# Short Courses and single talks

Most of the significant and exciting areas of Mathematics are not taught in universities in Pakistan and possibly in other developing countries. To promote these Mathematical areas in Pakistan and other developing countries, we offer introductory courses on such areas of Mathematics to allow students to learn the fundamentals. After completing one of these courses, a student can continue to learn more advanced topics in this thread through our short projects.

The classes are taught in English and are completely free. The participants can receive a certificate if they complete 75% of home work. Please write to us at themathetf@gmail.com for more details.

Courses:

Single Talks:

Action of Groups and Homogeneous Spaces [April 20, 2024]

Speaker: Irfan Ullah

Affiliation: ASSMS, GCU, Lahore.

Abstract: Beginning with the basics of Lie groups and their actions on sets, we will provide a gentle introduction to the concept of a group acting smoothly on a manifold. This will include foundational definitions and examples. The seminar will then explore homogeneous spaces - spaces that are the orbits of a continuous group action of a Lie group, showcasing their uniformity and structure. Special attention will be given to the geometric and algebraic implications of Lie group actions, introducing participants to pivotal concepts such as orbits, stabilizers, and the quotient spaces that lead to homogeneous spaces. Designed for beginners, this seminar aims to demystify the advanced topics of Lie group actions and homogeneous spaces, making them accessible to a broader audience. Participants will emerge with a clear

understanding of these key mathematical concepts, ready to apply them in their studies or to further explore the rich interconnections between mathematics and physics.

Geometry of Manifolds and the Gauss-Bonnet theorem [March 30, 2024]

Speaker: Ayodeji Farominiyi

Affiliation: University of Calabria, Italy

Abstract: This presentation delves into the fascinating realm of smooth manifolds, investigating their intricate geometry through a comprehensive exploration of tangent spaces, connections, curvature, and the Gauss-Bonnet theorem, a crucial result in the theory of surfaces. This theorem establishes a connection between a surface’s geometric and topological properties and acts as a model for similar statements that hold in higher-dimensional contexts. The presentation is organized into five parts, each addressing key aspects of this topic and culminating with an illustrative example featuring the sphere.

The first part serves as an introduction to the theory of differential manifolds, beginning with a comprehensive overview of the underlying concepts. The second part provides an examination of structures on a differentiable manifold such as tangent spaces and tangent bundles. Vector fields and vector bundles are then introduced, enabling a deeper understanding of the interplay between smooth functions and smooth vector fields.

Part 3 focuses on the study of connections, and curvature on Riemannian manifolds. This part delves into the properties and construction of connections, revealing their crucial role in measuring differentiation along curves on manifolds. Next, in part 4, we introduce the Gauss-Bonnet theorem, a classical result that highlights the interaction between the topology and geometry of surfaces.

Finally, in the last part, the theoretical framework established in the previous parts is applied to a specific example - the 2-sphere. We apply the Gauss-Bonnet formula to compute the Euler characteristic of the 2-sphere, a topological invariant using the knowledge of the curvature which is a geometric data.

Slides

STABILITY ANALYSIS, BIFURCATION THEORY AND APPLICATIONS [February 03, 2024]

Speaker: Hardik Poptani, University of Liverpool, UK

ABSTRACT: It is well known that under a variation of parameters, the analysis of ordinary differential equations (ODEs) is known as Bifurcation analysis, which applies to many fields such as chemistry, fluid mechanics, etc. Analyzing the stability point, we can check the properties of such ODEs as they predict their behaviour (i.e. whether the system remains in equilibrium after perturbation).

Indeed, we can do such analysis by using linear analysis which will provide enough information for the behaviour of such a system. For this presentation, we will first look at the definition of stability along with definitions of bifurcations for one variable, then move on to two variables. Finally, we will look at a Biological application of such bifurcation analysis.

Tropical Convexity and Phylogenetic Trees [August 26, 2023]

Speaker: Andrei Comăneci, Berlin Mathematical School, Germany

ABSTRACT: The notion 'tropical convexity' was introduced by Develin and Sturmfels in 2004 to give a combinatorial-geometric view on max-linear algebra. We will give an introduction to this topic and show its connection to phylogenetics. In particular, we will show how we can exploit tropical convexity to obtain averages of phylogenetic trees which gives a new approach to the consensus tree problem. The talk is based on the paper Tropical medians by transportation.

For slides of this talk click HERE

Reference book: Joswig, Michael. Essentials of tropical combinatorics. Vol. 219. American Mathematical Society, 2021.

Fixed Point Theory [August 19, 2023]

Speaker: Wahid Ullah, University of Trieste, Italy

Abstract: This lecture on “Fixed point theory” will be for the Bachelor (last year) and Master students. I will start this talk from a well-known Banach contraction principle together with some history behind it. I will try to discuss some steps required for the proof of Banach contraction principle in brief. After that, I will talk on some active research topics in the field of metric fixed point theory which will be of more interest especially for research students. There are two types of generalizations of the Banach contraction principle. Some people generalize metric space, where they introduce new metric type spaces, while others extend the idea of contraction. I will try to discuss some of these generalizations in detail.

Insights on Fundamental Models of Fluids [May 27, 2023]

Speaker: Muhammad Bilal

Affiliation: Dubai Arabian American Private School

Abstract: This session will provide insights into the fundamental models of fluids, including both Newtonian and non-Newtonian fluid models. We will first introduce the concept of fluids and their properties, such as viscosity, density, and pressure. Then the Newtonian fluid model will be presented, which assumes that the viscosity of a fluid is constant and independent of the shear rate or shear stress. The presentation will then delve into the non-Newtonian fluid models, which account for variations in fluid viscosity with changes in shear rate or stress. These models include the power-law model, Herschel-Bulkley model, and Bingham plastic model, among others.

Correspondence between Lie Groups and Lie Algebra [April 29, 2023]

Speaker: Mehmood Ur Rehman

Affiliation: The University of Beira, Portugal

Abstract: A Lie group is a smooth manifold obeying group properties and satisfies an additional condition that the group operations are differentiable. The tangent space at the identity of Lie group has the has the structure of Lie algebra and this Lie algebra determine the structure of the Lie group via the exponential map. In this talk, I will discuss this correspondence.

Weak and Weak* Topologies [April 01, 2023]

Speaker: Assad Ullah

Affiliation: The University of Beira, Portugal

Abstract: In this talk, we will discuss strong, weak and weak star topologies and connection between them. Moreover, we will discuss some results about the convergence in the these topologies.

Modelling Drug Release from Eroding Porous Medium [July 16, 2022]

Speaker: Maniru Ibrahim

Affiliation: University of Limerick, Ireland

Evolutionary Game Theory [July 23, 2022]

Speaker: Malgorzata Fic

Affiliation: Max Plank Institute Germany

Basic Analysis of NFT Markets [July 23, 2022]

Speaker: Laura Tresso

Affiliation: University of Torino, Italy