Short Courses and Single Talks

Speaker: Dr Rizwan Jahangir

Affiliation: Department of Mathematics, Uppsala University, Upssala, Sweden

Abstract:  

It is widely believed that if one compresses enough mass/energy in a small enough volume, gravitational collapse becomes unavoidable and a black hole will form. In the 1970s, Kip Thorne proposed the hoop conjecture in an attempt to make this idea a bit more precise: roughly speaking, if you can wrap a circular hoop of a certain critical size around a mass in every direction, then no matter what the details look like, a black hole will form. In his original formulation, however, the words "hoop", "size", and "mass" are left deliberately vague. Turning this into a sharp mathematical statement has remained a long-standing problem. 

Within the last twenty years, a group of mathematicians resolved this conjecture under the assumption that the spacetime is spherically symmetric. In this talk I’ll give a short, visual introduction to the spacetime picture of gravity and the modern notion of black-hole formation. With that in hand, I’ll explain Kip Thorne’s hoop conjecture: roughly, a horizon should form precisely when a mass can be wrapped by a sufficiently small hoop in every direction. Finally, I’ll sketch how mathematicians have made this slogan precise and proved a sharp version in the spherically symmetric setting. No background in relativity or differential geometry will be assumed - just a bit of calculus, linear algebra, and curiosity about how stars die.

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Speaker: Dr Rizwan Jahangir

Affiliation: National University of Sciences and Technology (NUST), Islamabad, Pakistan

Abstract: 

Hilbert series play a central role in describing the structure of graded algebras, and their computation often reveals deep connections between algebraic and combinatorial objects. This talk provides a brief and accessible introduction to Hilbert series of monomial and binomial ideals, emphasizing geometric intuition and standard computational techniques. The discussion then focuses on polyomino ideals, a class of toric ideals associated with polyominoes in the integer grid, highlighting how algebraic properties reflect combinatorial features such as convexity, holes, and decomposability. Representative examples illustrate the influence of the underlying shape on the form of the Hilbert series, and recent developments and open questions in the study of polyomino ideals are outlined. The presentation is designed to be self-contained and accessible to a broad mathematical audience.

(recording to be uploaded)

Speaker: Dr Shah Faisal

Affiliation: Postdoctoral Researcher, IRMA , Strasbourg, Frace 

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Speaker: Faraz William 

Affiliation: University of Uddine, Italy

Abstract: 

Mathematical models describing a real life process are often expressed as differential equations, where the evolution of a system depends only on its current state and initial condition. In many real-world situations, however, the present state is also influenced by past states. Such models are known as delay equations. 

An important example comes from finance: the dynamics of a stock price cannot be described by deterministic factors alone, but also by unpredictable influences such as company decisions or market shocks. To account for both deterministic behavior and randomness, one employs stochastic delay differential equations. 

These equations are highly relevant in applications, yet they are rarely solvable analytically. This talk will therefore also present numerical methods for their solution, combining theoretical aspects with practical considerations.

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Speaker: Dr Muhammad Izhar 

Affiliation: GPGC Mardan, Pakistan

Abstract: 

This talk presents a structured overview of cryptography from its historical origins to its modern mathematical foundations. We begin with the classical Caesar Cipher, one of the earliest systematic methods for secure communication, and examine how such elementary schemes motivate the need for more sophisticated approaches. The discussion then turns to public-key cryptography, focusing on the Discrete Logarithm Problem as a hardness assumption and its application in the Diffie–Hellman key exchange protocol, a seminal breakthrough that enabled two parties to establish a shared secret over an insecure channel. Building on this, we introduce Elliptic Curve Cryptography (ECC), where the group law on elliptic curves defined over finite fields provides the algebraic structure necessary for cryptographic schemes. Through examples, we illustrate the elliptic curve group operation and demonstrate how the Diffie–Hellman protocol adapts in this setting, yielding equivalent or stronger security with dramatically reduced key sizes.

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Speaker: Dr. Shah Faisal 

Affiliation: IRMA , Strasbourg, Frace

Abstract: 

The aim of this Maths Club Meeting is two-fold. First, we explain to high school students (grades 9–12) and the wider community some intersecting applications of Mathematics in our daily lives.  Second, we highlight fascinating areas of mathematical research that undergraduate students can explore and potentially pursue as future career paths.

Speaker: Shazia Fazal

Mathamatician & Frelancer 

Details: Freelancing offers excellent opportunities for Mathematics students and graduates to earn online and gain experience. This presentation highlights how platforms like Upwork, Preply, and Fiverr can be used for tutoring, academic writing, statistical analysis, and content creation. It provides practical steps to get started, tools to learn (like LaTeX and Desmos), and tips to build a strong profile. Real examples of successful freelancers are shared to inspire attendees. Whether you're a student looking for side income or a graduate aiming to build a career, freelancing can open doors to global opportunities. 

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Speaker: Atif Ali

Affiliation: University of Liverpool, Liverpool, UK

Details: Generative Adversarial Networks (GANs) have transformed how computers generate realistic images, but at their core, they rely on powerful mathematical ideas. In this talk, we will explore the essential math behind GANs—covering probability theory, linear algebra, and optimisation. You will see how two neural networks—the generator and the discriminator—use these mathematical concepts to compete, learn, and improve. This session will unpack the mathematics that drives one of AI’s most creative tools 

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Speaker: Muhammad Awais 

Affiliation: SISSA & University of Trieste, Italy 

Details: The fundamental group is one of the most basic yet powerful tools in algebraic topology, capturing essential information about the shape and structure of a topological space. This presentation offers an accessible introduction to the concept of the fundamental group, its geometric intuition. We will explore key examples , and highlight how the fundamental group connects algebra and topology.  We will also see the applications such as  Brouwer's fixed-point theorem. 

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Speaker: Dr. Imran Anwar

Affiliation: Lahore University of Management Sciences, Lahore, Pakistan

Details: Click here to view recordings

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. 

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.

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• 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.

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• 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.

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.  

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• 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. 

• 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. 

• 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. 

• Speaker: Ahsan Ali

• Affiliation: Montana State University, USA

• For slides click HERE

• For video recording click HERE

• Speaker: Ahsan Ali

• Affiliation: Montana State University, USA

• For slides click HERE

• In response to a question, to look at the proof of the continuity of Thomae's Function, click HERE.

• For video recording click HERE

• Speaker: Maniru Ibrahim

• Affiliation: University of Limerick, Ireland 

• Speaker: Malgorzata Fic

• Affiliation: Max Plank Institute Germany

• Speaker: Laura Tresso

• Affiliation: University of Torino, Italy