a pair of conjugate torsion-free affine connections. These geometric objects can be encoded into a 2-point function, termed contrast function. More general statistical structures have recently been considered where the torsion-free assumptions are relaxed or wholly removed. Even in this case, one can capture the geometrical structures through a single object, termed bi-form. In this talk, we review bi-form formalism and explain how it can be exploited to generate general statistical structures. Paradigmatic examples from classical and quantum information theory are discussed.Link for online session (Active on request): https://eu.bbcollab.com/guest/22f7877a774148a3aee3e398a4a86380
David Martinez Crespo (Universidad de Zaragoza)
Geometric Flavours of Quantum Field Theory: Modifications to the Schrödinger Equation and Applications to Cosmology
Tuesday the 26th of November, 2024, 13:00, Room 2.2.D08
In this talk, we will apply the geometrization of quantum field theory to explore its implications in curved backgrounds. In this program, the Schrödinger-like Hamiltonian equations of QFT are understood as a Hamiltonian flow in the manifold of pure states. This is an infinite-dimensional Kähler manifold representing the Schrödinger-like wavefunctions. In the case of curved backgrounds, this Kähler structure acquires a dependence on the Hamiltonian time parameter that, by relying solely on the Hamiltonian flow, jeopardizes the probabilistic interpretation of quantum theory. To overcome this difficulty, we describe the program of geometric quantum field theory (not to be confused with the geometrization program), explore its different representations, and propose modifications to the family of Schrödinger-like equations. The mathematical foundations of this study depend not only on differential geometry but also on the theory of Gaussian integration in topological vector spaces. As a consequence of this study, we will present applications of the formalism to cosmological Friedman-Lemaître-Robertson-Walker solutions of General Relativity.Link for online session (Active on request): https://eu.bbcollab.com/guest/22f7877a774148a3aee3e398a4a86380
Niel Van Buggenhout (UC3M)
Rational functions in quantum computing
Tuesday the 12th of November, 2024, 13:00, Room 2.2.D08
In classical computing, rational functions form an essential tool for developing effective algorithms for solving interior eigenvalue problems and approximating matrix functions.In quantum computing, the use of rational functions is underexplored. In this talk we introduce new methods for computing with rational functions on a quantum computer based on real-time evolution. We focus on computing matrix resolvents, since any rational function of a matrix can be written as a sum of resolvents.The matrix resolvent can be represented as an integral in terms of a complex matrix exponential. We propose two techniques for the approximation of this integral. The first technique discretizes the integral into the sum of time samples of the matrix exponential. We give a detailed quantification of the computational cost of this technique. The second technique expands the weight function of the integral as a sum of Gaussians, splitting the integral into the sum of easy-to-compute integrals.Link for online session (Active on request): https://eu.bbcollab.com/guest/d3ed81bffda3423692c5df53a2d584f4
Patrick Kreitzberg (University of Montana)
Property A vs. coarse embedding via graphs
Wednesday the 30th of October, 2024, 13:00, Room 2.2.D08
Since the creation of the coarse Baum-Connes conjecture there has been great interest in finding metric spaces which coarsely embed into Hilbert space. Of particular interest is when a metric space coarsely embeds into a Hilbert space yet does not possess G. Yu's 'property A'.A useful tool for coarsely embedding a graph $X$ into Hilbert space is the wall metric, $d_w$, which differs from the standard graph metric $d$. We will discuss an example of a metric space which coarsely embeds into Hilbert space and does not have property A and what we can say about $(X,d)$ based on results obtained from studying $(X,d_w)$.In this talk we will not assume prior familiarity with property A, coarse embeddings, and the wall metric.Link for online session (Active on request): https://eu.bbcollab.com/guest/d3ed81bffda3423692c5df53a2d584f4