Dylan J. Altschuler

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I am currently a math postdoc at Carnegie Mellon, supervised by Konstantin Tikhomirov. I completed my PhD in mathematics in May 2023 at New York University (Courant), where I was fortunate to be advised by Jonathan Niles-Weed. Prior to that, I graduated from Princeton University 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.

I study probability, especially high-dimensional phenomena with applications to geometry, statistical physics, and combinatorics.

Given a graph and a function mapping vertices to vectors, the standard discrete Poincare inequality bounds the Euclidean norm of the function in terms of the Euclidean norm of its discrete gradient. A natural generalization is to replace the usage of the Euclidean norm with an arbitrary norm, giving rise to the notion of ``nonlinear'' (i.e. non-Euclidean) spectral gaps. We introduce a combinatorial framework for proving quantitative estimates on nonlinear spectral gaps and show that random graphs actually satisfy a dimension-free discrete Poincare inequality for any(!) normed space having an unconditional basis and finite cotype. Additionally, we obtain optimal polynomial dependence of the Poincare constant on the cotype, improving over previous super-exponential bounds. As a corollary, we generalize a celebrated result of Matousek: expander graphs do not admit low-distortion embeddings into ℓq-like spaces.

The spectra of Wigner matrices is quite well understood. Less is known when the assumption of a uniform variance profile is dropped. We consider a Wigner matrix with sub-Gaussian atomic distribution, scaled entry-wise to have a doubly stochastic variance profile. (E.g. adjacency matrices of deterministic sparse graphs with random edge weights). The largest variance of any entry is a natural proxy for "sparsity". Our contribution is sharp conditions for the presence of eigenvalue outliers, depending only on the sparsity proxy rather than the fine-grained structure of the variance profile. This can be interpreted as a "structural universality" principle.

A note on an elementary argument to show the capacity of the half-space binary perceptron is at most .847 (conjectured is .833, the best recorded previous bound is .996, and one is the trivial upper bound). Not intended for journal publication.

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.

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).