## Coloring Graphs and Beyond

Colloquium Talk, Hilles 109, Haverford College

Colloquium Talk, Hilles 109, Haverford College

49th Annual New York State Regional Graduate Mathematics Conference, Carnegie Library, Syracuse University, NY

Slides and conference packet.

Machine Learning Seminar Talk, LOV103, Florida State University

**Abstract**: 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.

Poster Presentation, Math Honors Day, Florida State University, Tallahassee, FL

I presented the algorithm to find simplices within a directed graph via the adjacency approach. The poster may be found here

Graduate Student Seminar Talk, LOV103, Florida State University

Slides of the talk.

**Abstract**: 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.

Research Sharing Luncheon, Honors, Scholars and Fellows Society, Florida State University

Slides here. This was a public talk, open to members of the *Honors, Scholars and Fellows Society* of Florida State University

**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.

Topology Seminar Talk, LOV103, Florida State University

Notes of the talk.

**Abstract**: 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.

Algebra Seminar Talk, Zoom, Florida State University

Slides of the talk.

**Abstract**: 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.

Candidacy Exam Talk, Zoom, Florida State University

Slides of the talk here.

**Abstract**: 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.

Graduate Student Seminar Talk, LOV201, Florida State University

Draft of the talk here.

**Abstract**: 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.

Class Project, Dover, Delaware, Delaware State University

Slides here. Presentation of two papers as a final project for a course on Functional Analysis

3rd National Conference on Mathematical Physics,, International Islamic University, Islamabad

Invited speaker. Website of the conference. Slides of presentation here.

Seminar Talk, Islamabad, Quaid-e-Azam University

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

Seminar Talk, Islamabad, COMSATS Institute of Information Technology, Virtual Campus

Slides here. The same talk was presented in Quaid-e-Azam University on 12/4/2015.

**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

Seminar Talk, Islamabad, COMSATS Institute of Information Technology, Virtual Campus

Embarrassed to admit it, but you can find Powerpoint slides here. This talk focuses on the life and struggles of Alexander Grothendieck.

Seminar Talk, Islamabad, COMSATS Institute of Information Technology, Virtual Campus

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

Seminar Talk, Islamabad, COMSATS Institute of Information Technology, Virtual Campus

Slides of the talk available here. As the title suggests, we talked about some fixed point theorems and their applications.

Seminar Talk, Islamabad, COMSATS Institute of Information Technology, Virtual Campus

Different proofs of the divergence of harmonic series are presented, along with some applications

Seminar Talk, Islamabad, COMSATS Institute of Information Technology, Virtual Campus

See draft of talk here

Conference Talk, International Conference and Workshop on Nanoscience and Technology 2012, Quaid-e-Azam University, Islamabad

Webpage of the conference. I presented my undergrad thesis. Certificate of participation here.

Seminar Talk, Islamabad, COMSATS Institute of Information Technology, Islamabad

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