Quantitative Reasoning and Mathematical Thinking

The course project is a collaborative piece of work that carries the themes of the course beyond the seminar room. Working in small groups, you will take up one of the topics below, read into it, and present your findings to the class.

Quantitative Reasoning around You:

1. As the student population at Ashoka University grows, it becomes unclear how different aspects of the campus ought to scale. While certain elements clearly do not increase proportionally (for instance, one does not require more Chancellors), others such as majors, faculty, course offerings, classrooms, and administrative capacity raise complex questions. Explore how student population numbers are related to these parameters, and how they should scale.
2. CDO often reports high placement rates as indicators of institutional success. Explore and examine how such statistics are constructed and critically evaluate their validity. In particular, investigate the definitions, assumptions, and data selection processes underlying “placement” figures, including who is counted, who is excluded, and what qualifies as a successful outcome.
3. University rankings claim to provide a comparative measure of institutional quality, but their construction involves subjective choices regarding metrics, weightings, and data aggregation. Explore how these rankings are produced and how sensitive they are to underlying assumptions. Critically evaluate whether such rankings constitute meaningful measures, or whether they obscure more than they reveal through the simplification of complex educational realities into single numerical scores.
4. Faculty feedback systems claim to provide a quantitative report of teaching quality, yet they rely on voluntary responses, subjective perceptions, and aggregated scores. Students do not have incentives to respond truthfully. Explore how these evaluations are constructed and how factors such as response rates, variance in student opinions, and extreme ratings influence the final outcome.
5. Grades are often used to compare student performance across different instructors and academic years, despite variations in grading standards, course difficulty, and cohort ability. Explore how such comparisons can be made more rigorous through statistical normalization and standardization. Critically evaluate whether it is possible to construct a fair and consistent framework for comparing grades across contexts, or whether such comparisons remain fundamentally limited by underlying differences.
6. Policies addressing drug use on campus often rely on deterrence through monitoring and penalties, yet the effectiveness of such measures is difficult to observe directly. Explore how deterrence can be modeled quantitatively in terms of probability of detection, expected cost, and behavioural response. Critically evaluate how one might measure the prevalence of hidden activities and assess whether a given policy meaningfully reduces them, given the challenges of incomplete and biased data.
7. Design an ideal structure for students to specify preferences for FC allocation.
8. Wages for housekeeping staff are often determined within a complex system involving outsourcing, labour markets, institutional budgets, and regulatory constraints. Explore how such salaries are set, taking into account factors such as minimum wage laws, contractor margins, bargaining power, and labour supply. Critically evaluate what constitutes a “fair” wage in this context, and how different benchmarks lead to different quantitative claims about appropriate compensation.
9. Ashoka currently does not implement a formal reservation system, raising questions about how access to limited seats should be structured under conditions of socio-economic, linguistic, and caste inequality. Explore whether a reservation policy can be justified within a quantitative framework by modelling the allocation of seats across groups. Critically evaluate how different definitions of merit, equity, and diversity affect the optimal allocation, and assess whether the introduction of reservations would improve or distort outcomes under these competing criteria.
10. Claims about student mental health are often based on anecdotal evidence or limited data, making it difficult to assess how outcomes compare to other institutions. Explore how mental health can be measured quantitatively and what indicators might serve as proxies (and their limitations). Critically evaluate the challenges of comparing such data across contexts, including issues of reporting bias, selection effects, and differences in institutional environments, and assess whether it is possible to make meaningful comparisons at all.
11. Academic accommodations (extensions, make-up exams, and deadline flexibility) are intended to account for individual circumstances, yet they introduce variation into a system that also seeks uniform evaluation. Explore how such accommodations are granted and how they can be modeled as a problem of fairness under unequal constraints. Evaluate whether these policies create a level playing field or introduce new forms of inequity, and how different quantitative frameworks (such as equal treatment versus equitable adjustment) lead to different conclusions about what is fair.
12. Electricity consumption on campus often includes significant inefficiencies arising from everyday practices. Estimate how much electricity is wasted at Ashoka due to factors such as emergency lighting, floodlighting, air-conditioning with open doors, inefficient elevators, overcooling and other sources of inefficiency. Construct a quantitative model by breaking each source into measurable components, and evaluate the relative contribution of each. Critically assess how sensitive your estimates are to underlying assumptions, and identify which interventions would yield the greatest reduction in waste.
13. University fests require significant expenditure of money, time, and organizational effort, yet their benefits are difficult to quantify. Construct a framework to estimate the total cost of a fest (including hidden costs such as student time and opportunity cost) and compare it to measurable benefits (attendance, engagement, or external reach). Come up with methods to evaluate when an event constitutes a justifiable and efficient use of resources, and how conclusions change under different assumptions about value.
14. Universities often support a large number of student groups, each competing for limited funding and attention, yet there is little clarity on how many such groups are optimal or how resources should be allocated among them. Construct a quantitative framework to model the creation and funding of student groups, taking into account factors such as participation, fixed costs, overlap in activities, and diminishing returns to scale. Analyse how total welfare changes as the number of groups increases, and evaluate how different funding rules (equal allocation, performance-based, or demand-driven) affect outcomes.
15. In many real-world queues, not all customers take the same amount of time to be served. Even if the average service time remains the same, this variation can significantly affect how long people wait. Explore how such variability affects queuing time and the experience of waiting.
16. Minimum Support Prices are designed to stabilize farmer incomes by guaranteeing a price floor, yet their broader market effects remain debated. Explore how MSP influences production decisions, market prices, and government procurement. Critically evaluate whether such interventions improve welfare or lead to inefficiencies such as overproduction, fiscal strain, or market distortions.
17. After the new census, parliamentary seats must be reallocated across states based on population, yet no apportionment method can perfectly balance proportionality, fairness, and political constraints. Explore how seats can be distributed in India and how alternative mathematical methods would change the outcome. Evaluate how different apportionment rules embody trade-offs.
18. Taxation and redistribution aim to balance efficiency with equity, yet determining what constitutes a “fair” distribution of resources remains contested. Explore how tax systems can be modeled as problems of fair division, taking into account differences in income, need, and contribution. Critically evaluate how alternative frameworks lead to different allocation outcomes, and assess whether any system can be justified as both efficient and equitable under competing criteria.
19. The rapid growth of artificial intelligence systems has raised concerns about their environmental impact, particularly in terms of energy consumption and carbon emissions. Explore how the environmental cost of AI can be estimated quantitatively, considering factors such as computational requirements, data center energy use, and energy sources. Critically evaluate the assumptions involved in such estimates, including system boundaries, usage patterns, and indirect effects, and assess whether current metrics meaningfully capture the true environmental footprint of AI technologies.
20. Air quality indices are widely used to communicate pollution levels, yet they aggregate multiple pollutants into a single number based on specific thresholds and weightings. Explore how air quality is measured and how indices are constructed from underlying data. Construct a model to estimate an individual’s true exposure to air pollution based on time spent in different environments (indoors, outdoors, commuting). Compare this with standard AQI-based assessments and evaluate how much information is lost through averaging.
21. Estimate the population of Asawarpur using satellite images by approximating housing density, building size, and occupancy rates. Construct a model that translates visible structures into population estimates, and evaluate how sensitive your estimate is to assumptions about household size and building usage.
22. Temperature alone does not determine human comfort (humidity, airflow, etc. play a role). Construct a quantitative model of “thermal comfort” using a combination of measurable variables. Analyse how different factors interact, and evaluate whether comfort can be meaningfully captured by a single index or whether it is inherently context-dependent.
23. Lighting requirements vary across tasks and environments, yet standards often prescribe fixed illumination levels. Construct a model relating light intensity to task performance or visibility. Analyse whether increasing light leads to proportional improvements, and evaluate at what point additional light yields diminishing or negligible returns.
24. Aryabhata gives an algorithm for computing square roots by a method resembling long division. Reconstruct this algorithm, generalise it to the extraction of nth roots, and write proofs of the correctness of both.
25. Write a report on methods for fast multiplication of large integers, covering the Karatsuba algorithm (1960), the Toom–Cook algorithm (1963), the Schönhage–Strassen algorithm (1971), and the Harvey–van der Hoeven algorithm (2019). Explain how each improves on its predecessors and compare their asymptotic complexities.