Publications

An obstacle is immersed in an externally driven 2D Stokes or Navier-Stokes fluid. We study the self-equilibration conditions for that obstacle under steady state assumptions on the flow. We then seek to optimize the translational and/or angular velocity of the obstacle by varying its shape. To allow general variations, we must consider a very large class of obstacles for which the notion of trace is meaningless. This forces us to revisit the notion of self-equilibration for both Stokes and Navier-Stokes in a measure theoretic environment.

A key problem in deep learning and computational neuroscience is relating the geometrical properties of neural representations to task performance. Here, we consider this problem for continuous decoding tasks where neural variability may affect task precision. Using methods from statistical mechanics, we study the average-case learning curves for ɛ-insensitive support vector regression and discuss its capacity as a measure of linear decodability. Our analysis reveals a phase transition in training error at a critical load, capturing the interplay between the tolerance parameter ɛ and neural variability. We uncover a double-descent phenomenon in the generalization error, showing that ɛ acts as a regularizer, both suppressing and shifting these peaks. Theoretical predictions are validated both with toy models and deep neural networks, extending the theory of support vector machines to continuous tasks with inherent neural variability.

DNA methylation, a covalent modification, fundamentally shapes mammalian gene regulation and cellular identity. This review examines methylation's biochemical underpinnings, genomic distribution patterns, and analytical approaches. We highlight three distinctive aspects that separate methylation from other epigenetic marks: its remarkable stability as a silencing mechanism, its capacity to maintain distinct states independently of DNA sequence, and its effectiveness as a quantitative trait linking genotype to disease risk. We also explore the phenomenon of methylation clocks and their biological significance. The review addresses technical considerations across major assay types—both array-based technologies and sequencing approaches—with emphasis on data normalization, quality control, cell proportion inference, and the specialized statistical models required for next-generation sequencing analysis.

A great deal is known about biochemical aspects of transcription, but we still lack an understanding of how transcription is causally regulated in space and time. A major unanswered question is the extent to which transcription at different locations in the nucleus are independent from each other or, instead, are spatially coordinated. We propose two classes of models of coordination: 1) the shared environment model, in which neighboring loci exhibit coordinated transcriptional dynamics due to sharing the same local biochemical environment; 2) the mechanical crosstalk model, in which forces propagate from one actively transcribing locus to affect transcription of another. Determining the prevalence of the spatial coordination of transcription, and the underlying mechanisms when it occurs, is an exciting challenge in nuclear biophysics.

Selective excitation of vibrational modes using strong laser pulses has emerged as a powerful material engineering paradigm. However, to realize deterministic control over material properties for device applications, it is desirable to have an analogous scheme without a drive, operating in thermal equilibrium. We here propose such an equilibrium analog of the light-driven paradigm, leveraging the strong coupling between lattice degrees of freedom and the quantum fluctuations of the electric field of a THz micro-cavity. We demonstrate this approach by showing, using ab initio data, how electric field fluctuations can induce a sub-dominant ferromagnetic order, on top of the dominant zig-zag antiferromagnet order, in FePS

How thousands of microtubules and molecular motors self-organize into spindles remains poorly understood. By combining static, nanometer-resolution, large-scale electron tomography reconstructions and dynamic, optical-resolution, polarized light microscopy, we test an active liquid crystal continuum model of mitotic spindles in human tissue culture cells. The predictions of this coarse-grained theory quantitatively agree with the experimentally measured spindle morphology and fluctuation spectra. These findings argue that local interactions and polymerization produce collective alignment, diffusive-like motion, and polar transport which govern the behaviors of the spindle's microtubule network, and provide a means to measure the spindle's material properties. This work demonstrates that a coarse-grained theory featuring measurable, physically-interpretable parameters can quantitatively describe the mechanical behavior and self-organization of human mitotic spindles.

The male fruit fly produces ~1.8 mm long sperm, thousands of which can be stored until mating in a ~200 micron sac, the seminal vesicle. While the evolutionary pressures driving such extreme sperm (flagellar) lengths have long been investigated, the physical consequences of their gigantism are unstudied. Through high-resolution three-dimensional reconstructions of in vivo sperm morphologies and rapid live imaging, we discovered that stored sperm are organized into a dense and highly aligned state. The packed flagella exhibit system-wide collective 'material' flows, with persistent and slow-moving topological defects; individual sperm, despite their extraordinary lengths, propagate rapidly through the flagellar material, moving in either direction along material director lines. To understand how these collective behaviors arise from the constituents' nonequilibrium dynamics, we conceptualize the motion of individual sperm as topologically confined to a reptation-like tube formed by its neighbors. Therein, sperm propagate through observed amplitude-constrained and internally driven flagellar bending waves, pushing off counter-propagating neighbors. From this conception, we derive a continuum theory that produces an extensile material stress that can sustain an aligned flagellar material. Experimental perturbations and simulations of active elastic filaments verify our theoretical predictions. Our findings suggest that active stresses in the flagellar material maintain the sperm in an unentangled, hence functional state, in both sexes, and establish giant sperm in their native habitat as a novel and physiologically relevant active matter system

Recent breakthroughs in protein structure prediction have led to a surge in high-quality 3D models, highlighting the need for efficient computational solutions. In our work, we examine the structural clusters from the AlphaFold Protein Structure Database (AFDB), a high-quality subset of ESMAtlas, and the Microbiome Immunity Project (MIP). We create a single cohesive low-dimensional representation of the resulting protein space. We show that, while each database occupies distinct regions, they collectively exhibit significant overlap in their functional profiles. High-level biological functions tend to cluster in particular regions, revealing a shared functional landscape despite the diverse sources of data. By creating a representation of protein structure space, localizing functional annotations within this space, and providing an open-access web-server for exploration, this work offers insights for future research concerning protein sequence-structure-function relationships, enabling biological questions to be asked about taxonomic assignments, environmental factors, or functional specificity. This approach is generalizable, thus enabling further discovery beyond findings presented here.

We study the problem of learning a structured approximation (low-rank, sparse, banded, etc.) to an unknown matrix $A$ given access to matrix-vector product (matvec) queries of the form $x \rightarrow Ax$ and $x \rightarrow A^Tx$. This problem is of central importance to algorithms across scientific computing and machine learning, with applications to fast multiplication and inversion for structured matrices, building preconditioners for first-order optimization, and as a model for differential operator learning. Prior work focuses on obtaining query complexity upper and lower bounds for learning specific structured matrix families that commonly arise in applications.
We initiate the study of the problem in greater generality, aiming to understand the query complexity of learning approximations from general matrix families. Our main result focuses on finding a near-optimal approximation to $A$ from any finite-sized family of matrices, $\mathcal{F}$. Standard results from matrix sketching show that $O(\log|\mathcal{F}|)$ matvec queries suffice in this setting. This bound can also be achieved, and is optimal, for vector-matrix-vector queries of the form $x,y\rightarrow x^TAy$, which have been widely studied in work on rank-$1$ matrix sensing.
Surprisingly, we show that, in the matvec model, it is possible to obtain a nearly quadratic improvement in complexity, to $\tilde{O}(\sqrt{\log|\mathcal{F}|})$. Further, we prove that this bound is tight up to log-log this http URL covering number arguments, our result extends to well-studied infinite families. As an example, we establish that a near-optimal approximation from any \emph{linear matrix family} of dimension $q$ can be learned with $\tilde{O}(\sqrt{q})$ matvec queries, improving on an $O(q)$ bound achievable via sketching techniques and vector-matrix-vector queries.

Integrating task-relevant information into neural representations is a fundamental ability of both biological and artificial intelligence systems. Recent theories have categorized learning into two regimes: the rich regime, where neural networks actively learn task-relevant features, and the lazy regime, where networks behave like random feature models. Yet this simple lazy-rich dichotomy overlooks a diverse underlying taxonomy of feature learning, shaped by differences in learning algorithms, network architectures, and data properties. To address this gap, we introduce an analysis framework to study feature learning via the geometry of neural representations. Rather than inspecting individual learned features, we characterize how task-relevant representational manifolds evolve throughout the learning process. We show, in both theoretical and empirical settings, that as networks learn features, task-relevant manifolds untangle, with changes in manifold geometry revealing distinct learning stages and strategies beyond the lazy-rich dichotomy. This framework provides novel insights into feature learning across neuroscience and machine learning, shedding light on structural inductive biases in neural circuits and the mechanisms underlying out-of-distribution generalization.