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This book contains selected chapters on perfectoid spaces, their introduction and applications, as invented by Peter Scholze in his Fields Medal winning work. These contributions are presented at the conference on “Perfectoid Spaces” held at the International Centre for Theoretical Sciences, Bengaluru, India, from 9–20 September 2019. The objective of the book is to give an advanced introduction to Scholze’s theory and understand the relation between perfectoid spaces and some aspects of arithmetic of modular (or, more generally, automorphic) forms such as representations mod p, lifting of modular forms, completed cohomology, local Langlands program, and special values of L-functions. All chapters are contributed by experts in the area of arithmetic geometry that will facilitate future research in the direction.

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DEBARGHA BANERJEE is Associate Professor of mathematics at the Indian Institute of Science Education and Research (IISER), Pune, India. He earned his Ph.D. from the Tata Institute of Fundamental Research, Mumbai, in 2010, under the guidance of Prof. Eknath Ghate. He worked at the Australian National University, Canberra, and the Max Planck Institute for Mathematics, Germany, before joining the IISER, Pune. He works in the theory of modular forms. He published several articles in reputed international journals and supervised several students for their Ph.D. and master’s degree at the IISER, Pune. KIRAN KEDLAYA is Professor and Stefan E. Warschawski Chair in Mathematics at the University of California San Diego, USA. He did his Ph.D. from Massachusetts Institute of Technology (MIT), USA. He is an Indian–American Mathematician, and he held several visiting positions at several eminent universities like the Institute of Advanced studies, Princeton; the University of California, Berkeley; and MIT. He is an expert in p-adic Hodge theory, p-adic/non-Archimedean analytic geometry, p-adic differential equations, and algorithms in arithmetic geometry. He gave an invited talk at the ICM 2010. EHUD DE SHALIT is Professor of Mathematics, The Einstein Institute of Mathematics, Hebrew University, Giv'at-Ram, Jerusalem, Israel. A number theorist, Prof. Shalit has worked on topics related to class field theory, Iwasawa theory of elliptic curves, modular forms, p-adic L-functions, and p-adic analytic geometry. Current projects involve studying the cohomology of p-adic symmetric domains and the varieties uniformized by them.  CHITRABHANU CHAUDHURI is Assistant Professor at the National Institute of Science Education and Research, Bhubaneswar, Odisha, India. His research revolves around the topology and geometry of the moduli of curves. The moduli of curves parametrize algebraic curves or Riemann surfaces up to isomorphisms. He did his Ph.D. from Northwestern University, USA.

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