Publications

Topological semimetals with massless Dirac and Weyl fermions represent the forefront of quantum materials research. In two dimensions, a peculiar class of fermions that are massless in one direction and massive in the perpendicular direction was predicted fifteen years ago. These highly exotic quasiparticles - the semi-Dirac fermions - ignited intense theoretical interest but remain undetected. Using magneto-optical spectroscopy, we demonstrate the defining feature of semi-Dirac fermions - B

We introduce an efficient numerical method for second-order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory phase function that obeys the nonlinear Riccati equation. We propose a defect correction iteration that gives an asymptotic series for such a phase function; this is numerically approximated on a Chebyshev grid with a small number of nodes. For analytic coefficients we prove that each iteration, up to a certain maximum number, reduces the residual by a factor of order of the local frequency. The algorithm adapts both the stepsize and the choice of method, switching to a conventional spectral collocation method away from oscillatory regions. In numerical experiments we find that our proposal outperforms other state-of-the-art oscillatory solvers, most significantly at low to intermediate frequencies and at low tolerances, where it may use up to \(10^6\) times fewer function evaluations. Even in high-frequency regimes, our implementation is on average 10 times faster than other specialized solvers.

Studies in Drosophila were essential in delineating the highly conserved RAS signaling pathway. Indeed, some pathway components, such as Son of sevenless or Corkscrew, were named after mutant phenotypes in flies. Here, we discuss how Drosophila, with its ever-expanding arsenal of precise genetic manipulations and quantitative phenotypic assays, can be harnessed for investigating how RAS signaling is genetically deregulated in human diseases. The general approach is based on analyzing how disease mutations affect well-studied RAS-dependent developmental processes. Focusing on our work in the fly embryo and larval trachea, we illustrate this approach for missense mutations in MEK, a central kinase in the RAS cascade, which is deregulated in developmental abnormalities and cancers. The established approach provides clear insights into genotype/phenotype associations and can be extended to other signaling systems.

Ice in nature is dynamic at all scales, from glacial sheets that deform and flow to icebergs that melt down and capsize [1,2]. For the latter, much of the ice and much of the action is unseen beneath the surface [3–5]. Here we study laboratory-scale icebergs that freely float and melt, where direct visualizations show interesting and interconnected changes in the shape of the ice, its posture, and the flows of the surrounding water.

Our experiments reveal that free-floating ice persistently melts into unstable geometries, causing it to repeatedly capsize. Figure 1 shows the shape progression for a cylindrical piece of ice floating at the surface of room temperature water. It locks to an orientation, melts in place for several minutes, then abruptly rotates to a new posture and again locks. This process repeats for about 10 to 15 flips over the 30 minutes it takes to melt away. The photographs sample some of the locked orientations. Figure 2 displays the flows of the melt waters beneath the iceberg, where the two photos capture views along the axis and from the side, respectively. Below we describe the specialized techniques that enabled these images.

Electron cryomicroscopy (cryo-EM) is a technique in structural biology used to reconstruct accurate volumetric maps of molecules. One step of the cryo-EM pipeline involves solving an inverse-problem. This inverse-problem, referred to as \textit{ab-initio} single-particle reconstruction, takes as input a collection of 2d-images -- each a projection of a molecule from an unknown viewing-angle -- and attempts to reconstruct the 3d-volume representing the underlying molecular density.
Most methods for solving this inverse-problem search for a solution which optimizes a posterior likelihood of generating the observed image-data, given the reconstructed volume. Within this framework, it is natural to study the Hessian of the log-likelihood: the eigenvectors and eigenvalues of the Hessian determine how the likelihood changes with respect to perturbations in the solution, and can give insight into the sensitivity of the solution to aspects of the input.
In this paper we describe a simple strategy for estimating the smallest eigenvalues and eigenvectors (i.e., the `softest modes') of the Hessian of the log-likelihood for the \textit{ab-initio} single-particle reconstruction problem. This strategy involves rewriting the log-likelihood as a 3d-integral. This interpretation holds in the low-noise limit, as well as in many practical scenarios which allow for noise-marginalization.
Once we have estimated the softest modes, we can use them to perform many kinds of sensitivity analysis. For example, we can determine which parts of the reconstructed volume are trustworthy, and which are unreliable, and how this unreliability might depend on the data-set and the imaging parameters. We believe that this kind of analysis can be used alongside more traditional strategies for sensitivity analysis, as well as in other applications, such as free-energy estimation.

Gastrulation is a critical process during embryonic development that transforms a single-layered blastula into a multilayered embryo with distinct germ layers, which eventually give rise to all the tissues and organs of the organism. Studies across species have uncovered the mechanisms underlying the building blocks of gastrulation movements, such as localized in-plane and out-of-plane epithelial deformations. The next challenge is to understand dynamics on the scale of the embryo: this requires quantifying strain tensors, which rigorously describe the differences between the deformed configurations taken on by local clusters of cells at time instants of observation and their reference configuration at an initial time. We present a systematic strategy for computing such tensors from the local dynamics of cell clusters, which are chosen across the embryo from several regions whose morphogenetic fate is central to viable gastrulation. As an application of our approach, we demonstrate a strategy of identifying distinct Drosophila morphological domains using strain tensors.

A high-order quadrature scheme is constructed for the evaluation of Laplace single and double layer potentials and their normal derivatives on smooth surfaces in three dimensions. The construction begins with a harmonic approximation of the density on each patch, which allows for a natural harmonic polynomial extension in a volumetric neighborhood of the patch in the ambient space. Then by the general Stokes theorem, singular and nearly singular surface integrals are reduced to line integrals preserving the singularity of the kernel, instead of the standard origin-centered 1-forms that require expensive adaptive integration. These singularity-preserving line integrals can be semi-analytically evaluated using singularity-swap quadrature. In other words, the evaluation of singular and nearly singular surface integrals is reduced to function evaluations at the vertices on the boundary of each patch. The recursive reduction quadrature largely removes adaptive integration that is needed in most existing high-order quadratures for singular and nearly singular surface integrals, resulting in exceptional performance. The scheme achieves twelve-digit accuracy uniformly for close evaluations and offers a speedup of five times or more in constructing the sparse quadrature-correction matrix compared to previous state-of-the-art quadrature schemes.

Higher brain functions depend on experience-dependent representations of relevant information that may be organized by attractor dynamics or by geometrical modifications of continuous “neural manifolds”. To explore these scenarios we analyzed odor-evoked activity in telencephalic area pDp of juvenile and adult zebrafish, the homolog of piriform cortex. No obvious signatures of attractor dynamics were detected. Rather, olfactory discrimination training selectively enhanced the separation of neural manifolds representing task-relevant odors from other representations, consistent with predictions of autoassociative network models endowed with precise synaptic balance. Analytical approaches using the framework of manifold capacity revealed multiple geometrical modifications of representational manifolds that supported the classification of task-relevant sensory information. Manifold capacity predicted odor discrimination across individuals, indicating a close link between manifold geometry and behavior. Hence, pDp and possibly related recurrent networks store information in the geometry of representational manifolds, resulting in joint sensory and semantic maps that may support distributed learning processes.

We introduce a new class of computationally tractable scattering problems in unbounded domains, which we call decomposable problems. In these decomposable problems, the computational domain can be split into a finite collection of subdomains in which the scatterer has a "simple" structure. A subdomain is simple if the domain Green's function for this subdomain is either available analytically or can be computed numerically with arbitrary accuracy by a tractable method. These domain Green's functions are then used to reformulate the scattering problem as a system of boundary integral equations on the union of the subdomain boundaries. This reformulation gives a practical numerical method, as the resulting integral equations can then be solved, to any desired degree of accuracy, by using coordinate complexification over a finite interval, and standard discretization techniques.

Since Merrifield’s development of solid-phase peptide synthesis, we have seen explosive growth in the number of synthetic building-blocks that can be incorporated into peptides. This has created a problem: the number of possible molecules that could be synthesized is many orders of magnitude greater than the largest conceivable combinatorial libraries. Computational design, based on combinatorial optimization algorithms, addresses this problem by proposing sequences likely to have desired folds and functions. These computational methods complement experiments by reducing astronomically large numbers of combinatorial possibilities to experimentally tractable shortlists. This presentation describes our robust, versatile methods, made available to peptide scientists in the Rosetta and Masala software suites, for designing peptides that fold into rigid conformations. Our physics-based methods generalize to exotic chemical building blocks poorly amenable to machine learning-based methods for want of training data. Our pipeline has produced experimentally-validated mixed-chirality peptides that bind to targets of therapeutic interest, and peptides that diffuse across cell membranes. Ongoing research is mapping the sequence optimization problem (which grows intractable even for supercomputers as the number of candidate chemical building blocks grows very large) to current and near-future quantum computers, allowing use of quantum algorithms in the context of the existing, widely-used design protocols.