Writings

Graduate Writings

1. Buser's Inequality for Abelian Cayley Graphs (Description) (Slides) (Presentation)
   
   This is a self-contained expository note on Buser's inequality for Abelian Cayley Graphs, following the blog posts written by Luca Trevisan describing the proof of Buser's inequality due to himself and Shayan Oveis Gharan.
   This write-up is joint work with Rohit Agarwal and Anand Brahmbhatt.

Undergraduate Writings

1. Log-Concave Polynomials (Description)
   
   Prepared an expository report on the technique of log-concave polynomials, especially as pioneered by Shayan Oveis Gharan, Nima Anari, Kuikui Liu, and Cynthia Vinzant. Covered a proof of Mason's conjecture, and deterministic and randomized matroid base counting algorithms, including a proof of the Mihail-Vazirani conjecture.
2. Sum-of-Squares Hierarchy (Description)
   
   Prepared a report of the Sum-of-Squares Hierarchy from Pravesh Kothari's lecture series on the same, and covered Goemans-Williamson's Max-Cut algorithm, Nesterov's π/2-theorem, Arora-Rao-Vazirani's conductance algorithm, Global Correlation Rounding, Grigoriev's lower bounds on the k-XOR problem through SoS, and SoS vs. spectral refutation algorithms.
   Some of the content, and style, was inspired by Amit Rajaraman's notes on the same.
3. Percolation Theory (Description)
   
   Exposited the calculation of the critical probability for Bernoulli percolation on Z^2, one of the most fundamental results of percolation theory.
4. Coding Theory (Description)
   
   Prepared a report on coding theory from Guruswami, Rudra, and Sudan's book on the same, and covered Derivative, Folded Reed-Solomon codes, Algebraic-Geometric Codes, and BCH codes, and also covered List Decoding of Reed-Solomon codes, Elias-Bassalygo and Johnson bounds.
5. Boolean Function Analysis (Description)
   
   Prepared a report on Boolean Function Analysis from Ryan O'Donnell's book and lecture series on the same, covering the BLR Test, Arrow's theorem, KKL theorem, hypercontractivity on the boolean hypercube, and the Margulis-Russo formula.
6. Geodesic Convexity (Description) (Slides)
   
   Studied geodesic convexity from Nisheeth Vishnoi's exposition and Nicolas Boumal's book on the same. Stated the Geodesic equation, calculated geodesics for the positive orthant, and the space of symmetric positive definite matrices equipped with the affine-invariant metric. Stated and proved theorems about geodesic convexity and Riemannian gradient descent. Gave examples of geodesically convex functions on aforementioned manifolds, and explored applications in calculating the Brascamp-Lieb constant, and operator capacity for square operators.
7. Quantum Computing (Description)
   
   Studied the basics of quantum computation from Scott Aaronson and Ryan O'Donnell and John Wright's notes on the same.
8. Dirichlet Energy (Description) (Slides) (Advanced)
   
   Studied the use of Dirichlet Forms on reversible ergodic Markov chains to derive bounds regarding their relaxation time from this monograph by Aldous and Fill.
   More than a year later, I revisited this topic again, and understood it better. I rewrote some of the earlier material, and formally introduced the Poincaré and Modified Log-Sobolev constants.
9. Stochastic Differential Equations (Description)
   
   Covered the basics of Itô calculus and stochastic differential equations (from Bernt Oksendal's book on the same), and prepared a "cheatsheet" of sorts, summarizing the main notions and formulae.
10. Representation Theory (Description)
    
    Studied Representation Theory from Steinberg's book. Covered character theory for finite groups (and computed character tables for the quaternion and dihedral groups), and Fourier analysis over finite groups. Calculated the eigenvalues of Cayley graphs as an application.
11. Infinite Galois Theory (Description)
    
    Covered Krull topology, the Galois correspondence for infinite Galois groups, included a brief introduction to profinite groups and showed that all Galois groups are profinite.
12. Galois Theory Solved Exercises (Description)
    
    Solved an exhaustive variety of field theory and Galois theory exercises from Lang's Graduate Algebra.
13. Group Theory Solved Exercises (Description)
    
    Solved an exhaustive variety of group theory and ring theory exercises from I.N. Herstein and Dummit and Foote's classic books on the same.
14. The Graham-Pollak Theorem (Description)
    
    Made a brief survey of the Graham-Pollak theorem and various associated results, such as Alon's bounds on t-biclique coverings of complete graphs, and the Szemerédi-Katona theorem.
15. The Goldreich-Levin Theorem (Description) (Slides)
    
    Studied the proof of the Goldreich-Levin theorem, and presented it.
16. Exact Exponential Algorithms (Description) (Slides)
    
    Studied Zamir's work on breaking the 2^n-barrier for 5-coloring, in the process picking up tools such as Yates' fast zeta transform, and inclusion-exclusion methods for improving exponential algorithms, and presented it.
17. Ehrenfeucht-Fraïssé games (Description)
    
    Explored the Methodology theorem and Hanf's theorem (and their applications in proving the non-FO expressibility of various predicates) from the book Descriptive Complexity by Neil Immerman.
18. A DFA Trick (Description)
    
    I came up with a useful trick to prove the regularity of languages, which simplifies many typical exercises in an automata theory course.

High School and Early Undergraduate Writings

1. Dense Divisors of the Null Matrix (Description)
   
   Proved quick bound regarding the density (defined as the number of non-zero entries of a matrix divided by the size of total number of entries in the matrix) of matrices which multiply with each other to give the null matrix.
2. A Number Theoretic Identity (Description)
   
   Proved an amusing number theoretic identity by counting the number of edges of a certain poset.
3. Fullerenes (Description)
   
   Wrote an article explaining why all fullerenes have exactly 12 pentagonal faces.
4. Silicates (Description)
   
   Calculated the exact formulae of various silicate polymers.
5. Copper Complexes (Description)
   
   Verified the predictions of Crystal Field Theory regarding the colors of copper complexes.