Dylan J. Altschuler

I am currently a math postdoc at Carnegie Mellon, mentored by Konstantin Tikhomirov. I completed my PhD in mathematics in May 2023 at New York University (Courant), 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.

I study probability, especially high-dimensional phenomena, with applications to statistical physics, combinatorics, algorithm design, and (metric and asymptotic convex) geometry.

14. Metric dimension reduction modulus for logarithmic distortion
With Konstantin Tikhomirov.
Preprint, 2025. (arXiv)

A sharp characterization of the dimension reduction modulus for logarithmic distortion is given. The upper bound was was obtained in a series of advances, initiated by Bourgain. We finally give the matching lower bound, closing the long-standing gap. This also resolves a question of Naor's 2018 ICM plenary lecture.

13. Discrete Poincare inequalities and universal approximators for random graphs
With Pandelis Dodos, Konstantin Tikhomirov, Konstantinos Tyros.
Preprint, 2025. (arXiv // video)

The standard Euclidean Poincare inequality gives dimension-free bounds on the expansion of functions from an expander (e.g., random) graph into Euclidean space. We show an analogous inequality for functions from a random graph into another random graph. This gives a complete resolution to an open question of Jon Kleinberg (2013). As a corollary, we obtain a stochastic construction of a "universal approximator" for random graphs, answering a question of Mendel and Naor (2015). (A universal approximator is a remarkable data structure that allows for fast, approximate computation of certain functions).

12. A universal threshold for geometric embeddings of trees
With Pandelis Dodos, Konstantin Tikhomirov, Konstantinos Tyros.
Submitted, 2025. (arXiv)

When is it possible to embed a given tree into a given normed space? We show that there is a "universal" characterization: any bounded-degree tree on n vertices embeds into any space of dimension at least 64 log(n)/log(log(n)), and complete trees do not embed into any space of dimension less than .5 log(n)/log(log(n)). This phase transition is universal in the sense that it does not depend on the structure of the tree or host space. The proof proceeds by way of a randomized construction, which is analyzed using recent breakthroughs on KLS and slicing.

11. Universal geometric non-embedding of random regular graphs
With Konstantin Tikhomirov.
Submitted, 2025. (arXiv)

It is a classical result that degree-regular expander graphs cannot be embedded as geometric graphs into Euclidean space of dimension that is sublogarithmic in the number of vertices. We show that this dimensional obstruction is universal with respect to the choice of norm. For asymptotically almost every degree-regular graph, there is no normed space of sublogarithmic dimenison which admits a geometric embedding.

10. A combinatorial approach to nonlinear spectral gaps
With Pandelis Dodos, Konstantin Tikhomirov, Konstantinos Tyros.
Submitted, 2024. (arXiv)

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. As a corollary, we generalize a celebrated non-embeddability result of Matousek for expander graphs.

9. On spectral outliers for inhomogenous random symmetric matrices
With Patrick O. Santos, Konstantin Tikhomirov, Pierre Youssef.
Submitted, 2024. (arXiv)

The spectra of Wigner matrices is quite well understood. Less is known when the assumption of a uniform variance profile is dropped. Our contribution is sharp conditions for the presence of eigenvalue outliers, depending only on the ``sparsity'', rather than the fine-grained structure, of the variance profile. This provides a "structural universality" principle.

8. A note on the capacity of the binary perceptron
With Konstantin Tikhomirov.
Permanent manuscript, 2024. (arXiv).

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.

7. Zero-one laws for random feasibility problems
Annals of Applied Probability, 2025. (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.

6. Critical window of the symmetric perceptron
Electronic Journal of Probability, 2023. (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.

5. Discrepancy of random rectangular matrices
With Jonathan Niles-Weed.
Random Structures and 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.

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

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

3. Critical long range percolation: scaling limits for small β
Advised by Allan Sly.
Princeton Senior Thesis, 2018. (manuscript 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 regime considered is where the connection probability has the critical exponent, but is multiplied by a small leading constant, β.)

2. The developmental rules of neural superposition in Drosophila
With Marion Langen, Egemen Agi, 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.

1. The zoo of solitons for curve shortening in R^n
With 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).

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.