David Spivak

Last updated
David Isaac Spivak
David MIT 20190110.jpg
Spivak in December 2019
Born
Los Angeles, California
NationalityAmerican
Alma mater
Known for Ologs
Scientific career
Fields Mathematics
Category theory
Applied category theory
Institutions Topos Institute

David Isaac Spivak is an American mathematician and senior scientist at the Topos Institute. [1] He has worked on applications of category theory, in particular ologs and operadic compositionality of dynamical systems. He authored and coauthored the introductory texts on category theory and its applications, Category Theory for the Sciences and An Invitation to Applied Category Theory.

Contents

Early life and education

Spivak received his PhD in mathematics from UC Berkeley in 2007 under the supervision of Peter Teichner and Jacob Lurie. [2] His thesis was on derived manifolds, [3] Spivak worked as a postdoc at the University of Oregon and Massachusetts Institute of Technology. [4]

Work

Spivak and Robert Kent developed a human-readable categorical system of knowledge representation called ologs. [5] These were applied, in a series of collaborations with the materials scientist Markus Buehler, to different problems in that materials science. [6] [7] [8] Ologs have been also used by researchers at NIST. [9] The goal of ologs, and of Spivak's book, was to show that category theory can be made relatively easy and thus be understood by a wider audience. Piet Hut endorsed the book saying, "This is the first, and so far the only, book to make category theory accessible to non-mathematicians." [10]

Spivak has also studied dynamical systems and operads. [11] [12] [13]

Spivak and Brendan Fong wrote a book that summarizes the developments in applied category theory for a wide audience, and started a nonprofit applied category theory research institute called Topos Institute, located in Berkeley, California. [14]

Spivak is an editor of a diamond open access journal, Compositionality. [15]

Bibliography

Related Research Articles

<span class="mw-page-title-main">Category theory</span> General theory of mathematical structures

Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in almost all areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.

In mathematics, a bicategory is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative up to an isomorphism. The notion was introduced in 1967 by Jean Bénabou.

Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.

<span class="mw-page-title-main">Markus J. Buehler</span> American materials scientist and engineer

Markus J. Buehler is an American materials scientist and engineer at the Massachusetts Institute of Technology (MIT), where he holds the endowed McAfee Professorship of Engineering chair. He is a member of the faculty at MIT's Department of Civil and Environmental Engineering, where he directs the Laboratory for Atomistic and Molecular Mechanics (LAMM), and also a member of MIT's Center for Computational Science and Engineering (CCSE) in the Schwarzman College of Computing. His scholarship spans science to art, and he is also a composer of experimental, classical and electronic music, with an interest in sonification. He has given several TED talks about his work.

Categorical set theory is any one of several versions of set theory developed from or treated in the context of mathematical category theory.

String diagrams are a formal graphical language for representing morphisms in monoidal categories, or more generally 2-cells in 2-categories. They are a prominent tool in applied category theory. When interpreted in the monoidal category of vector spaces and linear maps with the tensor product, string diagrams are called tensor networks or Penrose graphical notation. This has led to the development of categorical quantum mechanics where the axioms of quantum theory are expressed in the language of monoidal categories.

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology, where one studies algebraic invariants of spaces, such as the fundamental weak ∞-groupoid.

<span class="mw-page-title-main">Category of relations</span>

In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms.

<span class="mw-page-title-main">Dennis Sullivan</span> American mathematician (born 1941)

Dennis Parnell Sullivan is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the Graduate Center of the City University of New York and is a distinguished professor at Stony Brook University.

In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.

Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably monoidal category theory. The primitive objects of study are physical processes, and the different ways that these can be composed. It was pioneered in 2004 by Samson Abramsky and Bob Coecke. Categorical quantum mechanics is entry 18M40 in MSC2020.

<span class="mw-page-title-main">Olog</span>

The theory of ologs is an attempt to provide a rigorous mathematical framework for knowledge representation, construction of scientific models and data storage using category theory, linguistic and graphical tools. Ologs were introduced in 2012 by David Spivak and Robert Kent.

Igor Rivin is a Russian-Canadian mathematician, working in various fields of pure and applied mathematics, computer science, and materials science. He was the Regius Professor of Mathematics at the University of St. Andrews from 2015 to 2017, and was the chief research officer at Cryptos Fund until 2019. He is doing research for Edgestream LP, in addition to his academic work.

Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras, simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory of singular algebraic varieties and cotangent complexes in deformation theory, among the other applications.

Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations. As such it is closely related to the concept of metamodeling, with applications in all areas of mathematical modelling.

Applied category theory is an academic discipline in which methods from category theory are used to study other fields including but not limited to computer science, physics, natural language processing, control theory, probability theory and causality. The application of category theory in these domains can take different forms. In some cases the formalization of the domain into the language of category theory is the goal, the idea here being that this would elucidate the important structure and properties of the domain. In other cases the formalization is used to leverage the power of abstraction in order to prove new results about the field.

Charles Waldo Rezk is an American mathematician, specializing in algebraic topology and category theory.

<span class="mw-page-title-main">Dan Burghelea</span> Romanian-American mathematician

Dan Burghelea is a Romanian-American mathematician, academic, and researcher. He is an Emeritus Professor of Mathematics at Ohio State University.

DisCoCat is a mathematical framework for natural language processing which uses category theory to unify distributional semantics with the principle of compositionality. The grammatical derivations in a categorial grammar are interpreted as linear maps acting on the tensor product of word vectors to produce the meaning of a sentence or a piece of text. String diagrams are used to visualise information flow and reason about natural language semantics.

References

  1. ↑ "Team". Topos Institute . Retrieved March 30, 2023.
  2. ↑ David Spivak at the Mathematics Genealogy Project
  3. ↑ Spivak, David I. (2008). "Derived Smooth Manifolds". Duke Mathematical Journal. 153: 55–128. arXiv: 0810.5174 . Bibcode:2008PhDT.......449S. CiteSeerX   10.1.1.244.3704 . doi:10.1215/00127094-2010-021. S2CID   18483726.
  4. ↑ "David I. Spivak" (PDF). David Spivak. Retrieved July 10, 2023.
  5. ↑ Spivak, David I.; Kent, Robert E. (31 January 2012). "Ologs: A Categorical Framework for Knowledge Representation". PLOS ONE . 7 (1): e24274. arXiv: 1102.1889 . Bibcode:2012PLoSO...724274S. doi: 10.1371/journal.pone.0024274 . PMC   3269434 . PMID   22303434.
  6. ↑ Brehm, Denise (8 December 2011). "Researchers link patterns seen in spider silk, melodies". news.mit.edu.
  7. ↑ Chandler, David L. (28 November 2012). "The music of the silks". news.mit.edu.
  8. ↑ Damrad, Kelsey (11 September 2015). "A new molecular design approach". news.mit.edu.
  9. ↑ Padi, Sarala; Breiner, Spencer; Subrahmanian, Eswaran; Sriram, Ram D. (June 2018). "Modeling and Analysis of Indian Carnatic Music Using Category Theory". IEEE Transactions on Systems, Man, and Cybernetics: Systems. 48 (6): 967–981. doi:10.1109/TSMC.2016.2631130. S2CID   21722758.
  10. ↑ Spivak, David I. (2014). Category Theory for the Sciences. MIT Press. ISBN   978-0-262-02813-4.[ page needed ]
  11. ↑ Spivak, David I.; Tan, Joshua (4 September 2016). "Nesting of dynamical systems and mode-dependent networks". Journal of Complex Networks: cnw022. doi:10.1093/comnet/cnw022.
  12. ↑ Giesa, Tristan; Jagadeesan, Ravi; Spivak, David I.; Buehler, Markus J. (September 2015). "Matriarch: A Python Library for Materials Architecture". ACS Biomaterials Science & Engineering. 1 (10): 1009–1015. doi:10.1021/acsbiomaterials.5b00251. PMC   4996638 . PMID   27570830.
  13. ↑ Spivak, David I.; Ernadote, Dominique; Hammammi, Omar (2016). "Pixel matrices: An elementary technique for solving nonlinear systems". 2016 IEEE International Symposium on Systems Engineering (ISSE). pp. 1–5. arXiv: 1605.00190 . doi:10.1109/SysEng.2016.7753120. ISBN   978-1-5090-0793-6. S2CID   1156200.
  14. ↑ Fong, Brendan; Spivak, David I. (2019). An Invitation to Applied Category Theory: Seven Sketches in Compositionality. Cambridge University Press. doi:10.1017/9781108668804. ISBN   978-1-108-66880-4. S2CID   199139551.[ page needed ]
  15. ↑ Editorial Board, Compositionality. Accessed August 16, 2019.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.