Dylan J. Altschuler

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**site under construction**

About

I am a Ph.D. candidate at the Courant Institute of Mathematics (NYU) where I am fortunate to be advised by Jonathan Niles-Weed. For summer 2022, Afonso Bandeira is kindly hosting me as a visiting researcher at ETH Zürich. Prior to NYU, I graduated from Princeton with a B.A. in mathematics.

You can reach me at first name [dot] last name [at] courant.nyu.edu. My NYU office is WWH 1030. For the summer, I can be found in Rämistrasse 101 G21. I am always happy to chat about math; feel free to stop by!

Research

I study probability, especially high-dimensional phenomena and random combinatorics. My research is partially supported by an NSF Graduate Research Fellowship (NSF GRFP) and a MacCracken Fellowship.

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Publications
Fluctuations of the symmetric perceptron
Dylan J. Altschuler,
ArXiv Preprint 2022. (arXiv)

We establish the size of the critical window for the "combinatorial discrepancy of a Gaussian matrix". also called "the storage capacity of the symmetric binary perceptron". Perhaps quite surprisingly, the critical window corresponds to the addition of an (almost) constant number of rows! We also contribute exponential tail bounds.

Discrepancy of random rectangular matrices
Dylan J. Altschuler, Jonathan Niles-Weed,
Random Struct. Algorithms, 2021. (arXiv)

We give an exact trade-off of the discrepancy vs the dimension, for some canonical ensembles of integer matrices. There is a fundamental obstacle to understanding discrepancy of integer matrices below a certain sparsity. By combining Stein's method of exchangeable pairs with the second moment method, we are actually able to treat all sparsities simultaneously.

Localized radial roll patterns in higher space dimensions
Jason J. Bramburger, Dylan J. Altschuler, Chloe I. Avery, Tharathep Sangsawang, Margaret Beck, Paul Carter, Bjorn Sandstede
SIAM J. Applied Dynamical Systems, 2019. ( PDF)

We investigate the "snaking" phenomenon in the bifurcation diagrams of some PDE's.

Critical Long Range Percolation: Scaling Limits for Small β
Dylan J. Altschuler, Allan Sly
Princeton Senior Thesis (2018). (I plan to eventually write this up for publication. In the meanwhile, manuscript is available on request.)

We establish the almost sure convergence of 1-d critical long-range percolation to a random scaling limit.

The developmental rules of neural superposition in Drosophila
Marion Langen, Egemen Agi, Dylan J. Altschuler, Lani F. Wu, Steven J. Altschuler, Peter R. Hiesinger
Cell, 2015. ( PDF )

We investigate pattern formation in neural wiring during development.

The zoo of solitons for curve shortening in R^n
Dylan J. Altschuler, Steven J. Altschuler, Lani F. Wu, Sigurd B. Angenent
Nonlinearity, 2015. ( ArXiv)

We classify all solutions to the curve shortening equation that evolve by homotheti (any combination of translation, dilation, rotation).


Disambiguation

There are several other mathematicians with the same last name; if you are looking for an optimization expert (who shares some co-authors), you may be looking for Jason.