Dylan J. Altschuler
Google Scholar
About
I completed my PhD in mathematics in May 2023 at the Courant Institute of Mathematics (NYU), where I was fortunate to be advised by Jonathan NilesWeed. Next year, I will be joining Carnegie Mellon. Prior to NYU, I graduated from Princeton in 2018 with a B.A. in mathematics, with my senior thesis advised by Allan Sly.
You can reach me at first name [dot] last name [at] courant.nyu.edu. My new NYU office is WWH 1005A, feel free to stop by!
Research
I study probability, especially highdimensional phenomena and random combinatorics. My research is partially supported by an NSF Graduate Research Fellowship (NSF GRFP) and a MacCracken Fellowship.


Critical window of the symmetric perceptron
Dylan J. Altschuler,
Under review (revisions), 2022. (arXiv // video).
We establish the size of the fluctuations in the "combinatorial discrepancy of a Gaussian matrix" (equivalently, the critical window of 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 NilesWeed,
Random Struct. Algorithms, 2021. (arXiv)
We give an exact tradeoff between discrepancy, dimension, and sparsity, for some canonical ensembles of integer matrices. This tradeoff was previously only known in very limited parameter regimes: there is a fundamental obstruction to obtaining concentration results for integer matrices below a certain sparsity. By combining Stein's method of exchangeable pairs with the second moment method, we move past this obstruction and essentially characterize all regimes 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 a.s. convergence of 1d critical longrange percolation to a random scaling limit, with respect to the "GromovHausdorff" metric.

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 the neural wiring process of the drosophila (fruit fly) compound eye 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 coauthors), you may be looking for my brother Jason.
