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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.
The William Lowell Putnam Mathematics Competition is the most
prestigious undergraduate mathematics problem-solving contest in
North America, with thousands of students taking part every year.
This volume presents the contest problems for the years 2001-2016.
The heart of the book is the solutions; these include multiple
approaches, drawn from many sources, plus insights into navigating
from the problem statement to a solution. There is also a section
of hints, to encourage readers to engage deeply with the problems
before consulting the solutions. The authors have a distinguished
history of engagement with, and preparation of students for, the
Putnam and other mathematical competitions. Collectively they have
been named Putnam Fellow (top five finisher) ten times. Kiran
Kedlaya also maintains the online Putnam Archive.
Now in its second edition, this volume provides a uniquely detailed
study of $P$-adic differential equations. Assuming only a
graduate-level background in number theory, the text builds the
theory from first principles all the way to the frontiers of
current research, highlighting analogies and links with the
classical theory of ordinary differential equations. The author
includes many original results which play a key role in the study
of $P$-adic geometry, crystalline cohomology, $P$-adic Hodge
theory, perfectoid spaces, and algorithms for L-functions of
arithmetic varieties. This updated edition contains five new
chapters, which revisit the theory of convergence of solutions of
$P$-adic differential equations from a more global viewpoint,
introducing the Berkovich analytification of the projective line,
defining convergence polygons as functions on the projective line,
and deriving a global index theorem in terms of the Laplacian of
the convergence polygon.
Introduced by Peter Scholze in 2011, perfectoid spaces are a bridge
between geometry in characteristic 0 and characteristic $p$, and
have been used to solve many important problems, including cases of
the weight-monodromy conjecture and the association of Galois
representations to torsion classes in cohomology. In recognition of
the transformative impact perfectoid spaces have had on the field
of arithmetic geometry, Scholze was awarded a Fields Medal in 2018.
This book, originating from a series of lectures given at the 2017
Arizona Winter School on perfectoid spaces, provides a broad
introduction to the subject. After an introduction with insight
into the history and future of the subject by Peter Scholze, Jared
Weinstein gives a user-friendly and utilitarian account of the
theory of adic spaces. Kiran Kedlaya further develops the
foundational material, studies vector bundles on Fargues-Fontaine
curves, and introduces diamonds and shtukas over them with a view
toward the local Langlands correspondence. Bhargav Bhatt explains
the application of perfectoid spaces to comparison isomorphisms in
$p$-adic Hodge theory. Finally, Ana Caraiani explains the
application of perfectoid spaces to the construction of Galois
representations associated to torsion classes in the cohomology of
locally symmetric spaces for the general linear group. This book
will be an invaluable asset for any graduate student or researcher
interested in the theory of perfectoid spaces and their
applications.
This third volume of problems from the William Lowell Putnam
Competition is unlike the previous two in that it places the
problems in the context of important mathematical themes. The
authors highlight connections to other problems, to the curriculum
and to more advanced topics. The best problems contain kernels of
sophisticated ideas related to important current research, and yet
the problems are accessible to undergraduates. The solutions have
been compiled from the American Mathematical Monthly, Mathematics
Magazine and past competitors. Multiple solutions enhance the
understanding of the audience, explaining techniques that have
relevance to more than the problem at hand. In addition, the book
contains suggestions for further reading, a hint to each problem,
separate from the full solution and background information about
the competition. The book will appeal to students, teachers,
professors and indeed anyone interested in problem solving as a
gateway to a deep understanding of mathematics.
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