Dylan J. Altschuler

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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 Niles-Weed. 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!

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.

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.

We give an exact trade-off between discrepancy, dimension, and sparsity, for some canonical ensembles of integer matrices. This trade-off 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.

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

We establish the a.s. convergence of 1-d critical long-range percolation to a random scaling limit, with respect to the "Gromov-Hausdorff" metric.

We investigate pattern formation in the neural wiring process of the drosophila (fruit fly) compound eye during development.

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