Nilava Metya

About me

I am a second year PhD student in Mathematics at Rutgers University - New Brunswick. My advisors are Arunesh Sinha and Paul Feehan. Email me at nilava.metya@rutgers.edu if you have any question (about anything: life in general, the math community at Rutgers, ...) Something I want to clarify for readers of this webpage: This is my professional webpage written from my personal point of view and I have different optinions on learning vs research. For me, learning includes research; I refer to learning from a book/paper/others as second-hand learning and learning from my own insights/research/results as first-hand learning. So if you see the word 'learn', it does include research sometimes. I have a strong urge to do second-hand learning from experts only, because of my strong belief that second-hand learning, by definition, cannot be done third-hand. My learning interests lie in applied areas of Distributionally Robust Optimization, Homotopy Continuation methods for optimization, Polynomial Optimization, Applied Algebraic Geometry and theoretical Machine Learning. I love solving problems (and mathematical puzzles), especially those similar to olympiad style and competitive coding problems. Here is a CV.

News: [09/24] I am now a doctoral candidate after having passed my oral qualifying exam on 9/11/24. Here's the syllabus. And here's the exam conversation. [02/24] Paper accepted to the Journal of Algebraic Statistics.

I'm actively seeking internships for summer 2025. It could be either in quant research or game theory and AI research or a combination. The former is due to my avid interest and longterm skill in problem solving (like, elementary number theory, combinatorics, probability, inequalities, Euclidean geometry). The latter is for research during my PhD. Here's some of my skillset and experience:

• Mathematical competition-style problem solving.
  This is due to the heavy number of hours I spent during my high school preparing for math and informatics olympiad (the latter covered combinatorics). I love cute and short problems like the ones that appear in ISI BStat objective section of the entrance exam. In 2019, I solved 28/30 problems correctly in the exam hall and have been solving those questions for the subsequent years.
  
  
• Coding in C++.
  As a high-schooler, I was preparing for the national informatics olympiad without any resource. So I practised coding in CodeChef. Here is my CodeChef profile. I've started practising on Codeforces again recently.
  
  
• Coding in JAVA.
  I had JAVA as a part of my high-school curricullum and I've extensively learnt OOP on this language. Here is an example of my code written for a school project.
  
  
• Coding in Python.
  I generally use this language now. My first extensive use was during an undergrad class where we solved solved problems implementing algorithms based on DP, graphs, greedy. Here are some Jupyter notebook examples of using Python for convex optimization (I used the CVXPY package): Image compression using SVD, Fuel Optimization with LP, Lasso, Converting to LP and solving, Stability of a pair of matrices, SDP bounds on stability number.
  
  
• Coding in MATLAB.
  I've used CVX and YALMIP on MATLAB for optimization. Here's some code for fitting increasing/convex curves/surfaces to data using sum-of-squares: best curve fit with plot, best increasing curve fit with plot, best surface fit with plot, best convex surface fit with plot.
  
  
• Coding in R.
  Here is my HW/exam repository from a data mining class where I coded in R. All assignments are mostly coding assignments to analyse some data.
  
  
• Protein data analysis.
  My team did a data-based analysis and prediction of symmetries based on certain features of proteins. This is the repository with code and results. This webpage contains my certificate of completion and the summary presentation.
  

Goals to reach within 2023:

• Read some of algebraic statistics. ✅
• Gain a general understanding of actual statistics. ✅^??

Goals to reach within 2024:

• Gain an understanding of optimization and apply algebra (not just linear algebra) to optimization. ✅
• Gain an understanding of Machine Learning (including implementation). ✅^??
• Learn about Distributioanlly Robust Optimization in depth. Apply to Game Theory.
• Get started with Deep Learning theory. Learn the fundamental -- building a neural network from scratch.

Goals to reach within 2025:

• Think deeply about SOS, polynomial optimization, algebraic techniques in optimization.
• Learn about the Polynomial Method and Incidence Theorems.
• Write an exposition on Distributionally Robust Optimization with clear definitions and conventions.

Goals to reach within 2027:

• Graduate with a PhD.
• Join a supportive research position (including academia and industry) to learn more about aspects of ML, and optimization, and use them in real-world problems.

Education

Activities

(Aug 2024) Princeton Machine Learning Theory Summer School at Princeton University.

(July 2024) Equilibrium Summer School at Rutgers University - New Brunswick.

(May 2024) DIMACS Workshop on Efficient Algorithms for High Dimensional Metrics: New Tools at DIMACS (Rutgers, New Brunswick).

(Dec 2023) Bayesian Statistics and Statistical Learning: New Directions in Algebraic Statistics at IMSI, Chicago.

(Oct 2023) Algebraic Statistics for Ecological and Biological Systems at IMSI, Chicago.

(Oct 2023) Apprenticeship Week at IMSI, Chicago.

(Sep 2023) Algebraic Statistics and Applications at IMSI, Chicago.

(Aug 2023) Permutations and Causal Inference at IMSI, Chicgo.

(Jun 2023) Algebraic Methods in Biochemical Reaction Networks at MPI, Leipzig (organized by MPI Leipzig and SLMath).

(May 2023) Attending as the head counsellor at PROMYS India.

(Jan-Apr 2023) Organizing ANGeLS - the Algebra and Geometry Learning Seminar for graduate students. Please drop by at HILL 525 at 9AM every Wednesday to enjoy bagels over some wonderful talks on Quiver Representations.

Talks/Write-ups

1. Inference on growth process of a network at Rutgers for the course on Data Mining
   Reference Paper.
   Slides. Report. Video.


2. Principal Components along Quiver representations at Rutgers for the course on Topological Data Analysis
   Reference Paper: Principal Components along Quiver Representations.
   Slides.


3. Representations as sections of line bundles at Princeton for the course on Algebraic Geometry
   Notes.


4. Complexity of Optimization at Rutgers Math department Pizza Seminar
   Slides.


5. Quvier representations - geometry and invariants at Rutgers Algebra aNd GEometry Learning Seminar
   References: a geometric view.
   Notes.


6. Quvier representations at Rutgers Graduate Algebra and Representation Theory Seminar
   References: Quiver representations via reflection functors, a geometric view, beyond Dynkin quivers.
   Notes.


7. Burnside's p^aq^b theorem at Rutgers Graduate Number Theory Seminar
   Reference: Abstract Algebra by Dummit and Foote.


8. Kneser Graph Coloring at Rutgers Graduate Combinatorics Seminar
   Reference: A new short proof of Kneser's conjecture.


9. Very Basic Lie Theory at Rutgers Geometry and Topology Seminar
   Reference: Very Basic Lie Theory by Howe.


10. Well-definedness of the Brauer group at Rutgers Algebra aNd GEometry Learning Seminar
    Reference: Associative Algebras by Pierce.


11. Fiedler Vector Methodat CMI for the course on Matrix Computations
    These are the slides, written report, and video based of a group project on the Fiedler vector method - an approximate way to find a balanced graph cuts.
    Slides. Report. Video.


12. Cantor Set
    Here are the notes for a talk on Cantor set I gave in a tutorial in Graduate Analysis I course.
    Notes.


13. Markov Chain Monte Carlo
    This is the presentation based on an internship with Prof R V Ramamoorthi.
    Presentation.


14. Quantum Computing
    These are the write-ups for a series of four talks I gave at a PROMYS counsellor seminar on Quantum Computing.
    Talk 1. Talk 2. Talk 3. Talk 4.


15. Lie Algebras and their Representations
    These are the slides and the writeup for a series of talks I gave at a PROMYS counsellor seminar on Representation theory of Lie algebras.
    Writeup. Talk 1. Talk 2. Talk 3.


16. Introduction to Hyperbolic Geometry
    These are the write-ups for a series of four talks I gave at a PROMYS counsellor seminar on the calculus on the upper half plane in Hyperbolic geometry.
    Writeup.


17. Computer Project in grade 12
    These are the project writeups for my project for grade 12 in high-school. One is a compilation of codes we did throughout the year, another is a larger project to imitate a retail shop and implement an inventory of items.
    Compilation. Inventory Code.