• Ran Cui, "The Real-Quaternionic Indicator" (2016) pdf
  • A shorter version of the first part of my thesis.

  • (Thesis) Ran Cui, "The Real-Quaternionic Indicator of Irreducible Self-Conjugate Representations of Real Reductive Algebraic Groups and A Comment on the Local Langlands Correspondence of GL(2,F)." (2016) pdf
  • In this thesis, we establish a relation between the Real-Quaternionic indicator and the Frobenius-Schur indicator for irreducible self-conjugate (g,K)-modules with real infinitesimal characters. The main tool we used to prove this relation is the c-invariant Hermitian form. This relation is given by the strong real form of G. We also provide formulas for the strong real forms for simple group G of some types.

    We also get into the subject of the local Langlands correspondence of GL(2,F). Many books has description of how to construct this correspondence, some using the theta correspondence such as "The Local Langlands Conjecture for GL(2)" by Bushnell and Henniart; and "Automorphic Forms and Representations" by Bump. We give a detailed computation of how the construction is done that is less circuitous. We use the full strength of the theta correspondence to prove the construction is well defined.

  • Ran Cui, "Explicit Construction of Local Langlangds Correspondence of GL(2,F) Using Theta Correspondence." (2014) arXiv
  • This paper is the second part of the thesis. It is available on arXiv.

  • Ran Cui, "Tamagawa Number for SL(2)" (2013) pdf
  • This is my notes on the Tamagawa number of SL(2). It contains the necessary theory for computing the Tamagawa number and detailed computations.

  • Ran Cui, "A Survey on the Finite Harmonic Oscillator and its Applications." (2012) pdf
  • A finite oscillator system was introduced by Gurevich, Hadani and Sochen. This is a survey of how the system is constructed using Weil representation on the group SL(2,Fp) and its applications on discrete radar and CDMA system. Finally, explicit algorithms for computing the finite split and non-split oscillator systems are described.