As part of the Ashoka Math Apprenticeship Programme, students had the opportunity to attend Apprenticeship Lectures offered by faculty members each week. These lectures were designed to go beyond regular coursework and introduce students to advanced or interdisciplinary topics in mathematics.
Students selected one lecture per week to attend. The lectures featured a mix of theoretical foundations, computational techniques, and research-style open-ended questions. Some of the articles in this year’s collection were directly inspired by these lectures.
In classical enumerative combinatorics, mathematicians typically ask the fundamental question, "How many?" and are satisfied with an integer answer. However, what happens when we change our perspective and ask, "How are these objects distributed?". Inspired by the mathematical phenomenon affectionately known by Richard Askey as the "q-disease" , this intensive five-day course introduces beginning undergraduates to the elegant world of q-analysis and structural combinatorics. The course's primary purpose is to teach students how to transition from simple counting to assigning polynomial weights—such as the number of inversions or the area under a lattice path-to combinatorial structures.
The study of symmetry is one of the principal motivations in mathematics. In this course, we examine the symmetries associated with modular forms—functions that sit at the intersection of number theory, complex analysis, geometry, and functional analysis.
We begin by examining how “symmetry” is captured through group actions and familiar “symmetric” functions, such as periodic functions on the real line. Using these as a guiding light, we develop the theory of functions exhibiting “non-abelian” symmetries: the modular forms.
Generative AI is rapidly revolutionizing industries from medicine to finance, driven by exponential gains in compute and data that automate human creative processes. At a foundational level, generation relies on extracting underlying patterns rather than merely copying or stitching raw data together. By analyzing distinct features—such as structural variations, colors, and proportions across sample data—a generative system learns the fundamental traits of a subject, allowing it to synthesize entirely original yet highly coherent outputs.
Understanding the mechanics behind these creations requires exploring the underlying statistical and mathematical framework. While mastering modern neural architectures can take a year or more, this material focuses on core principles: simple statistical approaches like Markovian fitting and random fields, along with brief primers on GANs, auto-encoders, and transformers. Ultimately, the emphasis is on why these models succeed, addressing open questions in the field and highlighting cutting-edge methods such as "drifting."
In contrast to our known Archimedean setting, where triangle inequality governs the notion of distance, we will introduce non-Archimedean spaces. Our journey will begin at the p-adic completion of integers. We will study field extensions of p-adic rational numbers. Moving to the algebraic closure and a subsequent completion, we will arrive at the p-adic complex numbers. Geometry and analysis on such non-Archimedean spaces will be discussed. We will begin this study with understanding the non-Archimedean topology, its mind-boggling anomalous nature and its surprisingly strong consequences.
Our overall aim is to build a theory parallel to complex analysis in the non-Archimedean setting. This course also aims to build a foundational understanding of geometric, analytic, topological and number theoretic properties of p-adic numbers.
The course is a brief introduction to the theory of L-functions, the following topics will be discussed: Riemann zeta function and the Dirichlet L-functions. Basic properties of L-functions: convergence, analytic continuation, and functional equations. Arithmetic applications of L-functions: the special values of L-functions, with attendant constructions like Gauss sums, Bernoulli numbers, and generalized Bernoulli numbers. Various theorems on infinitude of prime numbers in arithmetic progression.
Fourier series, transforms, and Fourier analysis in general is a vast field with countless applications both within mathematics and in physics and engineering. This "mini course" is designed to be beginner-friendly, favoring the concrete over the abstract by grounding theoretical concepts in practical examples and applications.
The journey begins with the study of Fourier Series, providing a foundation for understanding how complex periodic signals can be decomposed into simpler trigonometric components. From there, we transition to the Fourier transform on the real line. We will explore the mechanics of Fourier inversion and the elegance of Poisson Summation, which serves as a vital bridge between discrete and continuous analysis.
Complex Analysis is often described as the "perfection" of calculus. While real-valued functions can be erratic, complex-differentiable (holomorphic) functions possess a structural rigidity that borders on the magical. This course explores that rigidity, transitioning from the basic geometry of the complex plane to the profound connections between analysis and number theory.
We begin by breaking down the complex derivative not as a simple slope, but as a conformal mapping that preserves the local structure of space. This leads us to the heart of the subject: Cauchy’s integral formulas, which reveal that a holomorphic function is entirely determined by its values on a boundary. Finally, we apply these analytic tools to the "discrete" world of integers, exploring the Riemann Zeta function and the distribution of prime numbers. This course serves as the analytical bedrock for students pursuing high-level geometry or theoretical physics.
This course provides a comprehensive introduction to cryptology, tracing the evolution of secure communication from historical ciphers to modern encryption systems. Students begin by examining classical methods—such as the Caesar and Vigenère ciphers—and the fundamentals of cryptanalysis, learning how early cipher weaknesses were identified and exploited. To bridge these historical techniques with contemporary security, the course establishes essential mathematical foundations in number theory, focusing on modular arithmetic, prime numbers, and the Euclidean algorithm.
Building on this mathematical groundwork, the curriculum advances to modern cryptographic primitives and schemes, including stream and block ciphers, RSA, ElGamal, digital signatures, and advanced concepts like homomorphic and functional encryption. Emphasizing both conceptual understanding and practical problem-solving, the course prepares students to analyze how complex cryptographic systems operate, evaluate their vulnerabilities, and apply rigorous mathematical reasoning to ensure secure communication.
This intensive course explores how sequences of real numbers distribute across the unit interval $[0, 1)$ or unit circle $\mathbb{R}/\mathbb{Z}$, examining their behavior through both global and local perspectives. The first half covers the classical, macroscopic theory of Uniform Distribution Modulo 1 grounded in Hermann Weyl's foundational work. Students study key concepts such as Weyl’s Criterion, Kronecker’s theorem, and discrepancy to evaluate how deterministic sequences can achieve smooth, homogeneous coverage across an interval.
The second half transitions to a microscopic scale to analyze Local Distribution at window sizes of $1/N$. Focusing on fine-scale properties like consecutive spacing, pair correlation functions, and Poisson limits, the course highlights how structurally rigid linear sequences ($n\alpha$) evolve into chaotic, pseudo-random behaviors when slightly perturbed ($n^2\alpha$), providing a profound mathematical model for quantum-mechanical systems.
How can calculus detect a hole in space? This is the fundamental question of De Rham Cohomology. This course generalizes the line integrals of complex analysis into the universal language of differential forms. We move beyond the flat plane to study manifolds, where the act of integration becomes a tool for measuring topology.
The course is centered around the De Rham Complex: a sequence of spaces and operators where the failure of a "closed" form to be "exact" signals a fundamental topological feature of the manifold. We will unify the diverse theorems of vector calculus - Green’s, Stokes’, and Divergence - into a single, elegant Generalized Stokes’ Theorem. By the end of the course, students will understand how the analytic properties of differential forms are inextricably linked to the underlying shape of the universe, providing the necessary foundations for Hodge Theory and Algebraic Topology.