Applied Analysis Group
(GAMA)
Francisco Marcellán (pacomarc@ing.uc3m.es)
1. Polinomios ortogonales en espacios de Sobolev con pesos soportados en la recta real. Aplicaciones en problemas de valor inicial y problemas de frontera.
In the last years, Sobolev orthogonality has been one of the main generalisations of the classical idea of orthogonality for orthogonal polynomials, with on the real line and on the unit circle. It involves not only the polynomials themselves in the construction of the inner product that defines the orthogonality, but also their derivatives. As such, the construction is very suited for the approximation of solutions of initial value problems and boundary value problems in the theory of differential equations. This generalises the natural application of orthogonal polynomials (in particular classical orthogonal polynomials such as Hermite, Laguerre and Jacobi) in the construction of spectral methods for differential equations.
2. Transformaciones espectrales de funcionales lineales hermitianos. Aplicaciones a pares coherentes generalizados de medidas en el círculo unidad.
Hermitian linear functionals are natural objects in the study of orthogonal polynomials on the real line and on the unit circle. These linear functionals can be represented by Jacobi matrices, and there is a natural relation between the spectral measure of such Jacobi matrices and the measure of orthogonality of the corresponding sequence of orthogonal polynomials. When we consider classical factorizations of this Jacobi matrix (for example LU), and then operations with the resulting factors, we obtain modifications (polynomial or rational) of the measure of orthogonality. The main goal of this project is to investigate such transformations in the context of coherent pairs of measures on the unit circle.
José Manuel Rodríguez García (jomaro@math.uc3m.es)
Several proofs of Picard's small theorem (TFG or TFM)
Picard's small theorem is an important result in Complex Analysis that states that if f is an entire function (a complex function that is holomorphic everywhere in the complex plane) and if f omits two distinct complex values, then f must be a constant function.
The goal of this project is to present several proofs of this theorem, showing the great variety of approaches, ideas and methods that allow obtaining this result.
Each proof highlights different aspects of complex analysis and can be useful depending on the context and the tools available.
Integral inequalities involving convex functions (TFG or TFM)
Jensen's inequality is probably the most famous of all integral inequalities involving convex functions, but there are many others, which have been considerably improved in recent years, such as the inequalities of Petrovic, Giaccardi, Karamata, etc.
The goal of this project is to present detailed proofs of some of these inequalities, showing the great variety of approaches, ideas and methods that allow obtaining these inequalities.
Alfredo Deaño (alfredo.deanho@uc3m.es)
1.The Riemann zeta function (TFM).
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The Riemann zeta function is an example of special function.These are functions
that relevant in Mathematics and in applications and that cannot be written in
terms of elementary functions. In the case of the Riemann zeta function is central
in analytic number theory, and it is closely related to the distribution of prime
numbers. In this project, we explore this function and its properties, on the real line
and in the complex plane.
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​​​References:
[1] H. M. Edwards. Riemann's zeta function. Dover Publications, Inc., Mineola, NY, 2001.
[2] B. Mazur, W. Stein. Prime Numbers and the Riemann Hypothesis. Cambridge University Press,
Cambridge, 2015.
[3] T. Murphy. A Study of Bernhard Riemann's 1859 paper. Paramount Ridge Press, 2020.
[4] J. Stopple. A primer of analytic number theory. From Pythagoras to Riemann.
Cambridge University Press, Cambridge, 2003.
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2. Permutations and longest increasing subsequences (TFM).
Given a permutation of the N first natural numbers, a classical question in combinatorics is what is the length of the longest increasing subsequence. For example, (2 3 9 1 7 8 6 5 4) would yield 4. This example can be plotted as in the figure.
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One particularly intriguing problem is the behavior of such length as the parameter N grows large. This problem poses interesting challenges both computationally (e.g. how to sample permutations and calculate the longest increasing subsequence) and theoretically, and it shows remarkable connections with asymptotic analysis, random matrix theory, interacting particle systems and combinatorics.
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References:
[1] D. Romik. The surprising mathematics of longest increasing subsequences. Cambridge University Press, New York, 2015.
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3. Random tilings of plane figures (TFM).
This project investigates tiling of figures in the plane, using dominoes or lozenges. Some classical examples are the Aztec diamond and the regular hexagon. Relevant questions in this area are the total number of possible tilings or how to transform one tilings into another one. Moreover, If we assign a uniform probability over the configuration of tilings, one can observe interesting patterns as the size of the figures becomes large.
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References:
[1] N. Elkies, G. Kuperberg, M. Larsen, J. Propp. Alternating-sign matrices and domino tilings I and II. J. Algebraic Combin. 1 (1992), no. 2, 111–132 and no. 3, 219–234.
[2] K. Johansson. Random Matrices and Determinantal Processes. Mathematical Statistical Physics, Volume 83 (2006), 1--56.
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Tilings of the Aztec diamond. Code by C. Charlier.
Tilings of regular hexagon. Code by C. Charlier.
Yamilet Quintana (yaquinta@math.uc3m.es)
TFM proposal: Qi-type problems for hypergeometric Bernoulli polynomials
Among the applications of Bernoulli-type polynomials in number theory and analysis is their use in studying monotonicity, convexity, and inequalities involving special functions. Taking as a starting point the Qi’s problem on the monotonicity of ratios of Bernoulli polynomials, it is possible to investigate analogous questions for hypergeometric Bernoulli polynomials, a modern generalization defined through hypergeometric generating functions.
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These hypergeometric analogues retain structural properties such as Appell-type differential relations, inversion formulas, and recurrence identities, but they also introduce new parameters that enrich their algebraic and analytic behaviour.​
Under my supervision, a Master Thesis on the study of Qi-type monotonicity and inequality problems for hypergeometric Bernoulli polynomials is offered.
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Complementary information:
https://sites.google.com/view/yamiletquintana/research
G.-Z. Zhang, Z.-H. Yang, and F. Qi, On normalized tails of series expansion of generating function of Bernoulli numbers, Proc. Amer. Math. Soc. 153 (2025), no. 1, 131-141; available online at
1) Approximating Polynomials of High Degree (TFM or TFG)
Polynomials are everywhere: from numerical algorithms to modeling biological growth curves and economic data trends. In a course like Advanced Methods in Applied Analysis, you learn how to approximate functions using orthogonal polynomials. But what about the polynomials themselves? When the degree is very high, their behavior becomes subtle and challenging to handle. Several methods exist to approximate such polynomials, one of the most powerful being based on so-called Riemann-Hilbert problems. In this project, you will learn about these approximation methods and possibly apply them to explicit models drawn from physics, biology, economics, or other applied fields.
2) Universality in Random Matrix Models (TFM or TFG)
Randomness is often chaotic, yet in many complex systems (physics, finance, networks, even neural activity), statistical patterns repeat. In random matrix theory, such universal patterns are encoded in special kernels: the sine kernel (governing the bulk of spectra) and the Airy kernel (governing the spectral edge). The striking fact is that very different systems exhibit the same local behaviour, much like the central limit theorem shows for sums of random variables. In this project, you will study random matrices, explore the phenomenon of universality, and possibly apply the associated methods to explicit real-world models drawn from physics, biology, telecommunication, finance, or other applied fields.
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3) Point Processes in the Plane (2D Coulomb Gases) (TFM or TFG)
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Imagine particles in the plane that repel each other like electric charges: they spread out, yet form surprising collective patterns. Such systems, called 2D Coulomb gases, describe not only electrons in physics, but also eigenvalues of random matrices, the arrangement of plants or birds, and even the distribution of economic agents in markets. Mathematically, they are modeled by point processes, with rich connections to probability theory and statistical mechanics. In this project, you will learn about several classes of 2D Coulomb gases (and/or related models) and possibly apply the associated methods to explicit real-world models. (NB: Knowledge of physics is not required.)
1) Sobolev orthogonal polynomials for solving boundary value problems.
Spectral methods for solving differential equations are among the most efficient solvers for differential equations [1]. These spectral methods are typically based on classical orthogonal polynomials. In this project the goal is to develop a new class of spectral methods based on Sobolev orthogonal polynomails (SOPs). These SOPs can be generated numerically using linear algebra techniques [2]. In this project we will develop a solver for boundary value problems based on SOPs. Namely, we will generate a basis of SOPs which are closely related to the boundary value problem (BVP) and develop an automatic procedure to take the boundary conditions into account.
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Planning of the research project:
​1) Study the linear algebra algorithm [2] which is used for constructing SOPs.
2) For some interesting BVPs we will derive the weak formulation, which gives rise to an inner product that can be used to generate a good basis of SOPs.
3) Develop an algorithm that can take boundary conditions into account, this can be done either algebraically or analytically [1].
4) Compare the developed algorithm to other algorithms and perform an analysis of its numerical behavior.
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​​[1] Lloyd N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000. URL: https://doi.org/10.1137/1.9780898719598
[2] Niel Van Buggenhout, On generating Sobolev orthogonal polynomials. Numer. Math. 155, 415–443 (2023). URL: https://doi.org/10.1007/s00211-023-01379-3
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2) Rational quantum algorithms
Rational functions are essential in classical computing, they allow to compute interior eigenvalues of huge matrices and allow efficient approximations to matrix functions. Recent advancements in quantum computing have made it possible to compute with rational functions using a quantum computer [1].
Using a structured matrix approach, approximation theoretical results, and numerical linear algebra techniques we will develop a new quantum algorithm. The computational efficiency of these quantum algorithms for computing with rational functions strongly depend on the location of the poles of the rational function. Due to this dependency, classical rational algorithms cannot be translated directly to a quantum algorithm; it is necessary to rethink how these rational functions can be used in quantum computing.
Planning of the research project:
1) Study the structured matrix interpretation of important quantum algorithms for solving eigenvalue problems, for example [2].
2) Compare the algorithms using insights obtained from this interpretation.
3) Develop a novel quantum algorithm exploiting rational function computation [1].
4) Analyze the novel rational quantum algorithm and apply it to quantum chemistry applications.
[1] Yizhi Shen, Niel Van Buggenhout, Daan Camps, Katherine Klymko, and Roel Van Beeumen, Quantum Rational Transformation using Linear Combinations of Hamiltonian Simulations, ArXiv 2408.07742 (2024). URL: https://arxiv.org/abs/2408.07742
[2] Katherine Klymko, et al., Real-time evolution for ultracompact Hamiltonian eigenstates on quantum hardware, PRX Quantum, 3 (2022), p. 020323. URL: https://doi.org/10.1103/PRXQuantum.3.020323
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