There is a long history of interaction between number theory and combinatorics. In the past two decades, deep results in automorphic forms and number theory were used to construct (optimal) expanders, which are known to have wide applications in computer science and communication networks. These techniques were generalized to construct higher dimensional analogues. Very recently, zeta functions for graphs have been extended to complexes. They contain topological and spectral information of the combinatorial objects so that the Riemann hypothesis is satisfied if and only if the object is spectrally extremal. Furthermore, exciting developments in arithmetic combinatorics in recent years provide new tools to construct families of good expanders, which in turn are used to obtain deep number theoretic results. At the same time, the concept of expansion is extended in group theory and computer science to a different new context.
A workshop on graphs and arithmetic will take place March 8-12, 2010, at CRM, Montreal, Canada, to review recent exciting developments in expanders and number theory. The focus will be on the interconnections between combinatorics, group theory and number theory. Both theories and applications will be emphasized. Seventeen invited speakers are from the US. To cover the expenses of the invited speakers and partially support graduate students and recent doctorates, support from the NSF is sought to supplement the funds committed by the CRM.
organized by Eyal Goren (McGill), Andrew Granville (Montr'eal) and PI, held March 8-12, 2010, in CRM, Montr'eal, Canada, sponsored by CRM and NSF. $15437.16 was used to support 10 speakers from US. Several us speakers could not attend as planned. Intellectual Merit The connections between graphs and arithmetic emerged from several directions, via explicit constructions, uniformization, and arithmetic properties of subgroups of Lie groups. All these results rest on spectral or structural properties of algebraic groups, where the crucial input is often supplied by number theory and algebraic geometry. The development of expander graphs has benefited from number theory, geometry, group theory, and theoretical computer science; in return the theory of expanders has also fertilized these fields. The purpose of this workshop was to review recent exciting interactions among these areas. Main themes of the talks were: (1) Ramanujan complexes and Zeta functions of complexes, (2) expansion in finite simple groups and in Lie groups, (3) connection between expanders and geometry, (4) affine and combinatorial sieve methods, and (5) spectral gap and applications. Broader Impacts This workshop was attended by 52 participants from 13 countries. It gave a panorama view of the subject and important emerging directions. A one month course preceded the workshop. The invited speakers from US included 4 students, one postdoc, and 4 female. (II) Permitted by NSF, the remaining amount ($9562) was used to partially support 5 US speakers to attend the International Conference on Galois Representations, Automorphic Forms and Shimura Varieties, June 20-23, 2011, held in NCTS, Hsinchu, Taiwan. This conference was organized by the PI and KingFai Lai (National Sun Yet San Univ.), and cosponsored by NCTS and NSF. Intellectual Merit The Langlands philosophy predicts a connection between Galois representations and automorphic forms. Most of the progress has been from automorphic forms to Galois representations, and Shimura varieties played an important role in establishing such correspondences. The remarkable breakthrough by Wiles and Taylor-Wiles initiated the reverse direction. Tremendous progress has been made since then. The purpose of this conference is to show recent advances in this area from theoretical and computational aspects. Areas covered include: 1. values of L-functions, 2. Shimura varieties, 3. deformation of Galois representations, 4. Galois representations and noncongruence modular forms, 5. trace formula, 6. Galois representations and transcendence, and 7. differential equations and modular forms. Broader Impacts 47 people from 7 countries, including 9 speakers and 4 students from US, attended this conference. It was preceded by 2 short courses and followed by a one-day workshop. The conference talks showed interesting connections among various topics; the one-day workshop revealed a promising new development on congruences of Atkin-Swinnerton-Dyer type. Conference talks are posted in http://math.cts.nthu.edu.tw/Mathematics/GRAFSV2011.htm.
