著名数学家美国伊利诺斯大学Berndt 教授讲课日程安排

16 June, afternoon 15: 00-16: 00 : 地点: 1414

Ramanujan's Lost Notebook with particular attention to the
Rogers--Ramanujan and Enigmatic Continued Fractions

Abstract: In the spring of 1976, George Andrews visited the library
at Trinity College, Cambridge, and found a sheaf of 138 pages
containing approximately 650 unproved claims of Ramanujan. In view of the fame of Ramanujan's notebooks, Andrews called his finding "Ramanujan's Lost Notebook." I will provide a history and description of the lost notebook. I will then give a survey on several entries of the lost notebook pertaining to the Rogers-Ramanujan
and "enigmatic" continued fractions.



17 June, morning 9:30-10:30: 地点： 1４１４

Ramanujan's Contributions to Eisenstein Series Especially in his Lost Notebook

Eisenstein series are the building blocks of modular forms; in particular, every analytic modular form on the full modular group can be represented as a polynomial in two particular Eisenstein series. For Ramanujan, the primary Eisenstein series were, in his notation, $ P(q), Q(q)$, and $ R(q)$. We provide a survey of many of Ramanujan's discoveries about Eisenstein series; most of the theorems are found in his lost notebook. Some of the topics examined are formulas for the power series coefficients of certain quotients of Eisenstein series, the role of Eisenstein series in proving congruences for the partition function $ p(n)$, representations of Eisenstein series as sums of quotients of Dedekind eta-functions, a family of infinite series represented by polynomials in $ P, Q$, and $ R$, and approximations and exact formulas for $ pi $ arising from Eisenstein series.


17 June， after 15：00－16：00： 地点：１４１４


THE FIVE STRANGEST, MOST FASCINATING, MOST INTERESTING RESULTS IN RAMANUJAN'S LOST NOTEBOOK (IN THE SPEAKER'S MOST HUMBLE OPINION)

ABSTRACT: Many of Ramanujan's results, especially from his lost notebook, are so strange and surprising that it would seem that no one else, either in the present or the future, would have had the foresight to discover them. Five entries from Ramanujan's lost notebook have been chosen for presentation and detailed discussion. Each of them is surprising. All have been proved, except for one (as of this writing). At the conclusion of the lecture, members of the audience will be asked to rank on supplied paper ballots their choices from 1 to 5 as to which are the strangest, most fascinating, and most interesting.

20 June， morning 9:30-10:30: 地点： 1414

Ramanujan's Forty Identities for the Rogers-Ramanujan Functions

Abstract: The Rogers-Ramanujan identities are among the most famous
identities in combinatorics. Late in his stay in England, Ramanujan
derived 40 further identities relating the two Rogers-Ramanujan functions
at different arguments. Although almost all of the forty identities have now been proved, principally by L. J. Rogers, G. N. Watson, D. Bressoud, and A. J. F. Biagioli, an impenetrable fog still lies over the ideas which
led Ramanujan to derive these identities. In the past four years, the speaker and several of his former doctoral students have been attempting to find proofs in the spirit of Ramanujan's mathematics. At this moment, we have proofs of thirty-five of the identities that could have been given by Ramanujan. In this mostly expository talk, the various methods that have been used to prove the forty identities are discussed. It is possible that Ramanujan used asymptotic analysis of the Rogers-Ramanujan functions to discover, but not to prove, the identities. We also describe the ideas behind this approach.