Spectral theory of hyperbolic surfaces: arithmetic surfaces and Selberg's eigenvalue conjecture                   

 

                         연세대학교, 21, 28 May, 11 June 2014

                         장소: 과학관 247


                         발표연사

                         정준혁,    카이스트
                         

                        
                         일정 
 
5월 21일 수요일
  
5월 28 일 수요일
 
오후 1:00-2:00
 
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오후 2:00-2:30

 
휴식

휴식 
 
오후 2:30-3:30
 
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6월 11일 수요일
 
오후 3:00-4:00
 
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초록
 

Selberg's eigenvalue conjecture predicts that the lowest nonzero eigenvalue of Laplacian on arithmetic (hyperbolic)  surfaces is greater than or equal to 1/4. This is equivalent to the statement that every automorphic representation of GL_2 (principal series  representations, in particular) is tempered at archimedean places, hence is a special case of Generalized Ramanujan Conjecture. In this series of lectures, I will first go over spectral theory of Laplacian on hyperbolic surfaces, and introduce Kuznetsov trace formula. When the surface is arithmetic, I'll explain how one can prove the first non-trivial bound \geq 3/16 due to Selberg, using the trace formula and Weil's bound for Kloosterman sums. In the end, to emphasize the role of arithmeticity in Selberg's eigenvalue conjecture, I'm going to construct (non-arithmetic) hyperbolic surfaces which have arbitrarily small first nonzero eigenvalues. 
  






                                     

                       



                       



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