Thomas C. Spencer | |
---|---|

Born | December 24, 1946 75) | (age

Education | AB, University of California, Berkeley PhD, New York University |

Employer | Institute for Advanced Study |

Title | Professor |

Spouse | Bridget Murphy |

Awards | Henri Poincaré Prize (2015) Dannie Heineman Prize for Mathematical Physics (1991) |

**Thomas C. Spencer** (born December 24, 1946) is an American mathematical physicist, known in particular for important contributions to constructive quantum field theory, statistical mechanics, and spectral theory of random operators.^{ [1] } He earned his doctorate in 1972 from New York University with a dissertation titled *Perturbation of the Po2 Quantum Field Hamiltonian* written under the direction of James Glimm. Since 1986, he has been professor of mathematics at the Institute for Advanced Study. He is a member of the United States National Academy of Sciences,^{ [1] } and the recipient of the Dannie Heineman Prize for Mathematical Physics (joint with Jürg Fröhlich, "*For their joint work in providing rigorous mathematical solutions to some outstanding problems in statistical mechanics and field theory.*").^{ [2] }^{ [3] }

- Together with James Glimm and Arthur Jaffe he invented the cluster expansion approach to quantum field theory that is widely used in constructive field theory.
^{ [4] } - Together with Jürg Fröhlich and Barry Simon, he invented the approach of the infrared bound, which has now become a classical tool to derive phase transitions in various models of statistical mechanics.
^{ [5] } - Together with Jürg Fröhlich, he devised a 'multi-scale analysis' to provide, for the first time, mathematical proofs of: the Kosterlitz–Thouless transition,
^{ [6] }the phase transition in the one-dimensional ferromagnetic Ising model with interactions^{ [7] }and Anderson localization in arbitrary dimension.^{ [8] } - Together with David Brydges, he proved that the scaling limit of the self-avoiding walk in dimension greater or equal than 5 is Gaussian, with variance growing linearly in time.
^{ [9] }To achieve this result, they invented the technique of the lace expansion that since then has had wide application in probability on graphs.^{ [10] }

**Rudolf Haag** was a German theoretical physicist, who mainly dealt with fundamental questions of quantum field theory. He was one of the founders of the modern formulation of quantum field theory and he identified the formal structure in terms of the principle of locality and local observables. He also made important advances in the foundations of quantum statistical mechanics.

The **Berezinskii–Kosterlitz–Thouless transition** is a phase transition of the two-dimensional (2-D) XY model in statistical physics. It is a transition from bound vortex-antivortex pairs at low temperatures to unpaired vortices and anti-vortices at some critical temperature. The transition is named for condensed matter physicists Vadim Berezinskii, John M. Kosterlitz and David J. Thouless. BKT transitions can be found in several 2-D systems in condensed matter physics that are approximated by the XY model, including Josephson junction arrays and thin disordered superconducting granular films. More recently, the term has been applied by the 2-D superconductor insulator transition community to the pinning of Cooper pairs in the insulating regime, due to similarities with the original vortex BKT transition.

The **Yang–Mills existence and mass gap problem** is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 for its solution.

**Detlev Buchholz** is a German theoretical physicist. He investigates quantum field theory, especially in the axiomatic framework of algebraic quantum field theory.

In quantum field theory and statistical mechanics, the **Mermin–Wagner theorem** states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions *d* ≤ 2. Intuitively, this means that long-range fluctuations can be created with little energy cost and since they increase the entropy they are favored.

In mathematical physics, the **almost Mathieu operator** arises in the study of the quantum Hall effect. It is given by

In mathematical physics, the **quantum KZ equations** or **quantum Knizhnik–Zamolodchikov equations** or **qKZ equations** are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov equations for affine Kac–Moody algebras. They are a consistent system of difference equations satisfied by the *N*-point functions, the vacuum expectations of products of primary fields. In the limit as the deformation parameter *q* approaches 1, the *N*-point functions of the quantum affine algebra tend to those of the affine Kac–Moody algebra and the difference equations become partial differential equations. The quantum KZ equations have been used to study exactly solved models in quantum statistical mechanics.

**Jürg Martin Fröhlich** is a Swiss mathematician and theoretical physicist. He is best known for introducing rigorous techniques for the analysis of statistical mechanics models, in particular continuous symmetry breaking, and for pioneering the study of topological phases of matter using low-energy effective field theories.

**Leonid Andreevich Pastur** is a Ukrainian mathematical physicist and theoretical physicist, known in particular for contributions to random matrix theory, the spectral theory of random Schrödinger operators, statistical mechanics, and solid state physics. Currently, he heads the Department of Theoretical Physics at the B Verkin Institute for Low Temperature Physics and Engineering.

**Israel Michael Sigal** is a Canadian mathematician specializing in mathematical physics. He is a professor at the University of Toronto Department of Mathematics.

**Antti Kupiainen** is a Finnish mathematical physicist.

**Vadim V. Schechtman** is a Russian mathematician who teaches in Toulouse.

**David Chandos Brydges** is a mathematical physicist.

**Gian Michele Graf** is a Swiss mathematical physicist.

**Klaus Fredenhagen** is a German theoretical physicist who works on the mathematical foundations of quantum field theory.

**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.

**Giovanni Felder** is a Swiss mathematical physicist and mathematician, working at ETH Zurich. He specializes in algebraic and geometric properties of integrable models of statistical mechanics and quantum field theory.

**Alberto Sergio Cattaneo** is an Italian mathematician and mathematical physicist, specializing in geometry related to quantum field theory and string theory.

**Krzysztof Gawędzki** was a Polish-born French mathematical physicist.

**Stability of matter** refers to the problem of showing rigorously that a large number of charged quantum particles can coexist and form macroscopic objects, like ordinary matter. The first proof was provided by Freeman Dyson and Andrew Lenard in 1967–1968, but a shorter and more conceptual proof was found later by Elliott Lieb and Walter Thirring in 1975.

- 1 2 IAS website
- ↑ APS website
- ↑ 1991 Dannie Heineman Prize for Mathematical Physics Recipient, American Physical Society. Accessed June 24, 2011
- ↑ Glimm, J; Jaffe, A; Spencer, T (1974). "The Wightman axioms and particle structure in the quantum field model".
*Ann. of Math*.**100**(3): 585–632. doi:10.2307/1970959. JSTOR 1970959. - ↑ Fröhlich, J.; Simon, B.; Spencer, T. (1976). "Infrared bounds, phase transitions and continuous symmetry breaking".
*Comm. Math. Phys*.**50**(1): 79–95. Bibcode:1976CMaPh..50...79F. CiteSeerX 10.1.1.211.1865 . doi:10.1007/bf01608557. S2CID 16501561. - ↑ Fröhlich, J.; Spencer, T. (1981). "The Kosterlitz–Thouless transition in two-dimensional abelian spin systems and the Coulomb gas".
*Comm. Math. Phys*.**81**(4): 527–602. Bibcode:1981CMaPh..81..527F. doi:10.1007/bf01208273. S2CID 73555642. - ↑ Fröhlich, J.; Spencer, T. (1982). "The phase transition in the one-dimensional Ising model with 1/
*r*^{2}interaction energy".*Comm. Math. Phys*.**84**(1): 87–101. Bibcode:1982CMaPh..84...87F. doi:10.1007/BF01208373. S2CID 122722140. - ↑ Fröhlich, J.; Spencer, T. (1983). "Absence of diffusion in the Anderson tight binding model for large disorder or low energy".
*Comm. Math. Phys*.**88**(2): 151–184. Bibcode:1983CMaPh..88..151F. doi:10.1007/bf01209475. S2CID 121435596. - ↑ Brydges, D.; Spencer, T. (1985). "Self-avoiding walk in 5 or more dimensions".
*Comm. Math. Phys*.**97**(1–2): 125–148. Bibcode:1985CMaPh..97..125B. doi:10.1007/bf01206182. S2CID 189832287. - ↑ Slade, G. (2006).
*The lace expansion and its applications*. Lecture Notes in Mathematics. Vol. 1879. Springer. ISBN 9783540311898.

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