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James Tanis

Publications and source records attributed to James Tanis.

12 recordsLinked to original sources
arXiv2024

Anticipatory Understanding of Resilient Agriculture to Climate

With billions of people facing moderate or severe food insecurity, the resilience of the global food supply will be of increasing concern due to the effects of climate change and geopolitical events. In this paper we describe a framework to better identify food security hotspots using a combination of remote sensing, deep learning, crop yield modeling, and causal modeling of the food distribution system. While we feel that the methods are adaptable to other regions of the world, we focus our analysis on the wheat breadbasket of northern India, which supplies a large percentage of the world's population. We present a quantitative analysis of deep learning domain adaptation methods for wheat farm identification based on curated remote sensing data from France. We model climate change impacts on crop yields using the existing crop yield modeling tool WOFOST and we identify key drivers of crop simulation error using a longitudinal penalized functional regression. A description of a system dynamics model of the food distribution system in India is also presented, along with results of food insecurity identification based on seeding this model with the predicted crop yields.

cs.CV↗
arXiv2020

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

We define globally hypoelliptic smooth $\mathbb R^k$ actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When $k=1$, strong global rigidity is conjectured for such actions by Greenfield-Wallach and Katok: every such action is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces \cite{FFRH}. In this paper we conjecture that among homogeneous $\mathbb R^k$ actions ($k\ge 2$) on homogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic $\mathbb R^2$ actions on homogeneous spaces $G/Γ$, with at least one quasi-unipotent generator, where $G= SL(n, \mathbb R)$. We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds.

math.DS↗
arXiv2018

Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher rank Lie groups

Let $\mathbb{G}$ denote a higher rank $\mathbb R$-split simple Lie group of the following type: $SL(n,\mathbb R)$, $SO_o(m,m)$, $E_{6(6)}$, $E_{7(7)}$ and $E_{8(8)}$, where $m\geq 4$ and $n \geq 3$. We study the cohomological equation for discrete parabolic actions on $\mathbb G$ via representation theory. Specifically, we characterize the obstructions to solving the cohomological equation and construct smooth solutions with Sobolev estimates. We prove that global estimates of the solution are generally not tame, and our non-tame estimates in the case $\mathbb G=SL(n,\mathbb R)$ are sharp up to finite loss of regularity. Moreover, we prove that for general $\mathbb G$ the estimates are tame in all but one direction, and as an application, we obtain tame estimates for the common solution of the cocycle equations. We also give a sufficient condition for which the first cohomology with coefficients in smooth vector fields is trivial. In the case that $\mathbb G=SL(n,\mathbb R)$, we show this condition is also necessary. A new method is developed to prove tame directions involving computations within maximal unipotent subgroups of the unitary duals of $SL(2,\mathbb R)\ltimes\mathbb R^2$ and $SL(2,\mathbb R)\ltimes\mathbb R^4$. A new technique is also developed to prove non-tameness for solutions of the cohomological equation.

math.DS↗
arXiv2018

Cohomological equation and cocycle rigidity of discrete parabolic actions

We study the cohomological equation for discrete horocycle maps on $SL(2, \mathbb{R})$ and $SL(2,\mathbb R)\times SL(2, \mathbb{R})$ via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of $SL(2,\mathbb R)$. Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of $sl(2, \mathbb R)$, and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for $SL(2, \mathbb R)$ in which all cases of irreducible, unitary representations of $SL(2, \mathbb R)$ can be studied simultaneously. Finally, our results combine with those of a very recent paper by the authors to give cohomology results for discrete parabolic actions in regular representations of some general classes of simple Lie groups, providing a fundamental step toward proving differential local rigidity of parabolic actions in this general setting.

math.DS↗
arXiv2015

Effective equidistribution of twisted horocycle flows and horocycle maps

We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an application of our main theorems in the compact case we further improve on a result of A. Venkatesh, recently already improved by J. Tanis and P. Vishe, on a sparse equidistribution problem for classical horocycle flows proposed by N. Shah and G. Margulis, and in the general non-compact, finite area case we prove bounds on Fourier coefficients of cups forms which are off the best known bounds of A. Good only by a logarithmic term. Our approach is based on Sobolev estimates for solutions of the cohomological equation and on scaling of invariant distributions for twisted horocycle flows.

math.DS↗
arXiv2015

The distribution of $S$-integral points on $\mathrm{SL}_2$-orbit closures of binary forms

We study the distribution of $S$-integral points on $\mathrm{SL}_2$-orbit closures of binary forms and prove an asymptotic formula for the number of $S$-integral points of bounded height on $\mathrm{SL}_2$-orbit closures of binary forms. This extends a result of Duke, Rudnick, and Sarnak. The main ingredients of the proof are the method of mixing developed by Eskin-McMullen and Benoist-Oh, Chambert-Loir-Tschinkel's study of asymptotic volume of height balls, and Hassett-Tschinkel's description of log resolutions of $\mathrm{SL}_2$-orbit closures of binary forms.

math.NT↗
arXiv2015

Uniform bounds for period integrals and sparse equidistribution

Let $M=Γ\backslash\mathrm{PSL}(2,\mathbb{R})$ be a compact manifold, and let $f\in C^\infty(M)$ be a function of zero average. We use spectral methods to get uniform (i.e. independent of spectral gap) bounds for twisted averages of $f$ along long horocycle orbit segments. We apply this to obtain an equidistribution result for sparse subsets of horocycles on $M$.

math.DS↗
arXiv2014

Kaczmarz Algorithm and Frames

Sequences of unit vectors for which the Kaczmarz algorithm always converges in Hilbert space can be characterized in frame theory by tight frames with constant 1. We generalize this result to the context of frames and bases. In particular, we show that the only effective sequences which are Riesz bases are orthonormal bases. Moreover, we consider the infinite system of linear algebraic equations $A x = b$ and characterize the (bounded) matrices $A$ for which the Kaczmarz algorithm always converges to a solution.

cs.IT↗
arXiv2014

Cocycle Rigidity and Splitting for some Discrete Parabolic Actions

We prove trivialization of the first cohomology with coefficients in smooth vector fields, for a class of $\mathbb Z^2$ parabolic actions on $(SL(2, \mathbb R)\times SL(2, \mathbb R))/Γ$, where the lattice $Γ$ is irreducible and co-compact. We also obtain a splitting construction involving first and second coboundary operators in the cohomology with coefficients in smooth vector fields.

math.DS↗
arXiv2013

The Cohomological Equation and Invariant Distributions for Horocycle Maps

We study the invariant distributions for the horocycle map on $Γ\backslash SL(2, \mathbb{R})$ and prove Sobolev estimates for the cohomological equation of the horocycle map. As an application, we obtain an estimate for the rate of equidistribution for horocycle maps on compact manifolds.

math.DS↗