**Academic Calendar 2022/2023** here**Information about acceptance of candidates to MAP-PDMA can be consulted** here **and** here

**Information about the courses Syllabus_2022_2023****Information about the 1st semester Information_2022_2023_Sem1****Information about the 2nd semester Information_2022_2023_Sem2**

**SEMINARS***For information about this UC, visit the Course Description*

**Seminar #17: MAP-PDMA PhD Program 2022/2023 — January 27, 16:00**

Place: Via zoom at

https://videoconf-colibri.zoom.us/j/94769148463?pwd=YWRVYjEzMkk2bmtMY2Q2OEhFM0hUdz09

Speaker: Sofia J. Pinheiro, Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro

email: spinheiro@ua.pt

Title: **The spectrum of the $H$-join of arbitrary graphs – the walk matrix approach**

Abstract: The $H$-join of a family of graphs $\mathcal{G}={G_1, \dots, G_p}$, also called the generalized composition, $H[G_1, \cdots, G_p]$, where all graphs are undirected, simple and finite, is the graph obtained by replacing each vertex $i$ of $H$ by $G_i$ and adding to the edges of all graphs in $G$ the edges of the join $G_i \vee G_j$ , for every edge $ij$ of $H$. For a long time the known expressions for the determination of the entire spectrum of the $H$-join in terms of the spectra of its components and an associated matrix were limited to families of regular graphs. In this work, with an approach based on the walk-matrix, we extend such a determination, as well as the determination of the characteristic polynomial, to the adjacency matrix of the $H$-join of families of arbitrary graphs [1].

[1] D.M. Cardoso,H. Gomes, S.J. Pinheiro, The H-join of arbitrary families of graphs, arXiv:2101.08383v3, 2021.

**Seminar #16: MAP-PDMA PhD Program 2022/2023 — January 27, 14:30**

Place: Seminar Room of DMat-UMinho (3.08), and via zoom at

https://videoconf-colibri.zoom.us/j/94769148463?pwd=YWRVYjEzMkk2bmtMY2Q2OEhFM0hUdz09

Speaker: Inês Sousa, Centre of Mathematics, University of Minho

email: isousa@math.uminho.pt

Title: **Joint modeling for longitudinal and survival data**

Abstract: This paper will present the theory of joint models for longitudinal and survival data in the context of biostatistics. We will go through the different existing methodologies and present examples of real data to contextualize these models. Results of fitting joint models to a breast cancer database will be presented, where the longitudinal variable is the evolution of a tumour marker and the event of interest is death or relapse. The R package joineR is used to implement the models.

**Seminar #15: MAP-PDMA PhD Program 2022/2023 — January 20, 16:00**

* Place: Via zoom at

https://videoconf-colibri.zoom.us/j/92403741454?pwd=UE43T1c3M3g5Y3VoWlZENkMwaGcrUT09

* Speaker: Vera Afreixo, Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro

email: vera@ua.pt

* Title: **Meta-analysis: detecting the effect size not the bias**

* Abstract: Systematic reviews and meta-analyses have an important role to evidence-based clinical practice. Publication bias is an huge difficulty to to get around. Publication bias is a phenomenon that distorts and exaggerates the evidence to be compiled with a meta-analysis! A systematic review with meta-analysis with considerable publication bias can be useless. In this seminar will be discussed different manifestations of bias, tools for its detection and correction as tools of effect size sensitivity analysis.

**Seminar #14: MAP-PDMA PhD Program 2022/2023 — January 20, 14:30**

* Place: Seminar Room of DMat-UMinho (3.08), and via zoom at

https://videoconf-colibri.zoom.us/j/92403741454?pwd=UE43T1c3M3g5Y3VoWlZENkMwaGcrUT09

* Speaker: Carla M. A. Pinto, Center of Mathematics, University of Porto and ISEP, Polytechnic of Porto email: cap@isep.ipp.pt

* Title: **A beautiful journey towards an epidemiological model**

* Abstract: In this talk we will show how `simple’ ordinary differential equations can help our understanding of the dynamics of infectious diseases. We will do this with the help of a proposed within-host model for the co-existence of HIV infection and T1 diabetes [1].

[1] J. P. Chávez, K. P. Wijaya, C. M. Pinto & C. Burgos-Simón, A model for type I diabetes in an HIV-infected patient under highly active antiretroviral therapy, Chaos, Solitons & Fractals, 155, 111716, 2022.

**Seminar #13: MAP-PDMA PhD Program 2022/2023 — January 13, 14:30**

* Place: Seminar Room of DMat-UMinho (3.08), and via zoom at

https://videoconf-colibri.zoom.us/j/92403741454?pwd=UE43T1c3M3g5Y3VoWlZENkMwaGcrUT09

* Speaker: Wolfram Erlhagen, Centre of Mathematics, University of Minho

email: wolfram.erlhagen@math.uminho.pt

* Title: **Dynamic Neural Fields: Theory and Applications**

* Abstract: Dynamic Neural Fields (DNFs) formalized by nonlinear integro-differential equations have been originally introduced as a model framework for explaining basic principles of neural information processing in which the interactions of billions of neurons are treated as a continuum. The intention is to reduce the enormous complexity of neural interactions to simpler, population properties that are tractable by analytical mathematical tools. More recently, complex models consisting of several connected DNFs have been developed to explain higher level cognitive functions (e.g., memory, decision making, prediction and learning) and to implement these functionalities in autonomous robots. I will give an overview about the physiological motivation of DNFs, the mathematical analysis of their dynamic behaviors, and their application in cognitive robotics. As an example study, I focus on ”multi-bump” solutions that have been proposed as a neural substrate for a multi-item memory function. I show how the existence and stability properties of these solutions can be exploited to endow a robot with the capacity to efficiently learn the timing and the serial order of sequential events. I also discuss new mathematical challenges that are motivated by robotics applications.

**Seminar #12: MAP-PDMA PhD Program 2022/2023 — January 6, 14:30**

* Place: Seminar Room of DMat-UMinho (3.08), and via zoom at

https://videoconf-colibri.zoom.us/j/94769148463?pwd=YWRVYjEzMkk2bmtMY2Q2OEhFM0hUdz09

* Speaker: Ana Cristina Ferreira, Centre of Mathematics, University of Minho

email: anaferreira@math.uminho.pt

* Title: **Geodesic completeness on Lie groups**

* Abstract: In this talk we will discuss geodesic completeness of left- invariant metrics for real and complex Lie groups. We will start by establishing the Euler-Arnold formalism in the holomorphic setting. We will present a new method for reobtaining the well-known classification for the real Lie group SL(2,$\mathbb{R}$) and, as a new addition, how it can be used to investigate the maximum domain of definition of every single geodesic for every possible metric. We will also discuss the notion of geodesic completeness for holomorphic metrics and establish a full classification for the Lie group SL(2,$\mathbb{C}$) for which it can be seen that holomorphic complete metrics are rare.

This is joint work with Ahmed Elshafei and Helena Reis (University of Porto).

**Seminar #11: MAP-PDMA PhD Program 2022/2023 — December 16, 15:00**

* Place: Seminar Room of DMat-UMinho (3.08), and via zoom at

https://videoconf-colibri.zoom.us/j/94769148463?pwd=YWRVYjEzMkk2bmtMY2Q2OEhFM0hUdz09

* Speaker: Raquel Menezes, Center of Mathematics, University of Minho

email: rmenezes@math.uminho.pt

* Title: **Modelling observed data from a latent stochastic spatial process**

* Abstract: Geostatistical models become important when data is collected from different locations in space, and the variable of interest can (in theory) be measured at any location in the study area. One should assume an underlying spatial stochastic process indexed in a continuous domain, and spatial correlation must be taken into account. These models can be extended to include time, if one has data collected over space and time. In Portugal, the spatial distribution and abundance of several commercial fish species is mostly unknown, and there are many open questions about fishing sustainability. Geostatistical models, relying on information from scientific surveys or commercial fisheries, become

useful tools for the assessment of distribution species. Fishery-dependent data present advantages, namely easier to obtain and better coverage of the time dimension, but it leads to domain representation issues. The fishermen movements are guided by some prior-knowledge of the places where it is expected to find the target species, thus the sampled data do not equally represent the study area. This is coined preferential sampling [2]. In this talk, after the introduction of background concepts in spatial statistics, the differences between “preferential” and “clustering” sampling design issues are emphasized. The model in [2] is briefly presented, for which the locations of the observed points are assumed to be informative, ie, the presence of preferential sampling is assumed. Recent contributions that aim to overcome the computational limitations of the previous model are discussed [1], [3], [5], which allow creating the appropriate context to apply this model to large volumes of data. This research falls within the scope of FCT project PTDC/MAT-STA/28243/2017 – Prefferential, with a strong collaboration from the Department of the Sea and Marine Resources of the Portuguese Institute of Sea and Atmosphere (IPMA).

[1] Diggle P. and Giorgi E., Model-based geostatistics for global public health: Methods and applications, Chapman and Hall/CRC Press, 2019.

[2] Diggle P., Menezes R. and Su T.L., Geostatistical Inference under Preferential Sampling (with discussion), Journal of Royal Statistics Society, series C, 59(2), 191{232, 2010.

[3] Dinsdale D. and Salibian-Barrera M., Methods for preferential sampling in geostatistics, Journal of the Royal Statistical Society, Series C (Applied Statistics), 68(1), 2019.

[4] Lindgren F., Rue H. and Lindstrom J. (2011). An explicit link between gaussian fields and gaussian markov random fields: the stochastic partial differential equation approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73, 423-498, 2011.

[5] Monteiro A., Menezes R. and Silva M.E., Modelling preferential sampling in time, Boletin de Estadstica e Investigacon Operativa, 35(3), 180{196, 2019.

[6] Zuur A., Ieno E. and Saveliev A., Beginner’s Guide to Spatial, Temporal and Spatial-Temporal Ecological Data Analysis with R-INLA, Highland Statistics Ltd., 2017.

**Seminar #10: MAP-PDMA PhD Program 2022/2023 — December 9, 15:45**

* Place: Seminar Room of DMat-UMinho (3.08), and via zoom at

https://videoconf-colibri.zoom.us/j/94769148463?pwd=YWRVYjEzMkk2bmtMY2Q2OEhFM0hUdz09

* Speaker: Roman Chertovskih, Research Center for Systems and Technologies (SYSTEC), Engineering Faculty, University of Porto

email: roman@fe.up.pt

* Title:** Dynamo theory: From Physics and Engineering to Mathematics and Supercomputing**

* Abstract: We will discuss the dynamo problem [1] – magnetic field generation by flows of an electrically conducting fluid. We will survey the magnetic activity in the Universe: magnetic fields of planets, stars and galaxies [2], and consider an important engineering application – magnetic fields in liquid metals cooling a reactor. To simulate such magnetic phenomena, the governing equations will be introduced and the basics of magnetohydrodynamics [3] will be discussed. We also plan to overview the mathematical methods used in the analysis of such problems: ranging from the dynamical systems theory [4] andthe equivariant bifurcation theory [5] to the numerical spectral methods [6]. Finally, the use of high performance computers for the considered problems will be addressed.

[1] Moffatt K., Dormy E. Self-Exciting Fluid Dynamos. Cambridge Texts in Applied Mathematics, Cambridge University Press, 2019.

[2] Rudiger G., Hollerbach R. The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory. Wiley, 2004.

[3] Molokov S., Moreau R., Moffatt K. Magnetohydrodynamics: Historical Evolution andfTrends. Springer, 2007.

[4] Bohr T., Jensen M.H., Paladin G., Vulpiani A. Dynamical Systems Approach to Turbulence. Cambridge Nonlinear Science Series, Cambridge University Press, 2005.

[5] Chossat P., Lauterbach R. Methods in Equivariant Bifurcations and Dynamical Systems. Advanced Series in Nonlinear Dynamics, World Scientific, 2000.

[6] Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A. Spectral Methods: Fundamentals in Single

Domains. Scientific Computation series, Springer, 2006.

**Seminar #9: MAP-PDMA PhD Program 2022/2023 — December 9, 14:30**

https://videoconf-colibri.zoom.us/j/94769148463?pwd=YWRVYjEzMkk2bmtMY2Q2OEhFM0hUdz09

* Speaker:** **Ariel Martín Pacetti, Centro de Investigação e Desenvolvimento em Matemática e Aplicações, University of Aveiro

email: apacetti@ua.pt

* Title: **Zeta function of projective varieties**

* Abstract: The main goal of the present talk is to define the local and global zeta function of algebraic varieties, with special emphasis on particular examples. We will see how well known functions (like Riemann’s zeta function) appear in this way. We will state some hard open problems regarding zeta functions, and some important results obtained during the last years. The presentation is aimed at a general audience.

**Seminar #8: MAP-PDMA PhD Program 2022/2023 — November 18, 14:30**

MAP-PDMA PhD Program 2022/2023 — December 2, 14:30

* Place: Seminar Room of DMat-UMinho (3.08), and via zoom at

https://videoconf-colibri.zoom.us/j/94769148463?pwd=YWRVYjEzMkk2bmtMY2Q2OEhFM0hUdz09

* Speaker: Bruno M. P. M. Oliveira, FCNAUP and LIAAD – INESC TEC, University of Porto

email: bmpmo@fcna.up.pt

* Title: **A mathematical model of immune responses with CD4+ T cells and Tregs**

* Abstract:** **We use a a set of ordinary differential equations (ODE) to study mathematically the effect of regulatory T cells (Tregs) in the control of immune responses by CD4+ T cells. T cells trigger an immune response in the presence of their specific antigen, while regulatory T cells (Tregs) play a role in limiting auto-immune diseases due to their immune-suppressive ability, see Pinto et al. [5], Yusuf et al. [6] and references within.

We fitted this model to quantitative data regarding the CD4+ T cell numbers from the 28 days following the infection of mice with lymphocytic choriomeningitis virus LCMV. We observed the proliferation of T cells and, to a lower extent, Tregs during the immune activation phase following infection and subsequently, during the contraction phase, a smooth transition from faster to slower death rates, see Afsar et al. [1].

Furthermore, we have obtained explicit exact formulas that give the relationship between the concentration of T cells, the concentration of Tregs, and the antigenic stimulation of T cells, when the system is at equilibria, stable or unstable. We found a region of bistability, where 2 stable equilibria exist. Making a cross section along the antigenic stimulation of T cells parameter, we observe an hysteresis bounded by two thresholds of antigenic stimulation of T cells. Moreover, there are values of the slope parameter of the tuning, between the antigenic stimulation of T cells and the antigenic stimulation of Tregs, for which an isolacenter bifurcation appear and, for some other values, there is a transcritical bifurcation, see Yusuf et al. [6] and references within.

Time evolutions of this model were also used to simulate the appearance of autoimmunity both due to cross-reactivity or due to bystander proliferation, and to simulate the suppression of the autoimmune line of T cells after a different line of T cells responds to a pathogen infection, see Burroughs et al. [2, 3] and Oliveira et al. [4].

[1] A. Afsar, F. Martins, B. M. P. M. Oliveira, and A. A. Pinto. A fit of CD4 + T cell immune response to an infection by lymphocytic choriomeningitis virus. Mathematical Biosciences and Engineering, 16(6):70097021, 2019.

[2] N. J. Burroughs, B. M. P. M. Oliveira, and A. A. Pinto. Regulatory T cell adjustment of quorum growth thresholds and the control of local immune responses. Journal of Theoretical Biology, 241:134141, 2006.

[3] N. J. Burroughs, M. Ferreira, B. M. P. M. Oliveira, and A. A. Pinto. Autoimmunity arising from bystander proliferation of T cells in an immune response model. Mathematical and Computer Modelling, 53:13891393, 2011.

[4] B. M. P. M. Oliveira, R. Trinchet, M. V. Otero-Espinar, A. A. Pinto, and N. J. Burroughs. Modelling the suppression of autoimmunity after pathogen infection. Mathematical Methods in the Applied Sciences, 41(18):85658570, 2018.

[5] A. A. Pinto, N. J. Burroughs, F. Ferreira, and B. M. P. M. Oliveira. Dynamics of immunological models. Acta Biotheoretica, 58:391404, 2010.

[6] A. A. Yusuf, Isabel P. Figueiredo, A. Afsar, N. J. Burroughs, B. M. P. M. Oliveira, and A. A. Pinto. The effect of a linear tuning between the antigenic stimulations of CD4+T cells and CD4+ Tregs. Mathematics, 58:391404, 2010.

**Seminar #7: MAP-PDMA PhD Program 2022/2023 — November 25, 14:30**

* Place: Seminar Room of DMat-UMinho (3.08), and via zoom at

https://videoconf-colibri.zoom.us/j/92403741454?pwd=UE43T1c3M3g5Y3VoWlZENkMwaGcrUT09

* Speaker: Sílvio Gama, Centre of Mathematics, University of Porto

email: smgama@fc.up.pt

* Title: **Point vortices, regular islands and polynomials**

* Abstract: After a brief description of what point vortices and passive particles are – on the plane and on the sphere – and how they can mimic real flows, we will derive their dynamic equations from the two-dimensional incompressible Euler equation. Next, we establish the connection between the relative equilibria of identical point (plane) vortices and the first and second derivatives of the polynomial that has the positions of the vortices as roots. Finally, we will present some open problems, as well as simulations based on computational models.

**Seminar #6: MAP-PDMA PhD Program 2022/2023 — November 18, 15:45**

https://videoconf-colibri.zoom.us/j/94769148463?pwd=YWRVYjEzMkk2bmtMY2Q2OEhFM0hUdz09

* Speaker: Filipe Martins, Centre of Mathematics, University of Porto

email: luis.f.martins@fc.up.pt

* Title: **Bifurcations in evolutionary matrix models in population dynamics**

* Abstract: In this talk I will consider evolutionary game theoretic versions of a general class of matrix models frequently used in population dynamics. The evolutionary components model the dynamics of a vector of mean phenotypic traits subject to natural selection [1]. One fundamental question in population and mathematical biology is population extinction and persistence, i.e., stability/instability of the extinction equilibrium and of other non-extinction equilibria. I will discuss this question through the prism of dynamic bifurcations. When the model parameters, more precisely, the inherent population growth rate, dynamic bifurcations occur, opening possibility for population persistence and recurrence, or to possible extinction. The results present a complete answer to a general class of evolutionary matrix models often used in mathematical biology, the mathematical assumption being that the matrix is primitive. I will present an application of the general theoretical results to an evolutionary version of a classic Ricker model. This application illustrates the theoretical results and, in addition, several other interesting dynamic phenomena.

Most part of the results and conclusions that I will talk about in this seminar are presented in [2] (joint work with Jim M. Cushing, Alberto Pinto and Amy Veprauskas).

[1] Joel S. Brown and Thomas L. Vincent, Evolutionary Game Theory, Natural Selection and Darwinian Dynamics, Cambridge University Press, 2005.

[2] “A bifurcation theorem for evolutionary matrix models with multiple traits”, Journal of Mathematical Biology, Vol. 75, Issue 2, pp. 491–520, 2017.

**Seminar #5: MAP-PDMA PhD Program 2022/2023 — November 18, 14:30**

https://videoconf-colibri.zoom.us/j/94769148463?pwd=YWRVYjEzMkk2bmtMY2Q2OEhFM0hUdz09

* Speaker: António Machiavelo, Centre of Mathematics, University of Porto

email: ajmachia@fc.up.pt

* Title: **Counting families of combinatorial objects with complex analysis**

* Abstract: The methods of Analytic Combinatorics [1], namely using Complex Analysis to count families of combinatorial objects, have been extensively used and refined in the last years. We will give an overview to those methods based on the works [2] and [3], and will describe some of the challenges of use them in some intricate settings related to Theoretical Computer Science.

[1] P. Flajolet and R. Sedgewick. Analytic Combinatorics, Cambridge University Press, 2008.

[2] Sabine Broda, António Machiavelo, Nelma Moreira, Rogério Reis, A Hitchhiker’s Guide to Descriptional Complexity Through Analytic Combinatorics, Theoretical Computer Science 528 (2014) 85–100.

[3] Sabine Broda, António Machiavelo, Nelma Moreira, Rogério Reis, Analytic Combinatorics and Descriptional Complexity of Regular Languages on Average, ACM SIGACT News, 51(1):38–56, March 2020.

**Seminar #4: MAP-PDMA PhD Program 2022/2023 — November 11 **

https://videoconf-colibri.zoom.us/j/92403741454?pwd=UE43T1c3M3g5Y3VoWlZENkMwaGcrUT09

* Speaker: Thomas Kahl, Centre of Mathematics, University of Minho

email: kahl@math.uminho.pt

* Title: **Algebraic topology and concurrency theory**

* Abstract: It has been discovered relatively recently that concepts and methods from algebraic topology may be employed profitably in concurrency theory, the field of computer science that studies systems of simultaneously executing processes. A very expressive combinatorial-topological model of concurrency is given by higher-dimensional automata. In this talk, I will present a method to extract homological information from HDAs that is meaningful from a computer science point of view.

**Seminar #3: MAP-PDMA PhD Program 2022/2023 — October 28, 14:30**

https://videoconf-colibri.zoom.us/j/92403741454?pwd=UE43T1c3M3g5Y3VoWlZENkMwaGcrUT09

* Speaker: Pedro Patrício, Centre of Mathematics, University of Minho

E-mail: pedro@math.uminho.pt

* Title: **Applicable generalized inverses of matrices**

* Abstract: In 1906, Moore formulated the generalized inverse of a matrix in an algebraic setting, which was published in 1920. Kaplansky and Penrose, in 1955, independently showed that the Moore “reciprocal inverse” could be represented by four equations, now known as Moore- Penrose equations. Generalized inverses, as we know them presently, cover a wide range of mathematical areas, such as matrix theory, operator theory, c*-algebras, semi-groups or rings. They appear in numerous applications that include areas such as linear estimation, differential and difference equations, Markov chains, graphics, cryptography, coding theory, incomplete data recovery and robotics. In this seminar we will focus on the study of the generalized inverse of von Neumann, group, outer and Moore-Penrose in a purely algebraic setting and matrix setting. We will present some recent results dealing with the generalized inverse of certain types of matrices over rings, emphasizing the proof techniques used. We will address some applications.

**Seminar #2: MAP-PDMA PhD Program 2022/2023 — October 21, 14h30**

https://videoconf-colibri.zoom.us/j/94769148463?pwd=YWRVYjEzMkk2bmtMY2Q2OEhFM0hUdz09

* Speaker: Ana Jacinta Soares, Centre of Mathematics, University of Minho , E-mail: ajsoares@math.uminho.pt

* Title: **Kinetic models and applications to biological systems**

* Abstract: In many problems arising in Applied Mathematics, in particular in the interface of mathematics with natural and life sciences, one important aspect is the presence of different scaling regimes of evolution. In particular, when modeling biological systems,one should be able to describe the global behaviour of the cellular populations in terms of macroscopic equations and also the cellular dynamics and the biological expression of cells in terms of microscopic equations. The kinetic theory of mixtures, a branch of statistical mechanics, could provide a rather good approach to the microscopic description and, at the same time, it allows to obtain the corresponding macroscopic analogue as the hydrodynamic limit of the kinetic equations.

In this seminar, I will present some interesting problems and applications of the kinetic theory to biological systems.

**Seminar #1: MAP-PDMA PhD Program 2022/2023 — October 14, 14h30**

* Place: via zoom, https://videoconf-colibri.zoom.us/j/92403741454?pwd=UE43T1c3M3g5Y3VoWlZENkMwaGcrUT09

* Speaker: Eurica Henriques, DMat-UTAD and Centre of Mathematics, University of Minho: Pole CMAT-UTAD, E-mail: eurica@utad.pt

* Title: **A brief overview on differential equations**

* Abstract: Differential equations govern several phenomena and their study gives rise to some answers and several other questions. In this seminar we go on a tour starting at Newton’s cooling law (an ordinary differential equation), stoping briefly at some well known partial differential equations (PDE) and ending on a doubly nonlinear PDE. We will present recent results concerning regularity aspects of the weak solutions to the doubly nonlinear PDE

$$u_t-\textrm{div} \big(|u|^{m-1} |Du|^{p-2} Du\big)=0 , \qquad p>1 .$$