This volume stems from the Linde Hall Inaugural Math Symposium, held from February 22-24, 2019, at California Institute of Technology, Pasadena, California. The content isolates and discusses nine mathematical problems, or sets of problems, in a deep way, but starting from scratch. Included among them are the well-known problems of the classification of finite groups, the Navier-Stokes equations, the Birch and Swinnerton-Dyer conjecture, and the continuum hypothesis. The other five problems, also of substantial importance, concern the Lieb-Thirring inequalities, the equidistribution problems in number theory, surface bundles, ramification in covers and curves, and the gap and type problems in Fourier analysis. The problems are explained succinctly, with a discussion of what is known and an elucidation of the outstanding issues. An attempt is made to appeal to a wide audience, both in terms of the field of expertise and the level of the reader.

Lysosomes are key subcellular organelles that regulate the cell function. Many of the essential activities of the cell are dependent on lysosomes. Dysfunction is linked to multiple diseases - storage disorders, neurodegeneration, immunological diseases and cancer. This book discusses concepts and methods used to study lysosome ion and small molecule transport. The contents will not only attract accomplished investigators in need of a broad review and synthesis of this important subject but will also appeal to young investigators and trainees needing to acquire comprehensive knowledge and technical skills working with lysosomal ion channels and small molecule transporters. Key selling features: Summarizes the endocellular role that lysosomes play with respect to cellular waste disposal Reviews essential cellular functions of lysosomes Explores how lysosome dysfunction is the cause of many metabolic disorders Examines how lysomes are involved in storage diseases Describes various technologies and methods used in lysosome research

Lysosomes are key subcellular organelles that regulate the cell function. Many of the essential activities of the cell are dependent on lysosomes. Dysfunction is linked to multiple diseases - storage disorders, neurodegeneration, immunological diseases and cancer. This book discusses concepts and methods used to study lysosome ion and small molecule transport. The contents will not only attract accomplished investigators in need of a broad review and synthesis of this important subject but will also appeal to young investigators and trainees needing to acquire comprehensive knowledge and technical skills working with lysosomal ion channels and small molecule transporters. Key selling features: Summarizes the endocellular role that lysosomes play with respect to cellular waste disposal Reviews essential cellular functions of lysosomes Explores how lysosome dysfunction is the cause of many metabolic disorders Examines how lysomes are involved in storage diseases Describes various technologies and methods used in lysosome research

The rapid expansion of the TRP field has generated a large amount of excellent original work across many different research fields. However, investigators are not necessarily familiar with the pros and cons of the variety of methods used to study TRP channels. Because of functional and genetic diversity, as well as the different physiological roles they play, techniques used for studying TRP channels range from single molecular analysis to behavioral animal studies. Methods in multiple areas, such as molecular biology, fluorescence imaging, electrophysiology, cell biology, genetics, proteomics, pharmacology, system physiology, and behavioral assessment, are employed to investigate various aspects of these channels. Choosing among many possible topics in these broad areas was a daunting task. A comprehensive review of the field, TRP Channels spans the information gap by providing broad coverage of current methods and techniques commonly used in TRP channel research, and detailed protocols with thorough discussions of the advantages and disadvantages across methods. Some topics covered include Mammalian, Drosophila and C. elegans TRP channels Practical protocols for functional studies of TRP channels, including TRPC, TRPV, TRPA, TRPM and the intracellularly localized TRPML channels ThermoTRPs, including the new fast temperature jump apparatus and the high throughput random mutagenesis method for screening critical motifs involved in TRP channel regulation Cell-based high-throughput screening assays for TRP channels and their applications in drug discoveries TRP channel functions in native cells, including smooth muscles, neurons, and cancers Showcasing the current status of the field, TRP Channels covers the major techniques used in various areas of research. The majority of the chapters are protocol oriented, with the goal of providing clear directions for laboratory use. Because of the breadth of the TRP field, the applications of some methods are described in multiple chapters by experts working on a variety of channel types that serve different physiological functions, highlighting distinctive views on how the methodology can be utilized. Some chapters include discussion on the usefulness and pitfalls associated with the use of certain techniques. Together with chapters that offer comprehensive reviews on the functional regulation and other roles of TRP channels, students and investigators new to the field should find this book particularly informative.

The rapid expansion of the TRP field has generated a large amount of excellent original work across many different research fields. However, investigators are not necessarily familiar with the pros and cons of the variety of methods used to study TRP channels. Because of functional and genetic diversity, as well as the different physiological roles they play, techniques used for studying TRP channels range from single molecular analysis to behavioral animal studies. Methods in multiple areas, such as molecular biology, fluorescence imaging, electrophysiology, cell biology, genetics, proteomics, pharmacology, system physiology, and behavioral assessment, are employed to investigate various aspects of these channels. Choosing among many possible topics in these broad areas was a daunting task. A comprehensive review of the field, TRP Channels spans the information gap by providing broad coverage of current methods and techniques commonly used in TRP channel research, and detailed protocols with thorough discussions of the advantages and disadvantages across methods. Some topics covered include Mammalian, Drosophila and C. elegans TRP channels Practical protocols for functional studies of TRP channels, including TRPC, TRPV, TRPA, TRPM and the intracellularly localized TRPML channels ThermoTRPs, including the new fast temperature jump apparatus and the high throughput random mutagenesis method for screening critical motifs involved in TRP channel regulation Cell-based high-throughput screening assays for TRP channels and their applications in drug discoveries TRP channel functions in native cells, including smooth muscles, neurons, and cancers Showcasing the current status of the field, TRP Channels covers the major techniques used in various areas of research. The majority of the chapters are protocol oriented, with the goal of providing clear directions for laboratory use. Because of the breadth of the TRP field, the applications of some methods are described in multiple chapters by experts working on a variety of channel types that serve different physiological functions, highlighting distinctive views on how the methodology can be utilized. Some chapters include discussion on the usefulness and pitfalls associated with the use of certain techniques. Together with chapters that offer comprehensive reviews on the functional regulation and other roles of TRP channels, students and investigators new to the field should find this book particularly informative.