16:00 – 17:00
An introduction to stochastic integration
Salim Boukfal (UB)
The aim of this work is to extend the notion of stochastic integral with respect to Brownian motion, studied in the Stochastic Calculus course of the Master's in Advanced Mathematics at UB, to more general processes such as martingales. Once this task is completed, we see how we can define integrals with respect to random fields (processes that can depend on more than one parameter), starting with Gaussian white noise and, finally, moving on to martingale measures. Apart from carrying out the relevant constructions, we also present a couple of approximation results in law where we prove that the law of the stochastic integral with respect to Brownian motion or Gaussian white noise (in two dimensions) can be approximated by integrals with respect to random walks in one or two dimensions, respectively.
Stochastic differential equations driven by fractional Brownian motion
Òscar Burés (UB)
The work is a general study of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with Hurst index greater than 1/2. First, the stochastic integral with respect to fBm is defined and the existence and uniqueness of solutions to general SDEs is studied. Then, using Malliavin calculus adapted to fBm, the absolute continuity of the law of the solution to a general SDE is studied. Finally, using more powerful Malliavin calculus techniques, Gaussian bounds for the density of solutions to a family of SDEs are obtained.
Regularity of Lipschitz free boundaries in the Alt-Caffarelli problem
Joan Domingo Pasarin (UB)
In this work we study the regularity of Lipschitz free boundaries in the Alt-Caffarelli problem. We prove that these are C1,α using the scaling invariance of the problem and the initial Lipschitz regularity of the boundary. Furthermore, we also show that the boundaries C1,α are C∞, which, combined with the previous result, implies that Lipschitz free boundaries are C∞.
Spectral gap of generalized MIT bag models
Joaquim Duran i Lamiel (UPC)
We study spectral properties of generalized MIT bag models. These are Dirac operators acting in domains of ℝ3 with boundary conditions that generate confinement. Their lowest positive eigenvalue is of special interest, and it has been conjectured to be minimal for a ball among all domains with fixed volume.
17:30 – 18:30
Unified theory of decision-making dynamics: perspectives from optimal control and infinite horizon
Flàvia Ferrús (UB)
The main objective of this project is to develop a unifying theoretical framework for motor control and decision-making. To this end, an introduction to variational calculus and optimal control theory is first presented in order to build a solid theoretical basis as a means to understand the dynamics of the system. Then, based on the theoretical context presented, a linear generating dynamical system is proposed as an approximation to the studied physical system. Subsequently, a sequential model is developed to predict and simulate trajectories, approximating the experimental ones using the Infinite Horizon formulation. Given the complexity of finding an exact analytical solution for the considered dynamical system, the Kalman filter is presented and studied as an accurate method to find the best estimates for the real state vector of position and velocity profiles.
On the basins of attraction of root-finding algorithms
David Rosado Rodríguez (UB)
Root-finding algorithms have historically been used to numerically solve nonlinear equations of the form f(x)=0. Newton's method, one of the best-known techniques, began to be analyzed as a dynamical system in the complex plane at the end of the 19th century. This work explores the dynamics of the methods of the parameterized Traub family Tp,δ applied to polynomials. These methods include a range from Newton's method (δ=0) to Traub's method (δ=1). Our approach lies in investigating various topological properties of the basins of attraction, particularly their simple connectivity and unboundedness, which are crucial for identifying a universal set of initial conditions that ensure convergence to all roots of p. While these topological properties are already proven for Newton's method (δ=0), they remain open for δ≠0. We present results that contribute to addressing this open problem, including a proof for cases where δ is close to 0 and for the family of polynomials pd(z)=z(zd-1).
Modules of plane branches with a single characteristic exponent
María de Leyva Elola-Olaso (UPC)
We study the moduli space of plane branches with a single characteristic exponent via a stratification using the semimodule of values of the Jacobian ideal of the branch. In particular, we study how to approach the problem using different techniques. First, we provide an algorithmic procedure based on the Casas-Alvero procedure, which, under some assumptions, describes the strata. We include a Maple implementation of this algorithm. Furthermore, we compare our stratification with one previously studied by Peraire in 1998 based on the Zariski invariant. This allows us to make some reflections on the challenges of computing the dimension of our strata, which refine Peraire's strata, and to present some new tools to address the problem.
Geometric techniques in monogeneity
Francesc Pedret (UPC)
A number field K is monogenic if its ring of integers is generated by a single element as a ℤ-algebra. In the cubic case, determining whether K is monogenic or not is equivalent to solving the Diophantine equation |IK(X,Y)| = 1 over ℤ, where IK is the index form of the field. An integer solution determines a rational point on the genus 1 curve, IK(X,Y) = Z3. Using this construction, it can be shown that a cubic field K with discriminant D determines an 𝔽3-orbit in H1(ℚ, E[3]), where E is the elliptic curve defined by Y2 = 4X3 + D.
We give the explicit construction of this orbit for the case of pure cubic fields and characterize the sum of cocycles associated with non-isomorphic fields.
Ideals of pe–th roots of curves in the plane in positive characteristic
Pedro López Sancha (UPC)
The study of singularities in algebraic varieties has been a prosperous field of research and has become a central focus in modern algebraic geometry. A common approach to understanding these singularities is by attaching invariants that characterize them.
In this project, we compute several invariants of singularities in plane curves defined over fields of positive characteristic and compare them with their counterparts in characteristic zero.