Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy theory and type theory. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak $\infty$-groupoids. Homotopy type theory brings new ideas into the very foundation of mathematics. On the one hand, there is Voevodsky's subtle and beautiful univalence axiom. The univalence axiom implies, in particular, that isomorphic structures can be identified, a principle that mathematicians have been happily using on workdays, despite its incompatibility with the "official" doctrines of conventional foundations. On the other hand, we have higher inductive types, which provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory: spheres, cylinders, truncations, localizations, etc. Both ideas are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics - and convenient machine implementations, which can serve as a practical aid tothe working mathematician. This is the Univalent Foundations program. The present book is intended as a first systematic exposition of the basics of univalent foundations, and a collection of examples of this new style of reasoning - but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant. We believe that univalent foundations will eventually become a viable alternative to set theory as the "implicit foundation" for the unformalized mathematics done by most mathematicians.
C.07 Other editorial board
Ars mathematica contemporanea. Pisanski, Tomaž (founding and editor-in-chief 2008-). [Printed ed.]. Ljubljana: Slovenian society of mathematicians, physicists and astronomers, 2008-. ISSN 1855-3966. http://amc-journal.eu/index.php/amc. [COBISS.SI-ID 239049984]. The journal was accepted to SCI.
C.04 Editorial board of an international magazine
Prof. dr. Tomaž Pisanski was elected to prestigious membershipof Academio Europeae, section informatics.
D.03 Membership in foreign/international boards/committees
V. Batagelj and A. Mrvar have received INSNA William D. Richards Jr. Software Award , 2013 at XXXIII International Sunbelt Social Network Conference. The William D. Richards, Jr. Software Award is a "lifetime achievement award" to honor individuals who have created publically available social network analysis software without which it would be impossible to study social networks.
E.02 International awards
On the back cover page: Configurations can be studied from a graph-theoretical viewpoint via the so-called Levi graphs and lie at the heart of graphs, groups, surfaces, and geometries, all of which are very active areas of mathematical exploration. In this self-contained textbook, algebraic graph theory is used to introduce groups; topological graph theory is used to explore surfaces; and geometric graph theory is implemented to analyze incidence geometries. After a preview of configurations in Chapter 1, a concise introduction to graph theory is presented in Chapter 2, followed by a geometric introduction to groups in Chapter 3. Maps and surfaces are combinatorially treated in Chapter 4. Chapter 5 introduces the concept of incidence structure through vertex colored graphs, and the combinatorial aspects of classical configurations are studied. Geometric aspects, some historical remarks, references, and applications of classical configurations appear in the last chapter. With over two hundred illustrations, challenging exercises at the end of each chapter, a comprehensive bibliography, and a set of open problems, Configurations from a Graphical Viewpoint is well suited for a graduate graph theory course, an advanced undergraduate seminar, or a self-contained reference for mathematicians and researchers.
D.11 Other
COBISS.SI-ID: 16418137