The human brain is comprised of a complex network of specific cell types. Revealing the composition of these specific cells at a given point during development, in disease versus health and in specific brain regions may provide insight into the highly specialized and regulated organization of the brain.

We show finite time blow-up for strong solutions to the 3D Euler equations on the exterior of a cone. The solutions we construct has finite energy, and velocity is axisymmetric and Lipschitz continuous before the blow up time. We achieve this by first analyzing scale-invariant (radially homogeneous) solutions, whose dynamics is governed by a 1D system. Then we make a cut-off argument to ensure finiteness of energy.

Turbulent flows appear in many different phenomena and is of
fundamental importance in science and technology. Great part of the
conventional statistical theory of turbulence, however, is based on heuristic
arguments and empirical information, with the notion of ensemble average of
flows playing a fundamental role. The theory of statistical solutions aims
towards a rigorous foundation for the conventional statistical theory of
turbulence by rigorously defining the evolution of the probability
distributions of the velocity field within the framework of Leray-Hopf weak
solutions of three-dimensional incompressible Navier-Stokes equations. In this
talk we will review a few characteristics of turbulent flows and discuss the
concept of statistical solution. We then mention some rigorous results obtained
with this framework. If time permits, we discuss a generalization of the notion
of statistical solution to an abstract setting that easily applies to many
different equations.

Many important maps in algebraic combinatorics (the RSK bijection, the Schutzenberger involution, etc.) can be described by piecewise-linear formulas. These formulas can then be ``de-tropicalized,'' or ``lifted,'' to subtraction-free rational functions on an algebraic variety, and certain properties of the combinatorial maps become more transparent in the algebro-geometric setting. I will illustrate how this works in the case of the promotion map on semistandard tableaux of rectangular shape. I will also indicate how promotion can be viewed as the combinatorial manifestation of a symmetry coming from representation theory, and how its geometric lift fits into Berenstein and Kazhdan's theory of geometric crystals.

In this paper, we propose a new updating rule of the Levenberg–Marquardt (LM) parameter for the LM method for nonlinear equations. We show that the global complexity bound of the new LM algorithm is $O(\epsilon^{-2})$, that is, it requires at most $O(\epsilon^{-2})$ iterations to derive the norm of the gradient of the merit function below the desired accuracy $\epsilon$.

Starting in 1979, the physicist Giorgio Parisi wrote a series of ground
breaking papers introducing the idea of replica symmetry breaking, which
allowed him to predict a solution for the Sherrington-Kirkpatrick (SK)
model by breaking the symmetry of replicas infinitely many times. This
is known as full-step replica symmetry breaking (FRSB). In this talk, we
will provide a mathematical proof of Parisi's FRSB prediction at zero
temperature for the more general mixed p-spin model. More precisely, we
will show that the functional order parameter of this model is not a
step function. This talk is based on joint work with Antonio Auffinger
and Wei-Kuo Chen.

The revolution in geospatial data is transforming how we study the growth and development of cities. As improved satellite imagery becomes available, new remote-sensing methods and machine-learning approaches have been developed to convert terrestrial Earth-observation data into meaningful information about the nature and pace of change of urban landscapes and human settlements. Urban areas can be detected in satellite imagery using machine-learning approaches, which typically rely on reference ground-truth data that mark urban features, either for training or for validation. Reference data are fundamental not only for mapping and assessing cross-sectional urbanization across space, but also for classification of urbanization over time. However, because they are expensive to collect, large-scale reference datasets are scarce. We present a low-cost machine-learning approach for pixel-based image classification of built-up areas at a high-resolution and large scale. Our m
ethodology relies on data infusion of nighttime and daytime remotely sensed data for automatic collection of ground truth data, which we use for supervised pixel-based image classification of built-up land cover. We demonstrate the effectiveness of our methodology, which is implemented in Google Earth Engine, through the development of accurate 30m resolution maps that characterize built-up land cover in three diverse countries: India, Mexico, and the U.S. Our approach highlights the usefulness of data fusion techniques for studying the built environment and has broad implications for identifying the drivers of urbanization.

Let G be a smooth group scheme over the p-adic integers with reductive generic fiber.
We study the generic fiber of the universal lifting ring of a G-valued mod-p representation
of the absolute Galois group of an l-adic field. In particular, we show that it admits an
open dense regular locus, and is equidimensional of dimension dim G.
This is joint work with Stefan Patrikis.

Classical results in ergodic theory due to Dye and Ornstein--Weiss show that, for an arbitrary countable amenable group, any two free ergodic measure-preserving actions on the standard atomless probability space are orbit equivalent, i.e. their orbit equivalence relations are isomorphic. This motivates the question of what happens for nonamenable groups. Works of Ioana and Epstein showed that, for an arbitrary countable nonamenable group, the relation of orbit equivalence of free ergodic measure-preserving actions on the standard probability space has uncountably many classes. In joint work with Gardella, we strengthen these conclusions by showing that such a relation is in fact not Borel. The proof makes essential use of techniques from operator algebras, including cocycle superrigidity results due to Popa, and answers a question of Kechris.

Many eukaryotic cells chemotax, sensing and following chemical gradients. However, experiments find that even under conditions when single cells do not chemotax, small clusters may still follow a gradient. How can cell clusters sense a gradient that individual cells ignore? I will discuss possible ``collective guidance'' mechanisms underlying this motion, where individual cells measure the mean value of the attractant, but need not measure its gradient to give rise to directional motility for a cell cluster. I show that the collective guidance hypothesis can be directly tested by looking for strong orientational effects in pairs of cells chemotaxing. Collective gradient sensing also has a new wrinkle in comparison to single-cell chemotaxis: to accurately determine a gradient direction, a cluster must integrate information from cells with highly variable properties. When is cell-to-cell variation a limiting factor in sensing accuracy? I provide some initial answers, and discuss how cell clusters can sense gradients in a way that is robust to this variation. Interestingly, these strategies may depend on the cluster's mechanics: there is a fundamental bound that links the cluster's chemotactic accuracy and its rheology. This suggests that in some circumstances,
mechanical changes like fluidization can influence a cluster's sensing ability. Because of this effect, increasing the noise in a single cell's motion can actually increase the accuracy of cluster chemotaxis!

In this talk, we will pose some questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. To give some answers to these questions, we will construct some examples coming from smash copropucts of Sweedler Hopf algebras. One of the fundamental tools that we use to understand the modules of these Hopf algebras is the theory of support varieties. If time allows, we will mention the definition and some of the main properties of the support varieties for these examples.