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2. Riemann-Hilbert Problems


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      Problem 2.1.

      [Tamara Grava] Given \(Q\), produce \(Q(z^d)\) with multiple disjoint support sets for the droplet. Study the soft Riemann-Hilbert problem for orthogonal polynomials.


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          Problem 2.2.

          [Pavel Blaher] Study potential theory for \(k \times k\) Riemann-Hilbert problems for \(k>2\) when multiple orthogonal polynomials are involved.


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              Problem 2.3.

              [Manuela Girotti] In general, can you pose matrix Riemann Hilbert problems for KdV?


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                  Problem 2.4.

                  [Aron Wennman] For planar orthogonal polynomials, we know that \[\overline{\partial} Y=\overline{Y} \begin{bmatrix} 0 & w \\ 0 & 0 \\ \end{bmatrix}\] and \[\int p_j\overline{p_k}wdA=h_j\delta_{j,k}.\] Is there an analogue of small norm theory of such \(\overline{\partial}\)-problems (i.e. \(\overline{\partial} A=\overline{A} W +AV\)?) Or small norm theory for \(A_{+}-A_{-}=\overline{A_{-}}V_C+A_{-}V_n\)?


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                      Problem 2.5.

                      [Ken McLaughlin] Analyze the condition given by \[\overline{\partial}\phi=\overline{H_n}e^{-NQ},\] where \(Q=|z|^2+2t\text{Re}(z^2)\) and where \(\overline{H_n}\) is Hermite.


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                          Problem 2.6.

                          [Tomas Berggren] Consider \[\prod_{i=1}^N \begin{bmatrix} 1 & a_iz \\ b_i & 1 \\ \end{bmatrix},\] for constants \(a_i, b_i\). Suppose Wiener-Hopf factorization has no partial indices. Study the behavior as \(N \rightarrow \infty\). The asymptotics are known if the \(a_i\)’s and \(b_i\)’s are periodic. What happens in the more general case?


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                              Problem 2.7.

                              [Haakan Hedemaim] Consider \[Z_N=\int_{\Gamma} \cdots \int_{\Gamma} \prod_{i \leq j < k \leq N} |z_k-z_j|^{2\beta} \prod_{j=1}^N\Big(\frac{1}{1+|z_j|^2}\Big)^{(N-1)\beta+1} d|z_1| \cdots d|z_N|.\] Study \(Z_N\) as \(N \rightarrow \infty\), where \(\Gamma\) is a smooth loop on the Riemann sphere.


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                                  Problem 2.8.

                                  [Maksim Kosmakov] The discrete Bessel process is connected to Poissonized Plancherel measure. Consider the distance of the largest particle: \(\mathbb{P}[Q_{\text{max}}\leq x]=q(x)\), where \(q\) solves the cylindrical Toda equation. Take the limit to get cylindrical KdV. Is there a connection to Riemann-Hilbert Problems?


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                                      Problem 2.9.

                                      [Estelle Basor] Formulate and then analyze general finite matrix symbol with the analog of zeros and jumps.

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