Recent Research Talks in the UTC Department of Mathematics Colloquium Series

Since 1999, the department has sponsored the Colloquium Series. This provides both UTC math faculty as well as guest speakers the opportunity to highlight their most recent research in the field of mathematics.

Fall 2025

Vladimir Vladicic

• Date/Time: Fri., Sept. 5 at 2:30PM
• Location: Lupton 393
• Title: Inverse spectral problems: From Ambarzumian’s theorem to theorem of uniqueness
• Abstract: Inverse spectral theory its roots back to classical results such as Borg’s two spectra theorem and Ambarzumian’s theorem, which provide foundational insights into how the spectrum of a Sturm-Liouville operators determines its potential.
  In his famous paper ”Uber eine Frage der Eigenwerttheeorie” from (1929) Ambarzumian describes the exceptional case in wich the spectrum of boundary value problem generated with Sturm-Liouville operator under Neumann boundary conditions uniquely determines potential. In general, the specification of the spectrum does not uniquely determined potential. 
  Borg(1946) proved Theorem of uniqueness from two spectra of the boundary value problem defined with Sturm-Liouville operator under Robin boundary conditions. 
  In addition to this result, many mathematicians contributed to the determination potential functions or proved the uniqueness of the potential based on other spectral characteristics like spectral function, spectral data or Weyl function and we can say that inverse problem for Sturm-Liouville differential operators is fully solved.
  One of the most important direction for further research was BVPs generated with a Sturm-Liouville type differential equation with a constant delay and one of the first resulat was given by Freiling and Yurko (2012) and they proved Ambarzumian type theorem for differential operators defined with Sturm-Liouville equation with constant delay under Dirichlet-Neumann boundary conditions from two spectra. Compared with Ambarzumian’s theorem it could be expected that Borg-type inverse problem for this BVP will be more complicated than in the case of classical Sturm-Liouville operator. In several papers authors solve Borg-type inverse spectral problems for this class of the boundary value problems and they give complete answer for which value of the delay Theorem of uniqueness from two spectra is true or not true.
  There has recently been increasing interest in boundary value problems generated by Sturm–Liouville operators with more than one constant delay. In the case with two constant delays an attempt was made to answer the question of how many spectra are needed for Ambarzumian’s theorem to be correct, but until now we don’t have answer. Also, we don’t have complete
  answer when potential function is unique ordered.
  The focus of this research is on the boundary value problems generated with differential equation Sturm-Liouville-type two constant delay. The Ambarzumian type theorem for operator with two constant delays from four spectra will be proven and a partial answer to the question of when the Theorem of uniqueness is true and when it is not will be given.

Spring 2026

Coming Soon

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