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**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 entitled *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] }

In mathematical physics, **constructive quantum field theory** is the field devoted to showing that quantum theory is mathematically compatible with special relativity. This demonstration requires new mathematics, in a sense analogous to Newton developing calculus in order to understand planetary motion and classical gravity. Weak, strong, and electromagnetic forces of nature are believed to have their natural description in terms of quantum fields.

**Statistical mechanics** is one of the pillars of modern physics. It is necessary for the fundamental study of any physical system that has a large number of degrees of freedom. The approach is based on statistical methods, probability theory and the microscopic physical laws.

In mathematics, **spectral theory** is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.

- 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] }

**James Gilbert Glimm** is an American mathematician, former president of the American Mathematical Society, and distinguished professor at Stony Brook University. He has made many contributions in the areas of pure and applied mathematics.

**Arthur Michael Jaffe** is an American mathematical physicist at Harvard University, where in 1985 he succeeded George Mackey as the Landon T. Clay Professor of Mathematics and Theoretical Science.

In statistical mechanics, the **cluster expansion** is a power series expansion of the partition function of a statistical field theory around a model that is a union of non-interacting 0-dimensional field theories. Cluster expansions originated in the work of Mayer & Montroll (1941). Unlike the usual perturbation expansion, it converges in some non-trivial regions, in particular when the interaction is small.

- 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] }

**Jürg Martin Fröhlich** is a Swiss mathematician and theoretical physicist.

The **Berezinskii–Kosterlitz–Thouless transition** is a phase transition in the two-dimensional (2-D) XY model. 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.

In condensed matter physics, **Anderson localization** is the absence of diffusion of waves in a *disordered* medium. This phenomenon is named after the American physicist P. W. Anderson, who was the first to suggest that electron localization is possible in a lattice potential, provided that the degree of randomness (disorder) in the lattice is sufficiently large, as can be realized for example in a semiconductor with impurities or defects.

The **classical XY model** (sometimes also called **classical rotor****model** or **O model**) is a lattice model of statistical mechanics. It is the special case of the *n*-vector model for *n* = 2.

In mathematical physics, the **Yang–Mills existence and mass gap problem** is an unsolved problem and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 to the one who solves it.

The **Thirring model** is an exactly solvable quantum field theory which describes the self-interactions of a Dirac field in (1+1) dimensions.

**Detlev Buchholz** is a theoretical physicist. He investigates quantum field theory, especially algebraic quantum field theory. His contributions include the concept of infraparticles.

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.

**Roland Lvovich Dobrushin** was a mathematician who made important contributions to probability theory, mathematical physics, and information theory.

An important question in statistical mechanics is the dependence of model behaviour on the dimension of the system. The **shortcut model** was introduced in the course of studying this dependence. The model interpolates between discrete regular lattices of integer dimension.

**Subir Sachdev** is Herchel Smith Professor of Physics at Harvard University specializing in condensed matter. He was elected to the U.S. National Academy of Sciences in 2014, and received the Lars Onsager Prize from the American Physical Society and the Dirac Medal from the ICTP in 2018.

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.

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

**Quantum simulators** permit the study of quantum systems that are difficult to study in the laboratory and impossible to model with a supercomputer. In this instance, simulators are special purpose devices designed to provide insight about specific physics problems.

**Alexander Nikolaevich Varchenko** is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics.

The **conformal bootstrap** is a non-perturbative mathematical method to constrain and solve conformal field theories, i.e. models of particle physics or statistical physics that exhibit similar properties at different levels of resolution.

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

**Vladimir Georgievich Turaev** is a Russian mathematician, specializing in topology.

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

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

**Claude Georges Itzykson**, was a French theoretical physicist who worked in quantum field theory and statistical mechanics.

- 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. - ↑ 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. - ↑ 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. - ↑ 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. - ↑ 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. - ↑ Slade, G. (2006).
*The lace expansion and its applications*. Lecture Notes in Mathematics.**1879**. Springer. ISBN 9783540311898.

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