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1. Toeplitz Determinants and Toeplitz Operators


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      Problem 1.1.

      [Maurice Duits] The Borodin-Okounkov-Case-Geronimo identity is given by \[\text{det}(T_N(e^f))=(e^{N\hat{f}(0)+\sum k\hat{f}(k)\hat{f}(-k)})\text{det}(1-k)\] Find a useful adjustment to the right-hand side so that an identity holds in the Fisher-Hartwig case.


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          Problem 1.2.

          [Maksim Kosmakov] Investigate the connection between the \(\tau\) function from Painlevè equations and block Toeplitz determinants. Seek explicit examples in which \(\tau\) function asymptotics is known. Find the block Toeplitz determinant. What is it and what do we learn? Use Riemann-Hilbert to connect this to Fuchsian Riemann-Hilbert problems.


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              Problem 1.3.

              [Torsten Ehrhardt] Study merging singularities or double scaling limits for Toeplitz determinants with two symbols. Compare \(\text{det}(T_n(a_{\lambda}b_{\lambda}))\) and \(\text{det}T_n(a_{\lambda})T_n(b_{\lambda})\). Under what conditions does the ratio converge?

                •     Please see C.A. Berger and L.A. Coburn’s 1994 article "Heat Flow and Berlin-Toeplitz Estimates" for background on this conjecture.

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                  Conjecture 1.4.

                  [Michael Hitrik] (Berger-Coburr 1994): On the Fock/Bergman space, define the projection \(\prod:L^2(\mathbb{C},e^{-|z|^2}dA) \rightarrow \mathscr{F}_{\mathbb{C}}\). Take \(p: \mathbb{C} \rightarrow \mathbb{C}\) and define \(T_p(f)=\prod(p(z)f(z))\). \(T_p\) is bounded on the Fock space iff for all \((x,y) \in \mathbb{R}^2\), \(u_t=\Delta u,\) \(u\Big|_{t=0}=p\), and \(u(\frac{1}{4},x,y)\) are bounded.


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                      Problem 1.5.

                      [Roozbeh Gharakhloo] The determinant of Toeplitz \(+\) Handle is expressible in terms of the \(4 \times 4\) Riemann-Hilbert problem for Szego type symbols. The obstacle to asymptotic analysis is the existence of a solution to model problems. If \(d \tilde{d}=1\) on \(\{|z|=1\}\), where \(d=\frac{\phi(z)}{w(z)}\) and \(\tilde{d}=\frac{\phi(1/z)}{w(1/z)}\), then it is solvable. What if this doesn’t hold?

                          Cite this as: AimPL: Riemann-Hilbert problems, Toeplitz matrices, and applications, available at http://aimpl.org/riemhilberttoeplitz.