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In a world where we are constantly being asked to make decisions
based on incomplete information, facility with basic probability is
an essential skill. This book provides a solid foundation in basic
probability theory designed for intellectually curious readers and
those new to the subject. Through its conversational tone and
careful pacing of mathematical development, the book balances a
charming style with informative discussion. This text will immerse
the reader in a mathematical view of the world, giving them a
glimpse into what attracts mathematicians to the subject in the
first place. Rather than simply writing out and memorizing
formulas, the reader will come out with an understanding of what
those formulas mean, and how and when to use them. Readers will
also encounter settings where probabilistic reasoning does not
apply or where intuition can be misleading. This book establishes
simple principles of counting collections and sequences of
alternatives, and elaborates on these techniques to solve real
world problems both inside and outside the casino. Pair this book
with the HarvardX online course for great videos and interactive
learning: https://harvardx.link/fat-chance.
This book develops abstract homotopy theory from the categorical
perspective with a particular focus on examples. Part I discusses
two competing perspectives by which one typically first encounters
homotopy (co)limits: either as derived functors definable when the
appropriate diagram categories admit a compatible model structure,
or through particular formulae that give the right notion in
certain examples. Emily Riehl unifies these seemingly rival
perspectives and demonstrates that model structures on diagram
categories are irrelevant. Homotopy (co)limits are explained to be
a special case of weighted (co)limits, a foundational topic in
enriched category theory. In Part II, Riehl further examines this
topic, separating categorical arguments from homotopical ones. Part
III treats the most ubiquitous axiomatic framework for homotopy
theory - Quillen's model categories. Here, Riehl simplifies
familiar model categorical lemmas and definitions by focusing on
weak factorization systems. Part IV introduces quasi-categories and
homotopy coherence.
The language of -categories provides an insightful new way of
expressing many results in higher-dimensional mathematics but can
be challenging for the uninitiated. To explain what exactly an
-category is requires various technical models, raising the
question of how they might be compared. To overcome this, a
model-independent approach is desired, so that theorems proven with
any model would apply to them all. This text develops the theory of
-categories from first principles in a model-independent fashion
using the axiomatic framework of an -cosmos, the universe in which
-categories live as objects. An -cosmos is a fertile setting for
the formal category theory of -categories, and in this way the
foundational proofs in -category theory closely resemble the
classical foundations of ordinary category theory. Equipped with
exercises and appendices with background material, this first
introduction is meant for students and researchers who have a
strong foundation in classical 1-category theory.
The articles in this volume grew out of a 2019 workshop, held at
Johns Hopkins University, that was inspired by a belief that when
mathematicians take time to reflect on the social forces involved
in the production of mathematics, actionable insights result.
Topics range from mechanisms that lead to an inclusion-exclusion
dichotomy within mathematics to common pitfalls and better
alternatives to how mathematicians approach teaching, mentoring and
communicating mathematical ideas. This collection will be of
interest to students, faculty and administrators wishing to gain a
snapshot of the current state of professional norms within
mathematics and possible steps toward improvements.
In a world where we are constantly being asked to make decisions
based on incomplete information, facility with basic probability is
an essential skill. This book provides a solid foundation in basic
probability theory designed for intellectually curious readers and
those new to the subject. Through its conversational tone and
careful pacing of mathematical development, the book balances a
charming style with informative discussion. This text will immerse
the reader in a mathematical view of the world, giving them a
glimpse into what attracts mathematicians to the subject in the
first place. Rather than simply writing out and memorizing
formulas, the reader will come out with an understanding of what
those formulas mean, and how and when to use them. Readers will
also encounter settings where probabilistic reasoning does not
apply or where intuition can be misleading. This book establishes
simple principles of counting collections and sequences of
alternatives, and elaborates on these techniques to solve real
world problems both inside and outside the casino. Pair this book
with the HarvardX online course for great videos and interactive
learning: https://harvardx.link/fat-chance.
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