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This volume results from two programs that took place at the
Institute for Mathematical Sciences at the National University of
Singapore: Aspects of Computation — in Celebration of the
Research Work of Professor Rod Downey (21 August to 15 September
2017) and Automata Theory and Applications: Games, Learning and
Structures (20-24 September 2021).The first program was dedicated
to the research work of Rodney G. Downey, in celebration of his
60th birthday. The second program covered automata theory whereby
researchers investigate the other end of computation, namely the
computation with finite automata, and the intermediate level of
languages in the Chomsky hierarchy (like context-free and
context-sensitive languages).This volume contains 17 contributions
reflecting the current state-of-art in the fields of the two
programs.
Computability theory is a branch of mathematical logic and computer
science that has become increasingly relevant in recent years. The
field has developed growing connections in diverse areas of
mathematics, with applications in topology, group theory, and other
subfields. In A Hierarchy of Turing Degrees, Rod Downey and Noam
Greenberg introduce a new hierarchy that allows them to classify
the combinatorics of constructions from many areas of computability
theory, including algorithmic randomness, Turing degrees,
effectively closed sets, and effective structure theory. This
unifying hierarchy gives rise to new natural definability results
for Turing degree classes, demonstrating how dynamic constructions
become reflected in definability. Downey and Greenberg present
numerous construction techniques involving high-level nonuniform
arguments, and their self-contained work is appropriate for
graduate students and researchers. Blending traditional and modern
research results in computability theory, A Hierarchy of Turing
Degrees establishes novel directions in the field.
Computability theory is a branch of mathematical logic and computer
science that has become increasingly relevant in recent years. The
field has developed growing connections in diverse areas of
mathematics, with applications in topology, group theory, and other
subfields. In A Hierarchy of Turing Degrees, Rod Downey and Noam
Greenberg introduce a new hierarchy that allows them to classify
the combinatorics of constructions from many areas of computability
theory, including algorithmic randomness, Turing degrees,
effectively closed sets, and effective structure theory. This
unifying hierarchy gives rise to new natural definability results
for Turing degree classes, demonstrating how dynamic constructions
become reflected in definability. Downey and Greenberg present
numerous construction techniques involving high-level nonuniform
arguments, and their self-contained work is appropriate for
graduate students and researchers. Blending traditional and modern
research results in computability theory, A Hierarchy of Turing
Degrees establishes novel directions in the field.
This Festschrift is published in honor of Rodney G. Downey, eminent
logician and computer scientist, surfer and Scottish country
dancer, on the occasion of his 60th birthday. The Festschrift
contains papers and laudations that showcase the broad and
important scientific, leadership and mentoring contributions made
by Rod during his distinguished career. The volume contains 42
papers presenting original unpublished research, or expository and
survey results in Turing degrees, computably enumerable sets,
computable algebra, computable model theory, algorithmic
randomness, reverse mathematics, and parameterized complexity, all
areas in which Rod Downey has had significant interests and
influence. The volume contains several surveys that make the
various areas accessible to non-specialists while also including
some proofs that illustrate the flavor of the fields.
The theory relating algebraic curves and Riemann surfaces exhibits
the unity of mathematics: topology, complex analysis, algebra and
geometry all interact in a deep way. This textbook offers an
elementary introduction to this beautiful theory for an
undergraduate audience. At the heart of the subject is the theory
of elliptic functions and elliptic curves. A complex torus (or
"donut") is both an abelian group and a Riemann surface. It is
obtained by identifying points on the complex plane. At the same
time, it can be viewed as a complex algebraic curve, with addition
of points given by a geometric "chord-and-tangent" method. This
book carefully develops all of the tools necessary to make sense of
this isomorphism. The exposition is kept as elementary as possible
and frequently draws on familiar notions in calculus and algebra to
motivate new concepts. Based on a capstone course given to senior
undergraduates, this book is intended as a textbook for courses at
this level and includes a large number of class-tested exercises.
The prerequisites for using the book are familiarity with abstract
algebra, calculus and analysis, as covered in standard
undergraduate courses.
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