**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 first semester Information_2022_2023_Sem1**

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

**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**

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

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**

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

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 **

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

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**

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

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

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