# Vishesh Jain

 Massachusetts Institute of Technology Department of Mathematics Office: 2-232A Email:   visheshj at mit dot edu CV

I am a fifth-year Ph.D. candidate in mathematics at MIT, broadly interested in probability, combinatorics, and theoretical computer science. I am fortunate to be advised by Elchanan Mossel. Previously, I received a B.S. in mathematics from Stanford University in June, 2015. I spent the 2013-14 academic year at Math in Moscow.

## Publications and Preprints

### MCMC

1. Perfectly sampling $$k \geq (8/3 + o(1))\Delta$$-colorings in graphs, joint with Ashwin Sah and Mehtaab Sawhney. Submitted. [arXiv:2007.06360]
2. Fast and memory-optimal dimension reduction using Kac's walk, joint with Natesh Pillai, Ashwin Sah, Mehtaab Sawhney, and Aaron Smith. Submitted. [arXiv:2003.10069]

### Invertibility of random matrices

1. Quantitative invertibility of random matrices: a combinatorial perspective. Submitted. [arXiv:1908.11255]
2. Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices. To appear in Israel Journal of Mathematics. [arXiv:1904.10592]
3. On the counting problem in inverse Littlewood-Offord theory, joint with Asaf Ferber, Kyle Luh, and Wojciech Samotij. Submitted. [arXiv:1904.10425]
4. Singularity of random symmetric matrices -- a combinatorial approach to improved bounds, joint with Asaf Ferber. Forum of Mathematics, Sigma, vol. 7, e22, 29 pages (2019). [Journal][arXiv:1809.04718]

### Mean-field approximation

1. Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective, joint with Frederic Koehler and Andrej Risteski. 51st ACM Symposium on the Theory of Computing (STOC 2019). [Conference][arXiv:1808.07226]
2. The Vertex Sample Complexity of Free Energy is Polynomial, joint with Frederic Koehler and Elchanan Mossel. 31st Annual Conference on Learning Theory (COLT 2018). [Conference][arXiv:1802.06129]
3. The Mean-Field Approximation: Information Inequalities, Algorithms, and Complexity, joint with Frederic Koehler and Elchanan Mossel. 31st Annual Conference on Learning Theory (COLT 2018). [Conference][arXiv:1802.06126]

### Graph decompositions

1. Towards the linear arboricity conjecture, joint with Asaf Ferber and Jacob Fox. Journal of Combinatorial Theory, Series B, Volume 142, May 2020, Pages 56--79. [Journal][arXiv:1809.04716]
2. Number of 1-factorizations of regular high-degree graphs, joint with Asaf Ferber and Benny Sudakov. Combinatorica, 2020. [Journal][arXiv:1803.10360]
3. On the k-planar local crossing number, joint with John Asplund, Thao Do, and Arran Hamm. Discrete Mathematics, vol. 342, issue 4, pp. 927--933 (2019). [Journal][arXiv:1804.02117]
4. 1-factorizations of pseudorandom graphs, joint with Asaf Ferber. 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2018). Random Structures and Algorithms. [Conference][Journal][arXiv:1803.10361]

### Universality of ESDs

1. Universality and least singular values of random product matrices: a simplified approach, joint with Rohit Chaudhuri and Natesh Pillai. Submitted. [arXiv:2007.03595]
2. A note on the universality of ESDs of inhomogeneous random matrices, joint with Sandeep Silwal. Submitted. [arXiv:2006.05418]
3. The strong circular law: a combinatorial view. Submitted. [arXiv:1904.11108]

### Miscellaneous

1. On the real Davies' conjecture, joint with Ashwin Sah and Mehtaab Sawhney. Submitted. [arXiv:2005.08908]
2. Accuracy-Memory Tradeoffs and Phase Transitions in Belief Propagation, joint with Frederic Koehler, Jingbo Liu, and Elchanan Mossel. 32nd Annual Conference on Learning Theory (COLT 2019). [Conference][arXiv:1905.10031]
3. On the number of Hadamard matrices via anti-concentration, joint with Asaf Ferber and Yufei Zhao. Submitted. [arXiv:1808.07222]
4. On discontinuity of planar optimal transport maps, joint with Otis Chodosh, Michael Lindsey, Lyuboslav Panchev, and Yanir Rubinstein. Journal of Topology and Analysis, vol. 7, no. 2, pp. 239--260 (2015). [Journal][arXiv:1312.2929]

### Preprints not intended for publication

1. The probability of selecting k edge-disjoint Hamilton cycles in the complete graph, joint with Asaf Ferber and Kaarel Haenni. [arXiv:2001.01149]
2. Uniformity-independent minimum degree conditions for perfect matchings in hypergraphs, joint with Asaf Ferber. [arXiv:1903.12207]
3. A Counterexample to the "Majority is Least Stable" Conjecture. [arXiv:1703.07657]

## Expository

1. An expanded version of a MathOverflow comment by Nazarov, giving a simple proof of an inequality due to Feige.
2. Unedited and informal notes from a three-hour reading group talk on the Balogh-Morris-Samotij proof of the hypergraph containers theorem .
3. Unedited and informal notes from a two-hour reading group talk on Peter Keevash's Counting Designs.