Abstract: Imagine that you are running on a circular track. For company, you have a few  friends with you. No two of you run at the same speed and all of you  have a constant, non-zero speed. If you all run in the same direction, will there ever be a  moment when you are a set minimum distance apart from the friend nearest to you? Does the  same thing happen with each of your friends?  In this talk, we shall look at different formulations of the conjecture, and the various  approaches that have been taken to tackle the problem. I’ll then explain,  in brief, how the conjecture can be viewed geometrically using the concept of polyhedra.  Finally, we see some of the important known results, especially those proven using the  polyhedral theory approach.  

Date & Time : 4th April 2023, 07:00pm

Abstract: Profinite Groups are a fascinating family of topological groups which can be also  defined using the notion of an inverse limit. We start by discuss this equivalence, and using  this theory we try to understand various different kinds of profinite groups such as Pro-p  Groups, Powerful Pro-p Groups, and Uniform Pro-p Groups. Uniform Pro-p Groups have a  well-defined structure to them being torsion free and homoeomorphic to 𝐙pd, where d is the  number of topological generators of the group, and has far reaching applications in the field  of Number Theory. We then elaborate on the characteristics of various different families of  Uniform Pro-p Groups.  

Date & Time: 10th April 2023, 07:00pm

Abstract: Over the last century, knot theory has developed into an active area of research in  topology. This is partly due to its connections with several other areas of science and  technology. In the domain of mathematics, low-dimensional topologists look to results and  techniques from knot theory to gain a better understanding of 3-manifolds. One of the  fundamental structures studied in knot theory is a compact connected surface whose  boundary is the knot under consideration, called a Seifert surface. In this talk, we will learn  various properties related to two well-known simplicial complexes constructed using the  Seifert surfaces of knots - the incompressible complex and the Kakimizu complex (named  after Osamu Kakimizu). We will look at some classic properties of the Kakimizu complex,  the more popular of the two, like contractibility and local infiniteness. I will then talk about  my experiences of trying to extend some of these properties to the seemingly less cool  incompressible complex, who (SPOILER ALERT) may be a lot more interesting than we  originally thought. I will also recount my experiences of reasoning with (aka, researching)  these complexes.

Date & Time: 12th April 2023, 07:00pm

Abstract: Wireless networks is an essential technology for enabling flexible communication  and connectivity between individuals (or machines) across regions. In addition, it is  transforming every sector of the economy (transportation, healthcare, education, etc.), and  powerful new technologies (artificial intelligence, internet of things, etc.) are being built upon  it. Cellular networks is the technology that 1G to 5G relies on. However, as the number of  devices that depend on wireless communication networks continues to grow, each needing a  high connection rate and better coverage with minimal interference, this technology will no  longer be suitable. For future wireless communications (e.g., 6G), the key technology that has  the potential to enhance connectivity and provide better coverage for billions of users is  referred to as Cell-Free Massive Multiple-Input Multiple-Output (CF-mMIMO).    CF-mMIMO is accompanied by many challenges, one of which is how to efficiently manage  limited resources, giving rise to the resource allocation problem in wireless networks. This  talk will describe a major problem that hinders resource allocation in wireless networks,  namely the Pilot Assignment problem (PA). To design heuristics that can find reasonably  good solutions to PA, the problem has been mapped to several well-known graph theory  problems which fall in the class of NP-hard problems. However, there is no literature that  formally states the PA problem as a mathematical problem, let alone explicitly prove  complexity results for it. The aim of this talk is to look at the pilot assignment problem from  a theoretical computer science perspective and present a few results on the complexity and  approximability of this problem. We discuss our findings in the context of the algorithms  already proposed in the literature to tackle this problem. 

Date & Time: 14th April 2023, 07:00pm

Abstract: Minimal surfaces are zero mean curvature surfaces that appear in nature as  idealized soap films. Given two curves one can ask for a minimal surface connecting them. In  this talk, we would look at the shapes of such curves and how it controls the image of the  minimal surfaces interpolating them. Besides containing lots of creative arguments involving  beautiful pictures, this talk would also explore various other different branches such as  Fourier analysis, complex analysis, split-complex analysis, functional analysis, and so on.  This is a joint work with Prof. Rukmini Dey (ICTS Bangalore).  

Date & Time: 16th April 2023, 07:00pm