This thesis focuses on applications of classical tools from probability theory and convex analysis such as limit theorems to problems in theoretical computer science, specifically to pseudorandomness and learning theory. At first look, limit theorems, pseudorandomness and learning theory appear to be disparate subjects. However, as it has now become apparent, there's a strong connection between these questions through a third more abstract question: what do random objects look like. This connection is best illustrated by the study of the spectrum of Boolean functions which directly or indirectly played an important role in a plethora of results in complexity theory. The current thesis aims to take this program further by drawing on a variety of fundamental tools, both classical and new, in probability theory and analytic geometry. Our research contributions broadly fall into three categories. Probability Theory: The central limit theorem is one of the most important results in all of probability and richly studied topic. Motivated by questions in pseudorandomness and learning theory we obtain two new limit theorems or invariance principles. The proofs of these new results in probability, of interest on their own, have a computer science flavor and fall under the niche category of techniques from theoretical computer science with applications in pure mathematics. Pseudorandomness: Derandomizing natural complexity classes is a fundamental problem in complexity theory, with several applications outside complexity theory. Our work addresses such derandomization questions for natural and basic geometric concept classes such as halfspaces, polynomial threshold functions (PTFs) and polytopes. We develop a reasonably generic framework for obtaining pseudorandom generators (PRGs) from invariance principles and suitably apply the framework to old and new invariance principles to obtain the best known PRGs for these complexity classes. Learning Theory: Learning theory aims to understand what functions can be learned efficiently from examples. As developed in the seminal work of Linial, Mansour and Nisan (1994) and strengthened by several follow-up works, we now know strong connections between learning a class of functions and how sensitive to noise, as quantified by average sensitivity and noise sensitivity, the functions are. Besides their applications in learning, bounding the average and noise sensitivity has applications in hardness of approximation, voting theory, quantum computing and more. Here we address the question of bounding the sensitivity of polynomial threshold functions and intersections of halfspaces and obtain the best known results for these concept classes.

Improving agricultural water management is important for conserving water during dry seasons, using limited water resources in the most efficient way, and minimizing environmental risks (e.g., leaching, surface runoff). The understanding of water movement in different zones of agricultural production fields is crucial to developing an effective irrigation strategy. This work centered on optimizing field water management by characterizing the spatial patterns of soil hydraulic properties. Soil hydraulic conductivity was measured across different zones in a farmer’s field, and its spatial variability was investigated by using geostatistical techniques. Since direct measurement of hydraulic conductivity is time-consuming and arduous, pedo-transfer functions (PTFs) have been developed to estimate hydraulic conductivity indirectly through more easily measurable soil properties. Due to ignoring soil structural information and spatial covariance between soil variables, PTFs often perform unsatisfactorily when field-scale estimations of hydraulic conductivity are needed. The performance of PTFs in estimating hydraulic conductivity in the field was therefore critically evaluated. Due to the presence of structural macro-pores, saturated hydraulic conductivity (Ks) showed high spatial heterogeneity, and this variability was not captured by texture-dominated PTF estimates. However, the general spatial pattern of near-saturated hydraulic conductivity can still be reasonably generated by PTF estimates. Therefore, the hydraulic conductivity maps based on PTF estimates should be evaluated carefully and handled with caution. Recognizing the significant contribution of macro-pores to saturated water flow, PTFs were further improved by including soil macro-porosity and were proven to perform much better in estimating Ks compared with established PTFs tested in this study. Additionally, the spatial relationship between hydraulic conductivity and its potential influencing factors were further quantified by the state-space approach. State-space models outperformed current PTFs and effectively described the spatial characteristics of hydraulic conductivity in the studied field. These findings provided a basis for modeling water/solute transport in the vadose zone, and sitespecific water management.