Mathematics Colloquia and Seminars

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The Fisher-Hartwig conjecture concerns the asymptotic behavior of finite Toeplitz determinants with singular symbols. One of the key ingredients in the proof of the conjecture was the exact computation of the determinants for symbols of the form \[\phi_{\alpha, \beta}(e^{i\theta}) = (2 - 2\cos \theta)^{(\alpha +\beta)/2}e^{i(\theta -\pi)(\alpha - \beta)/2}, \,\,\,\, 0 < \theta < 2\pi .\] (NOTE: if you want only text, replace in the above "for symbols of the form ..." with "for symbols with one singularity of a particular type.") This was combined with a localization technique to find more general answers. In the case of finite Wiener-Hopf operators with singular symbols the same localization techniques were developed. However, there never has been a single example where the finite Wiener-Hopf determinants could be evaluated explicitly. This problem was overcome recently by finding exact formulas for the Toeplitz and Wiener-Hopf determinants and then showing that the formulas are asymptotically equal. The formulas are expressed in terms of Fredholm determinants and are obtained by using Borodin-Okounkov type identities for the operators with ''nice'' symbols. In both cases we introduce a parameter to regularize the symbol, apply the Borodin-Okounkov identity, and then take the limit. The talk will outline the above computations, describe analogues and proofs of the Borodin-Okounkov identities for other classes of operators and, if time permits, also extensions of the original Fisher-Hartwig conjecture.