References for approximate representation theory (Fall 2019).

Not a complete list of references.

Starred sections / papers might make good projects, as we unlikely to fully cover these subjects in class.

Stability:

• Dan Voiculescu. Asymptotically commuting finite rank unitary operators without commuting approximants. Acta Sci. Math. (Szeged) 45 (1983), no. 1-4, 429–431.
• (*) B. Blackadar. Shape theory for C*-algebras. Math. Scand. 56 (1985), 249-275.
• Ruy Exel and Terry Loring. Almost Commuting Unitary Matrices. Proceedings of the American Mathematical Society Vol. 106, No. 4 (Aug., 1989), pp. 913-915.
• T. A. Loring. C*-Algebras Generated by Stable Relations. Journal of Functional Analysis,Volume 112, Issue 1, 15 February 1993, Pages 159-203.
• (*) T. A. Loring. Stable Relations II: Corona semiprojectivity and dimension-drop C*-algebras. Pacific Journal of Mathematics, vol 172, no. 2, 1996.
• M. B. Hastings. Making Almost Commuting Matrices Commute. https://arxiv.org/abs/0808.2474
• Matthew B. Hastings and Terry A. Loring. Almost commuting matrices, localized Wannier functions, and the quantum Hall effect. J. Math. Phys., 51(1):015214, 2010.
• Peter Friis and Mikael Rørdam, Almost commuting self-adjoint matrices—a short proof of Huaxin Lin’s theorem, J. Reine Angew. Math. 479 (1996), 121–131.
• A. I. Shtern. Roughness and Approximation of Quasi-Representations of Amenable Groups. Mathematical Notes, Vol 65, No 6, 1999.
• (*) Tatiana Shulman. Semiprojectivity of universal C*-algebras generated by algebraic elements. https://arxiv.org/abs/0810.2497
• Burger, Ozawa, and Thom. On Ulam Stability. Israel Journal of Mathematics 193 (2013), 109-129.
• Kazhdan. On $\epsilon$-representations. Israel Journal of Mathematics, vol 43, issue 4, pp 315-323.
• (*) Kazhdan and Ziegler. Approximate cohomology. Selecta Math, Vol 24, Issue 1, pp 499-509.
• W. T. Gowers and O. Hatami. Inverse and stability theorems for approximate representations of finite groups. Sbornik: Mathematics, Volume 208, Number 12.
• Moore and Russell. Approximate Representations, Approximate Homomorphisms, and Low-Dimensional Embeddings of Groups SIAM J. Discrete Math., 29(1), 182-197.
• (*) De Chiffre, Ozawa, and Thom. Operator algebraic approach to inverse and stability theorems for amenable groups. Mathematika 65 (2019) pp 98-118.
• (*) Hadwin and Shulman. Tracial stability for C*-algebras. https://arxiv.org/abs/1607.04470
• (*) Don Hadwin and Tatiana Shulman. Stability of group relations under small Hilbert-Schmidt perturbations https://arxiv.org/abs/1706.08405
• Jung, K. Math. Ann. (2007) 338: 241. https://doi.org/10.1007/s00208-006-0074-y

Finite-dimensional approximation

• (*) Tsirelson's Problem and Asymptotically Commuting Unitary Matrices Narutaka Ozawa https://arxiv.org/abs/1211.2712
• Capraro and Lupini. Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture.
• Andreas Thom. Finitary approximations of groups and their applications. Proceedings of ICM 2018.
• (*) Andreas Thom. About the metric approximation of Higman's group. Journal of Group Theory.
• Marcus De Chiffre, Lev Glebsky, Alex Lubotzky, and Andreas Thom. Stability, cohomology vanishing, and non-approximable groups. https://arxiv.org/abs/1711.10238.
• (*) Oren Becker and Alexander Lubotzky. Group stability and Property (T). https://arxiv.org/abs/1809.00632
• (*) Alexander Lubotzky, Izhar Oppenheim. Non p-norm approximated Groups. https://arxiv.org/abs/1807.06790

(*) P-stability and sofic groups

• G.Elek and E. Szabo. On sofic groups. https://arxiv.org/abs/math/0305352
• G. Elek and E. Szabo. Sofic representations of amenable groups. https://arxiv.org/abs/1010.3424
• Lev Glebsky and Luis Manuel Rivera. Almost solutions of equations in permutations. https://arxiv.org/abs/0709.1134
• Goulnara Arzhantseva and Liviu Păunescu, Almost commuting permutations are near commuting permutations, J. Funct. Anal. 269 (2015), no. 3, 745–757.
• Oren Becker, Alexander Lubotzky, and Andreas Thom. Stability and Invariant Random Subgroups. https://arxiv.org/abs/1801.08381
• Arie Levit, Alexander Lubotzky. Infinitely presented stable groups and invariant random subgroups of metabelian groups. https://arxiv.org/abs/1909.11842.
• Adrian Ioana. Stability for product groups and property (τ). https://arxiv.org/abs/1909.00282

(*) Expanders, property T, derandomization

• Lubotzky. Discrete Groups, Expanding Graphs, and Invariant Measures.
• Shachar Lovett, Cristopher Moore, Alexander Russell. Group representations that resist random sampling. Random Structures and Algorithms, Volume 47, Issue 3, 2015, pages 605-614.

(*) Applications in quantum information

• Joel J. Wallman and Joseph Emerson. Noise tailoring for scalable quantum computation via randomized compiling. Phys. Rev. A 94, 052325.
• Joel J. Wallman. Randomized benchmarking with gate-dependent noise. https://arxiv.org/abs/1703.09835
• Seth T. Merkel, Emily J. Pritchett, Bryan H. Fong. Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors. https://arxiv.org/abs/1804.05951
• Steven T. Flammia, Joel J. Wallman. Efficient estimation of Pauli channels. https://arxiv.org/abs/1907.12976
• McKague, Yang and Scarani. Robust self-testing of the singlet. Journal of Physics A: Mathematical and Theoretical, Volume 45, Number 45.
• Dimiter Ostrev, Thomas Vidick. Entanglement of approximate quantum strategies in XOR games. https://arxiv.org/abs/1609.01652
• Matthew McKague. Self-testing in parallel with CHSH. https://arxiv.org/abs/1609.09584
• Matthew Coudron and Anand Natarajan. The Parallel-Repeated Magic Square Game is Rigid. https://arxiv.org/abs/1609.06306
• Rui Chao, Ben W. Reichardt, Chris Sutherland, Thomas Vidick. Overlapping qubits. https://arxiv.org/abs/1701.01062
• Andrea Coladangelo and Jalex Stark. Robust self-testing for linear constraint system games. https://arxiv.org/abs/1709.09267
• William Slofstra, Thomas Vidick. Entanglement in non-local games and the hyperlinear profile of groups. https://arxiv.org/abs/1711.10676
• Anand Natarajan, Thomas Vidick. Robust self-testing of many-qubit states. Proc. of STOC '17, pp. 1003-1015 (2017).
• Anand Natarajan, Thomas Vidick. Two-player entangled games are NP-hard. Proc. CCC 2018, pp. 20:1-20:18.
• Anand Natarajan, Thomas Vidick. Low-degree testing for quantum states, and a quantum entangled games PCP for QMA. Proc. FOCS 2018, pp. 731-742.