Deligne's conjecture on Hochschild cohomology

Last updated

In deformation theory, a branch of mathematics, Deligne's conjecture is about the operadic structure on Hochschild cochain complex. Various proofs have been suggested by Dmitry Tamarkin, [1] [2] Alexander A. Voronov, [3] James E. McClure and Jeffrey H. Smith, [4] Maxim Kontsevich and Yan Soibelman, [5] and others, after an initial input of construction of homotopy algebraic structures on the Hochschild complex. [6] [7] It is of importance in relation with string theory.

Contents

See also

Related Research Articles

<span class="mw-page-title-main">Maxim Kontsevich</span> Russian and French mathematician (born 1964)

Maxim Lvovich Kontsevich is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami. He received the Henri Poincaré Prize in 1997, the Fields Medal in 1998, the Crafoord Prize in 2008, the Shaw Prize and Fundamental Physics Prize in 2012, and the Breakthrough Prize in Mathematics in 2014.

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.

In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad , one defines an algebra over to be a set together with concrete operations on this set which behave just like the abstract operations of . For instance, there is a Lie operad such that the algebras over are precisely the Lie algebras; in a sense abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations.

In category theory, a weak n-category is a generalization of the notion of strict n-category where composition and identities are not strictly associative and unital, but only associative and unital up to coherent equivalence. This generalisation only becomes noticeable at dimensions two and above where weak 2-, 3- and 4-categories are typically referred to as bicategories, tricategories, and tetracategories. The subject of weak n-categories is an area of ongoing research.

John Willard Morgan is an American mathematician known for his contributions to topology and geometry. He is a Professor Emeritus at Columbia University and a member of the Simons Center for Geometry and Physics at Stony Brook University.

<span class="mw-page-title-main">Jack Morava</span> American mathematician

Jack Johnson Morava is an American homotopy theorist at Johns Hopkins University.

Steve Shnider is a retired professor of mathematics at Bar Ilan University. He received a PhD in Mathematics from Harvard University in 1972, under Shlomo Sternberg. His main interests are in the differential geometry of fiber bundles; algebraic methods in the theory of deformation of geometric structures; symplectic geometry; supersymmetry; operads; and Hopf algebras. He retired in 2014.

<span class="mw-page-title-main">Noncommutative algebraic geometry</span>

Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them.

<span class="mw-page-title-main">Yan Soibelman</span> Russian mathematician

Iakov (Yan) Soibelman born 15 April 1956 is a Russian American mathematician, professor at Kansas State University, member of the Kyiv Mathematical Society (Ukraine), founder of Manhattan Mathematical Olympiad.

<span class="mw-page-title-main">Christopher Deninger</span> German mathematician

Christopher Deninger is a German mathematician at the University of Münster. Deninger's research focuses on arithmetic geometry, including applications to L-functions.

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.

In mathematics, the Hodge–de Rham spectral sequence is an alternative term sometimes used to describe the Frölicher spectral sequence. This spectral sequence describes the precise relationship between the Dolbeault cohomology and the de Rham cohomology of a general complex manifold. On a compact Kähler manifold, the sequence degenerates, thereby leading to the Hodge decomposition of the de Rham cohomology.

In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra is a generalisation of the concept of a differential graded Lie algebra. To be a little more specific, the Jacobi identity only holds up to homotopy. Therefore, a differential graded Lie algebra can be seen as a homotopy Lie algebra where the Jacobi identity holds on the nose. These homotopy algebras are useful in classifying deformation problems over characteristic 0 in deformation theory because deformation functors are classified by quasi-isomorphism classes of -algebras. This was later extended to all characteristics by Jonathan Pridham.

This is a glossary of properties and concepts in symplectic geometry in mathematics. The terms listed here cover the occurrences of symplectic geometry both in topology as well as in algebraic geometry. The glossary also includes notions from Hamiltonian geometry, Poisson geometry and geometric quantization.

<span class="mw-page-title-main">Ralph Kaufmann</span> German mathematician

Ralph Martin Kaufmann is a German mathematician working in the United States.

Gabriele Vezzosi is an Italian mathematician, born in Florence (Italy). His main interest is algebraic geometry.

Anton Yurevich Alekseev is a Russian mathematician.

In mathematics, an algebra such as has multiplication whose associativity is well-defined on the nose. This means for any real numbers we have

<span class="mw-page-title-main">Alexander A. Voronov</span> Russian-American mathematician

Alexander A. Voronov is a Russian-American mathematician specializing in mathematical physics, algebraic topology, and algebraic geometry. He is currently a Professor of Mathematics at the University of Minnesota and a Visiting Senior Scientist at the Kavli Institute for the Physics and Mathematics of the Universe.

In mathematics, a piecewise algebraic space is a generalization of a semialgebraic set, introduced by Kontsevich and Soibelman. The motivation was for the proof of Deligne's conjecture on Hochschild cohomology. Hardt, Lambrechts, Turchin, and Volić later developed the theory.

References

  1. Tamarkin, Dmitry E. (1998). "Another proof of M. Kontsevich formality theorem". arXiv: math/9803025 .
  2. Hinich, Vladimir (2003). "Tamarkin's proof of Kontsevich formality theorem". Forum Math. 15 (4): 591–614. arXiv: math/0003052 . doi:10.1515/form.2003.032. S2CID   220814.
  3. Voronov, Alexander A. (2000). "Conférence Moshé Flato 1999". Conférence Moshé Flato 1999, Vol. II (Dijon). Dordrecht: Kluwer Acad. Publ. pp. 307–331. arXiv: math/9908040 . doi:10.1007/978-94-015-1276-3_23. ISBN   978-90-481-5551-4.
  4. McClure, James E.; Smith, Jeffrey H. (2002). "A solution of Deligne's Hochschild cohomology conjecture". Recent progress in homotopy theory (Baltimore, MD, 2000). Providence, RI: Amer. Math. Soc. pp. 153–193. arXiv: math/9910126 .
  5. Kontsevich, Maxim; Soibelman, Yan (2000). "Deformations of algebras over operads and the Deligne conjecture". Conférence Moshé Flato 1999, Vol. I (Dijon). Dordrecht: Kluwer Acad. Publ. pp. 255–307. arXiv: math/0001151 .
  6. Getzler, Ezra; Jones, J. D. S. (1994). "Operads, homotopy algebra and iterated integrals for double loop spaces". arXiv: hep-th/9403055 .
  7. Voronov, A. A.; Gerstenhaber, M. (1995). "Higher operations on the Hochschild complex". Funct. Anal. Its Appl. 29: 1–5. doi:10.1007/BF01077036. S2CID   121740728.

Further reading