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Wednesday 27th

• Paula Cerejeiras
  University of Aveiro
  
  Title:  q-rational functions - a new concept
  Abstract:  The topic of matrix-valued rational functions has several applications, such as transfer functions of finite dimensional linear systems and observability operators in control theory. In this talk, we present a new approach to introduce the concept of rational functions into quantum calculus, give different characterisations of such functions, and establish the foundations of a theory of matrix-valued q-rational functions.
  
  

• Carmen Judiht Vanegas
  Universidad Yachay Tech
  
  Title: Weighted Dirac Operators
  Abstract: The heat transfer problem in anisotropic media has been little studied in the context of Clifford analysis. As a first step in this direction, we introduce weighted Dirac operators belonging to the Clifford algebra A_n, which factor the second-order elliptic differential operator  \Delta_n= div (B \nabla), where B belongs to R^(n x n) is a positive definite symmetric matrix. For these weighted Dirac operators, we construct fundamental solutions and obtain certain integral representations of functions such as Cauchy-Pompeiu integral formulas and Cauchy integral formulas.
  
  

• Jaqueline Godoy Mesquita
  University  of Brasilia
  
  Title: Functional Differential Equations with state-dependent delays: an overview and new trends

Abstract: In this talk, we will introduce the class of equations with state-dependent delays. We will explain the importance of dealing with this class of equations from the perspective of applications, and we will discuss the open problems in the field.

• Claudia Garetto
  Queen Mary University of London
  
  Title: Schrödinger type equations with singular coefficients
  Abstract: In this talk, I will report on some recent work in collaboration with Alexandre Arias Junior (São Paulo), Alessia Ascanelli (Ferrara), and Marco Cappiello (Torino) on the well-posedness of the Cauchy problem for Schrödinger type equations with singular coefficients. The singular nature of the coefficients will be dealt via regularisation method, and solutions will be provided in the very weak sense.
  
                                                                                  Thursday  28th
  
  

• Irene Tubikanec
  Johannes Kepler University Linz
  
  Title: Network inference in a stochastic multi-population neural mass model via approximate Bayesian computation
  Abstract: The aim of this talk is to infer the connectivity structures of brain regions before and during epileptic seizure. Our contributions are fourfold. First, we propose a 6N-dimensional stochastic (ordinary) differential equation (SDE) for modelling the activity of N coupled populations of neurons in the brain. This model further develops the (single population) stochastic Jansen and Rit neural mass model, which describes human electroencephalography (EEG) rhythms, in particular signals with epileptic activity. Second, we construct a reliable and efficient numerical scheme for the SDE simulation, extending a splitting procedure proposed for one neural population. Third, we propose an adapted Sequential Monte Carlo Approximate Bayesian Computation algorithm for simulation-based inference of both the relevant real-valued model parameters as well as the 0,1-valued network parameters, the latter describing the coupling directions among the N modelled neural populations. Fourth, after illustrating and validating the proposed statistical approach on different types of simulated data, we apply it to a set of multi-channel EEG data recorded before and during an epileptic seizure. The real data experiments suggest, for example, a larger activation in each neural population and a stronger connectivity on the left brain hemisphere during seizure.
  
  

• Dzoara Nuñez Ramos
  University of Wuppertal
  
  Title: Cohomology of a certain wild quotient singularity
  Abstract: Let k be an algebraically closed field. In characteristic 2, the isolated wild quotient singularities arising from an action of the cyclic group Z/2Z on the power series ring in two variables k[u,v], were completely described by Artin as an explicit hypersurface singularity. Peskin extended Artin's result in characteristic 3. Moreover, she developed canonical forms for certain p-cyclic automorphism of formal power series rings over fields in positive characteristic. More recently, Schröer and Lorenzini introduced moderately ramified actions in any characteristic p>0 on the formal power series ring k[u_1,..., u_n] with n greater than or equal to 2, which give rise to a new class of wild quotient singularities.

Motivated by this construction, we will describe in this talk the resolution of singularities of the singularity arising from the moderately ramified action on k[u,v] in a particular case and we aim to compute the cohomology of the structure sheaf of this resolution which is an invariant of this singularity.

• Laura Miller
  University of Arizona
  
  Title: Flows around some species of soft corals
  Abstract: In this presentation, I will discuss the construction and results of numerical simulations quantifying flows around several species of soft corals. In the first project, the flows near the tentacles of xeniid soft corals are quantified for the first time. Their active pulsations are thought to enhance their symbionts’ photosynthetic rates by up to an order of magnitude. These polyps are approximately 1 cm in diameter and pulse at frequencies between approximately 0.5 and 1 Hz. As a result, the frequency-based Reynolds number calculated using the tentacle length and pulse frequency is on the order of 10 and rapidly decays as with distance from the polyp.

This introduces the question of how these corals minimize the reversibility of the flow and bring in new volumes of fluid during each pulse. We estimate the Péclet number of the bulk flow generated by the coral as being on the order of 100–1000 whereas the flow between the bristles of the tentacles is on the order of 10. This illustrates the importance of advective transport in removing oxygen waste. In the second project, the flows through the elaborate branching structures of gorgonian colonies are considered. As water moves through the elaborate branches, it is slowed, and recirculation zones can form downstream of the colony. At the smaller scale, individual polyps that emerge from the branches expand their tentacles, further slowing the flow. At the smallest scale, the tentacles are covered in tiny pinnules where exchange occurs. 

We quantified the gap to diameter ratios for various gorgonians at the scale of the branches, the polyp tentacles and the pinnules. We then used computational fluid dynamics to determine the flow patterns at all three levels of branching. We quantified the leakiness between the branches, tentacles and pinnules over the biologically relevant range of Reynolds numbers and gap-to-diameter ratios, and found that the branches and tentacles can act as either leaky rakes or solid plates depending upon these dimensionless parameters. The pinnules, in contrast, mostly impede the flow. Using an agent-based modeling framework, we quantified plankton capture as a function of the gap-to diameter ratio of the branches and the Reynolds number. We found that the capture rate depends critically on both morphology and Reynolds number.

• Thaís Jordão
  University of São Paulo
  
  Title:  An asymptotic analysis of entropy numbers of Reproducing Hilbert Spaces of zonal kernels
  Abstract: In this talk, I will present a weak asymptotic equivalence for the covering numbers of the unit ball for a general class of Reproducing Kernel Hilbert Spaces (RKHS) on compact two-point homogeneous spaces. This equivalence is a consequence of the upper and lower bounds for the covering numbers of the unit ball of the RKHS generated by a continuous, rotation-invariant, positive-definite kernel. These results are new and extend both the previously known estimates on the d-dimensional unit sphere and the weak asymptotic equivalence for the growth rate of covering numbers.
  
  

• Sylvie Paycha
  University of Postdam
  
  Special talk: During this special session, we are delighted to introduce Professor Sylvie Paycha, a mathematician whose remarkable contributions have greatly advanced the visibility of women in mathematics.

Professor Paycha will present her inspiring project, Women of Mathematics: A Gallery of Portraits. This exhibition beautifully highlights the lives and work of women mathematicians around the world. In her talk, she will share insights into her motivations for creating this project, its evolution over the years, and its journey across various countries, including the development of sister exhibitions. She will also discuss her vision for the project's future and the profound impact it has had on the mathematical community, particularly in encouraging women in mathematics.

                                                                                Friday 29th

• Rajula Srivastava
  University of Bonnh
  
  Title:  Counting Rational Points using Harmonic Analysis
  Abstract: How many rational points (fractions) with denominator of a given size lie within a certain distance from a compact, "non-degenerate" surface? This talk is about some recent progress towards answering this question. We shall describe how the geometric and analytic properties of the surface play a key role in determining this count, and present a heuristic for the same. We shall then discuss how harmonic analytic techniques can be exploited to establish the desired asymptotic for rational points near manifolds satisfying certain strong geometric conditions.
  
  

• Maria Gordina
  University of Connecticut
  
  Title:  Stochastic analysis and geometric functional inequalities
  Abstract: We start by recalling that on a Euclidean space there is a connection between the spectrum of the Laplacian and the speed of heat diffusion, which leads to several functional inequalities, such as Poincare, Nash etc. Moving to a curved space, we see that the geometry of the underlying space plays an important role in such an analysis. If, in addition, the state space is infinite-dimensional, the log-Sobolev inequality becomes a useful fact which can be applied to describe entropic convergence of the heat flow to an equilibrium. A probabilistic point of view comes from a path integral representation of the heat flow for stochastic differential equations driven by a Brownian motion. In particular, we will discuss how the Cameron-Martin-Girsanov type theorem is related to certain functional inequalities. The talk will review recent advances in the field, including elliptic and hypoelliptic settings over both finite-and infinite-dimensional spaces.
  
  

• Luz Roncal
  Basque Center for Applied Mathematics
  
  Title:  Directional singular integrals in codimension one
  Abstract: This talk is intended to give an overview of the theory of maximal directional averages and directional multipliers in dimensions two and higher. The motivation is found in the so-called Stein and Zygmund conjectures, and the main obstruction for the boundedness of these operators are Kakeya counterexamples. These maximal operators are connected to differentiation problems and maximally modulated singular integrals, such as generalizations of Carleson’s maximal operator. In particular, I will present a sharp estimate for maximal directional singular integrals in codimension one.
  
  Joint work with O. Bakas, F. Di Plinio, and I. Parissis.