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## Mathematical Physics

**Introduction:**

The Mathematical Physics group of LPTHE is a central actor in the study of fundamental structures of physical theories, especially quantum and classical integrability, Conformal Field Theory, random matrix theory and non-perturbative aspects of Quantum Field Theory. Our work tends to focus on these mathematical aspects, with connections to several areas in Mathematics: symplectic geometry, quantum algebra, representation theory, combinatorics, algebraic geometry, probability theory, etc. It also leads to concrete applications to physical problems, such as critical exponents and correlations for critical phenomena, quantum entanglement, quantum disordered systems, or dynamical systems.

**Staff members:**

M. Bellon

Y. Ikhlef

F. Smirnov

C. Viallet

P. Zinn-Justin (on leave)

J.-B. Zuber

**PhD students:**

A. Rotaru

**Recent PhDs:** E. Russo, C. Babenko, T. Dupic, A. Garbali, P. Clavier, S. Negro

**Research themes:**

*Combinatorics & Representation Theory:*
Littlewood-Richardson coefficients

Horn´s problem

Higher-genus partitions

*1d Quantum Systems:*
Exact correlations

Entanglement entropy

*Statistical Mechanics:*
Random geometries

Critical interfaces

*Dynamical systems:*
Discrete Painlevé
"Algebraic entropy"
Discrete integrable systems

Solitons

*Fundamental QFT:*
Form factors

Perturbed CFT

Quantum Toda chain

Alien calculus

Schwinger-Dyson equations

**Interactions with Mathematics:**

*Combinatorics:*
Enumeration of partitions

Higher-genus partitions

*Algebraic Geometry:*
Quantum Riemann bilinear id

Homology of dynamical systems

*Complex analysis:*
Resummation theory

Mould calculus

*Probability Theory:*
Proofs of conformal invariance

Dimers

Rigorous Liouville QFT

*Representation Theory:*
Lie algebras/groups

Quantum groups

Cellular algebras

Horn´s problem

*Number Theory:*
Algebraic dynamical systems

Complexity