Dylan J. Altschuler

About

I am currently a math postdoc at Carnegie Mellon, supervised by Konstantin Tikhomirov. 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. 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 daltschu [at] andrew.cmu.edu. My office is Wean Hall 8212.

Research

I study probability, especially high-dimensional phenomena and random combinatorics.

Publications

Zero-One Laws for Random Feasibility Problems
DJA,
arXiv

We introduce and study a random model of combinatorial optimization with geometric structure. A variety of problems are encoded in this model, including random versions of: linear programming, integer programming, closest vector problem, shortest vector problem, combinatorial discrepancy, matrix balancing, and generalized perceptron models. Our main result is a robust "sharp threshold" or "zero-one" law for the feasibility of this problem. This yields a number of new sharp threshold results in the mentioned problems.

Critical window of the symmetric perceptron
DJA,
To Appear Electron. J. Probab., 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
DJA, Jonathan Niles-Weed,
Random Struct. Algorithms, 2021. (arXiv)

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.

Localized radial roll patterns in higher space dimensions
Jason J. Bramburger, DJA, 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
DJA, 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 1-d critical long-range percolation to a random scaling limit, with respect to the "Gromov-Hausdorff" metric.

The developmental rules of neural superposition in Drosophila
Marion Langen, Egemen Agi, DJA, 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
DJA, 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 my brother Jason.