Posts by Collection


Introductory Topology


Topology I notes here, consisting of point set topology and metric spaces. Follows Munkres, mostly. Topology II notes consisting of basics of Algebraic Topology. Mostly from Hatcher. There are lots of topics missing!

Fuzzy Algebra


Some topics in Fuzzy Algebra. Notes are self-contained, starting with lattice theory, logic and fuzzy set theory. Notes here.

Measure Theory


Built from classics of Royden, Folland and some Rudin. See notes here



Engineering Deutsch-Jozsa Algorithm in Cavity QED via Bragg Regime

Published in COMSATS Institute of Information Technology, 2012

The theory of quantum mechanics, developed as a limiting case to classical mechanics, notwithstanding its interpretive difficulties, has with it the elegance for paving way to a variety of applications. One such application is the implementation of a working Quantum Computer. The push one receives for using quantum principles as a measure of information and execution of algorithms is from quantum parallelism. It seems as though nature hides its enormous calculations. One such realisation of the power of Quantum Parallelism can be seen with quantum optics when one considers engineering a Quantum Computer, choosing techniques of cavity QED amongst many other competitors. The Deutsch-Josza algorithm, although of little practical signi cance, is an encouraging example which greatly reduces the time required for a specific function to be determined completely, when compared with its classical counterpart. The Hadamard gate has been physically realised, and so has the other unitary transformations in the Deutsch-Jozsa algorithm using different times of interactions in the cavity. Also, a generalisation of the Deutsch-Jozsa algorithm has been discussed, which might pave way for a working model of a Quantum Computer.

Recommended citation: Malik, A.N. Engineering Deutsch-Jozsa Algorithm in Cavity QED via Bragg Regime. Diss. COMSATS Institute of Information Technology, Islamabad, 2012. the Deutsch-Jozsa Algorithm.pdf

Operator Algebras and the Foundations of Quantum Mechanics

Published in Quaid-i-Azam University, 2016

The purpose of this thesis is to analyse the Hilbert Space requirement for Quantum Mechanics. In particular, we justify sharp observables but question the requirement of completeness of the inner product space and the underlying field. We view our mathematical framework as a dynamical theory but with a mysterious probabilistic interpretation instead of the otherway round. Whenever we speak of Quantum Mechanics, we mean Non-relativistic Quantum Mechanics. To make things less messy, we assume associativity through-out. No attempt has been made to refer to QFT and statistical quantum mechanics and we use conventional mathematical symbols instead of Dirac’s formalism.

Recommended citation: Malik, A.N. Operator Algebras and Foundations of Quantum Mechanics. Diss. Quaid-e-Azam University, Islamabad, 2016 Algebras and the Foundations of Quantum Mechanics.pdf

Orthomodularity and the incompatibility of Relativity and Quantum mechanics

Published in Springer, 2016

We show that orthomodularity in general and non-existence of isotropic vectors in particular decisively yield the geometry of quantum mechanics and that a fundamental reason why quantum mechanics and relativity cannot be unified is because of the non-existence of isotropic vectors

Recommended citation: Malik, A.N., Kamran, T. Orthomodularity and the incompatibility of relativity and quantum mechanics. Quantum Stud.: Math. Found. 4, 171–179 (2017).

Effects of Gender on the Performance of Microenterprises in Pakistan

Published in The International Journal of Humanities & Social Studies, 2017

In Pakistan, there has been a surge in women entrepreneurship, either as a sole proprietorship or joint partnership mostly with male family members. Using non-randomized data, in this paper we compare the impact of gender of owner on the performance of the enterprise in Pakistan. Our results show an intricate association between the gender of the owner and the performance of the enterprise. While there is no significant difference in the performance with respect to profitability of the business, however, female owned enterprises have shown more employment growth as compared to male owned enterprises. Results of OLS regression on the basis of gender show that common factors that affect the performance of the enterprises have assorted effects for male-owned and female-owned enterprises. For female entrepreneurs, education in the most significant factor in their business success.

Recommended citation: Xing, Y. H., Farooq, M. U., & Malik, A. N. Effects of Gender on the Performance of Microenterprises in Pakistan. The International Journal of Humanities & Social Studies, 5(11). 2017

Technology Embedded Hybrid Learning

Published in, 2018

This preprint is a culmination of the proposal that marked the introduction of hybrid courses to COMSATS Institute of Information Technology, and the evolution of its model as an amalgam of the traditional class room model augmented with the aid of state-of-the-art online learning technologies. Two hybrid courses were offered to full-time students, with all the courses in traditional classroom mode, except one course offered as hybrid course, with both synchornous and asynchronous learning modalities. A survey and its results of the pilot program are presented.

Recommended citation: Behzad, M.; Adnan, N.; Malik, A.N..; Merchant, S.A. Technology-Embedded Hybrid Learning. Preprints 2018, 2018030229.


Limits and Randomness of Reason


The works of Kurt Godel, Alan Turing and Gregory Chaitin are summarised. I gave this talk as an undergrad in my alma mater. See draft of the talk

1, 2, …


Slides here. This presentation was a public talk about the countable and uncountable.

Mathematics and the formalism of Quantum Mechanics


Slides here. The same talk was presented elsewhere.

Abstract: We work our way by physical intuition to the popular axioms of Quantum Mechanics as defined by von Neumann. Directions come by appeal to the Double Slit Experiment and by using notions of functional analysis, as defined in E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1978

Mathematics and the formalism of Quantum Mechanics


Slides here

We work our way by physical intuition to the popular axioms of Quantum Mechanics as defined by von Neumann. Directions come by appeal to the Double Slit Experiment and by using notions of functional analysis, as defined in E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1978

Hilbert’s Sixth Problem and Topos Theory


Draft of the talk here

On August 8, 1900, David Hilbert presented the then community of mathematicians a list of problems in the Second International Congress in Paris. These problems shaped the mathematics of the proceeding years. One particular problem, called Hilbert's Sixth Problem, was a programmatic call for the axiomatization of physics. To date, no satisfactory axioms have been proposed. In this talk, we survey the mathematical reasons why this is the case and present a possible approach using Topos Theory.

The variational bicomplex and Takens’ theorem


Slides of the talk

All of mathematical physics either concerns infinitesimal descriptions of reality or global descriptions of reality. A simple example of the former is a differential equation, whereas fields like gravitational fields and electromagnetic fields exemplify the latter. Differential Geometry and Differential Topology provide a language that unfiies this description in the language of bundles. For instance, fields are defined as sections of appropriate bundles. In fact, a field theory, which is usually formalised as a variational calculus problem and its leading differential equation, can be prescribed in terms of special bundles called jet bundles. This formalism for field theory spans both classical and quantum field theory, after suitable modifications.

Axiomatization of Differential Cohomology


Slides of the talk

Differential Cohomology is a cohomological theory which extends topological cohomology theories. These latter generalized cohomology theories are those which satisfies Eilenberg-Steenrod axioms, an example of which is singular cohomology. Differential Cohomology extends topological cohomology by taking into account geometric data of the underlying topological space. In this talk, we review the basic construction of ordinary differential cohomology in the spirit of Simons and Cheeger and then talk about the axioms of Differential Cohomology. This document contains all of the details of the talk.

A Combinatorial Approach to Simplicial Sets


Notes of the talk

Simplicial Sets model weak homotopy types of topological spaces and ∞-groupoids, thanks to their particularly simple but powerful functorial definition. By their very nature, these models are very combinatorial. However, much of the combinatorics are hidden, and therefore, mostly forgotten in the above models. In this talk, we will delve into the details of the combinatorics of a simplicial set itself, working our way through the combinatorics of a simplicial complex and delta set, with their gluing data, thus motivating the need for a simplicial set in the first place. We will introduce terminology that lets us handle all this (infinite) data simultaneously, effectively bringing the combinatorics to the fore. Finally, time permitting, we will talk about the combinatorics of Kan complexes and simplicial homotopy.

Infinite Simplices: An Excursion into Simplicial Sets


Slides here

Public talk.

Abstract: The first use of a graph of a network can be traced back to Euler on his solution to the Bridges of Konigsberg problem. Since then, graphs have been mainly a mathematical curiosity. The acceptance of graphs by the wider scientific community has exploded in the past twenty years or so, especially after the very humbling and exciting endeavor of mapping of the network of the code of our origins a.k.a., the Human Genome Project. However, many of the common utilizations of graphs rely on a finite amount of data, however large. In this talk, we'll introduce the notion of a simplicial set, which generalizes the notion of a graph to account for an infinite amount of data. Time permitting, we'll look into an example of these infinite graphs applied to the modelling of misinformation.

An Introduction to Bell’s Theorem


Slides of the talk

Recent Nobel in Physics went to the pioneering experimentalists who helped lay foundations for Quantum Information Science. However, what the experiments also do is displace unsettling questions in philosophy out from the domain of social sciences, by bridging the physics of John Stuart Bell towards empirically verifiable claims. This is accomplished by exhibiting systems that violate an inequality, known as Bell's Inequality. In this talk, we will delve into the details of Bell's Inequality and discuss its assumptions, advancements that led to the development of Bell's Theorem, and its implications for the foundations of Quantum Mechanics.

Weisfeiler and Lehman use Simplicial Sets: Psuedotop Vertex Neural Networks


More information here

Graph neural networks are paradigms of computation that yield powerful results for structured data based on binary relationships. However, they are limited in their expressivity by the Weisfeiler-Lehman test of for graph isomorphism. The core idea behind machine learning community's circumvention of this limitation relies on identifying (and working with) higher relationships within the data. In this talk, we put forward an architecture closely based on the identification of such higher relationships via Kan Extensions of structured data built on binary relations. We will talk about its theoretical underpinnings based on the combinatorics of simplicial sets, and based off of it, introduce the notion of a pseudotop vertex as a proxy for these higher relations. We talk about how this choice respects the variance and bias trade off necessary for generalizability of the architecture.