Getting recognized and appreciated by the person you’ve always looked up to may sound fulfilling to you, but it was like a 'wish granted' for Atif. Muhammad Atif Zaheer, a first-year MS Mathematics student at LUMS, got his name mentioned in the preface of the book “Analysis I and II” by Terence Tao, a professor of mathematics at the University of California, Los Angeles. Recognized as one of the greatest living mathematicians today and widely referred to as "The Mozart of Mathematics", Prof. Terence is no lesser than a math deity to Atif.

 

 

The story begins with Atif stumbling upon Prof. Terence's online blog and casually recommending that in Analysis 1, a method that proves a series to be convergent while supposing the number to be a rational number can also be generalized for a real number. To clarify, the highlighted part in the picture below originally said “let q>0 be a rational number” instead of “real number”. This expansion of the proof’s scope was possible because rational numbers are essentially a subset of real numbers. As a result, the proof made it to the book and so did Atif’s name.  

 

Atif fondly recalls the first time he saw his name in the book's preface. He was in the library, showing the book to a friend and explaining how significant Prof. Terence Toa’s work is. To emphasize Prof. Terence's stature in the field, Atif shared a fascinating insight:  "There is this trend in mathematics nowadays, that if you're interested in solving a really difficult math problem, you need to get somehow a leading mathematician who works in that area to take an interest in it. Nowadays, it’s said that to solve a big problem, you need to get Terence Tao interested in it". If anyone were to ask Atif about his admiration for Prof. Terence, he would show his phone wallpaper, which displays a picture of the professor.

Atif’s budding interest in Mathematics dates to A-levels when he was asked to drop Composite Math as a subject by one of his teachers because of his matric background. While another teacher instead encouraged him to "go for it" and went out of their way to help him learn difficult concepts, even offering him free home tuition. With his hard work and this teacher’s support, Atif earned an A in Math and then pursued Further Math, scoring a stellar A* in that as well.  

This challenging journey encouraged Atif to opt for pure mathematics as his major in college, as he realized his keen interest in the art of problem-solving. He also remembers how mathematical symbols like the integral symbol always fascinated him, leading him to spend hours staring at old math books, wondering what they meant.

Atif’s journey started with mere curiosity, but he is now diving into the higher realms of Algebraic Geometry, Number Theory, and Analytic Number Theory. According to him, mathematics is like "a game you have to navigate through", and with this mindset, he believes one day he'll be able to solve a problem as complex as the Reimann Hypothesis. This problem is so difficult that even a highly qualified mathematician like David Hilbert once said, "If I were to awaken after having slept for 1000 years, my first question would be, has the Riemann hypothesis been solved?”

The Department of Mathematics at SSE which is also an Emerging Regional Centre of Excellence, supported by the European Mathematical Society, has also been now declared as an ICTP Affiliated Centre by the Abdus Salam International Centre for Theoretical Physics (ICTP). 

The International Mathematics Masters, as it is called, promises to offer rigorous training in fundamental and applied mathematics, preparing its students for a rapid-paced, highly complex world that awaits them. Importantly, this means that our existing master's in mathematics will now be co-taught by international Faculty who will visit LUMS every year. Furthermore, each cohort will comprise of local and international students. The initiative will be co-funded by the ICTP and marks a new beginning in offering mathematical learning of the highest order in the country. 

IMM-LUMS program is a joint effort of the School and the department for a paradigm qualitative shift; elaborating our role in promoting mathematics in the region as the Emerging Regional Centre of Excellence of Mathematics. We agree and committed to the following Common Core State Standards for Mathematics. 

Teaching and Learning

An excellent mathematics program requires effective teaching that engages students in meaningful learning through individual and collaborative experiences that promote their ability to make sense of mathematical ideas and reason mathematically.

Access and Equity

An excellent mathematics program requires that all students have access to a high-quality mathematics curriculum, effective teaching and learning, high expectations, and the support and resources needed to maximize their learning potential.

Curriculum

An excellent mathematics program includes a curriculum that develops important mathematics along coherent learning progressions and develops connections among areas of mathematical study and between mathematics and the real world.

Tools and Technology

An excellent mathematics program integrates the use of mathematical tools and technology as essential resources to help students learn and make sense of mathematical ideas, reason mathematically, and communicate their mathematical thinking.

Assessment

An excellent mathematics program ensures that assessment is an integral part of instruction, provides evidence of proficiency with important mathematics content and practices, includes a variety of strategies and data sources, and informs feedback to students, instructional decisions, and program improvement.

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Dr. Shaheen Nazir, will be the focal person of this international programme which is set to attract its first cohort of international students in Fall 2023. 

 

 

The honours variant of mathematics is arriving in Fall of 2022.

Mathematics is in your phone, computer, smart watch shopping mall and traffic lights, and so much more! It is the framework on which our understanding of the world is built, and the enabler that converts concepts to applications we can use and hold in our hands. It powers our economy, commerce, innovation, research and feeds so many other fundamental building blocks of growth— and essential for asking many fundamental questions about how our world works.

If you have an affinity for numbers or would like to embellish your curiosity for math with more flavour and rigour, the honours version is for you! The SSE version of honours in Math lets you explore the deeper meanings and problems. For example, differences between the two calculus course flavours are shared below:

MATH 101 is intended for students who took Pre-calculus in their first semester or students who have taken calculus before coming to LUMS and are primarily interested in a computational approach to the subject.

MATH 101A is intended for students who have done well in calculus before coming to LUMS and would like to see calculus done with a more conceptual approach and will not need repeated practice of differentiation and integration formulas.

We hope that this decision will empower our students. Some students will be able to tailor their math core trajectory to their interests and to enrich their mathematical ability. While the other students taking a different flavour covering the breadth of the subject, still maintaining desired level of rigour will be tilted towards interdisciplinary connections and applications.

 

 

 

 

 

LUMS is a Shangri La for interdisciplinary work. It offers the right mixture needed to create memorable experiences through the cross-pollination of different schools, courses, and faculty; a win-win situation for passionate learners. Two students of Dr. Zahra Lakdawala, who is a faculty at the Department of Mathematics at SBASSE, from the Quantitative Finance, FINN422, (sometimes referred to as QuantFinn in this story) course have been accepted to MIT, Oxford, and McGill. They are Ahmed Salman and Muhammad Ibrahim who hail from the Suleman Dawood School of Business (SDSB) and took this course from Dr. Zahra during the Spring 2021 semester.

“Both of them are SDSB students. Muhammad Ibrahim is also graduating as a CS minor. MIT is intense with its admission process and its round of interviews. This is where Ibrahim said the course and its content, particularly the modeling and applying it using python packages for solving really helped!”, said Dr. Zahra, recalling the work of one of her students involved in this story.

Ahmed Salman, while commenting on the course said "The course I took with Dr. Zahra was one of the most rigorous experiences of my degree. Besides managing an A+ grade, I saw myself intellectually challenged at every step of the way. With the intensive assignments and especially the final project, I used the reference of over 5 textbooks to get through them. Besides coursework, it also piqued my general interest in the subject, as I read books on Chaos Theory and Jim Simons, both topics that I discussed in her office hours.” Wrapping up his thoughts, he said “I can confidently say that this course gave me a solid grounding in quant-finance concepts and will surely complement my higher studies."

Muhammad Ibrahim, who got admission offers from MIT, Oxford and McGill said, "Applying to MIT and getting an admission wouldn't have been possible without taking QuantFinn from Dr. Zahra, and without her recommendation for admissions. I am extremely grateful for the help! "

Let’s take a closer look at the final year project of Ahmed Salman and Muhammad Ibrahim, which garnered the attention of Dr. Zahra and her colleagues, particularly the detail of work displayed by the two brilliant students.

First, the students had to construct a portfolio that maximizes shareholder returns. “We started with the stocks in the DJIA index as a well-diversified portfolio. We evaluated four approaches of weighing these stocks in our ideal portfolio.” The first one prioritized stock with the highest eigenvalues based on their principal component analysis. Because the largest vectors explain most of the stock variation, the hypothesis here was that this portfolio would match the market index while reducing risk.

Sure enough, the portfolio seemed to beat the market index while also exhibiting lower volatility.

 

They also compared this with weights assigned to stocks based on allowing short sales to minimize variance.

The two portfolios were also judged in performance by looking at their performance over the historical time series period, by dividing it into an in sample and an out sample (which consisted of the last 60 days). Clearly, the portfolio with the maximum eigenvalue (a special set of scalars associated with a linear system of equations) performed better but was also more volatile.

The students then compared these approaches with another portfolio based on maximizing Sharpe ratio weights in the chart below:

 

The second portfolio generated the highest returns at 9.4% while the portfolio with the maximum Sharpe ratio we computed from the efficient frontier, giving a return of 8.6%. However, the team conclude that the best strategy was to follow the weights of the maximum Sharpe ratio portfolio, as they will beat the market over the long run and give a good risk-adjusted return.

Regarding Option Pricing, the team said “In the second question, we approximated the value of a call option on Apple (AAPL) with the numerical solutions to the Black Scholes model. We used three approximation methods: binomial, Monte Carlo and finite differences. Our biggest takeaway here was the importance of being granular enough in using these approximations, such that they divide the target time period into smaller frames or ‘timesteps’ as call them – for them to accurately price options. For the binomial approximation, we found that it accurately matched the Black-Scholes model but only for smaller timesteps.”

 

Next, the Monte Carlo method modelled underlying prices using geometric Brownian motion. Although for smaller simulations, there are significant differences between the different models, they found that for larger simulations, all schemes converge to the Black-Scholes pricing.

Referring to the most challenging bit, “the finite difference scheme was the most challenging for us.” Here, they compared two schemes: explicit and implicit. While the explicit scheme calculates the state of a given system at a different time from the current time, the implicit scheme a solution by solving an equation that includes both the current and later states of the given system. They found that here too the explicit scheme came with a Neumann condition, which required more granular timesteps for it to generate a useful output, as shown below:

 

 

Reflections by Ahmed Salman

Before this course, I had taken courses in finance, CS and some in mathematics, but didn’t feel like these subjects could be integrated. But FINN422 was an incredible experience because it combined the three subjects together into a holistic experience for us. It forced us to reconcile the financial concepts, mathematical equations, and programming tools we already knew to build a coherent picture in our minds. I still remember being very overwhelmed by the initial assignments in this course, fumbling with endless equations and coding challenges, but looking back, I’d do it all again. While it was a very rigorous course, I came out of it with much more than a good grade. It gave me the confidence to adapt to unfamiliar subjects and really get comfortable in my own skin as I navigate them!