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SUMMARY:Mirror symmetry and cluster algebras - Mark Gross ( University Cal
 ifornia\, San Diego)
DTSTART:20130702T130000Z
DTEND:20130702T140000Z
UID:TALK45793@talks.cam.ac.uk
CONTACT:HoD Secretary\, DPMMS
DESCRIPTION: Mirror symmetry is a subject which originated in string theor
 y in 1990\,\nand has since developed mathematically in many different dire
 ctions.\nRoughly put\, mirror symmetry gives a correspondence between diff
 erent\nalgebraic varieties\, in which certain calculations on one variety 
 (usually\ninvolving counting curves) coincides with a different type of ca
 lculation\non the other (usually computing integrals of certain forms).\nJ
 oint work with Bernd Siebert over the last 10 years has explored the\nunde
 rlying geometry of mirror symmetry\, and the structures that have\nemerged
  in this work can now be applied to diverse situation.\n\n   In this talk 
 I will discuss joint ongoing work with Keel\, Kontsevich\nand Hacking. I w
 ill make a connection between structures called\nscattering diagrams\, or 
 wall-crossing structures\, which emerged in the\nstudy of mirror symmetry\
 , and cluster algebras of Fomin and Zelevinsky\,\nwhich were developed to 
 model aspects of canonical bases of Lusztig and\nKashiwara. In particular\
 , a construction of an analogue of theta functions\,\nmotivated by the hom
 ological mirror symmetry conjecture\, allows the\nconstruction of canonica
 l bases of cluster algebras in many cases\, as\nwell as the proof of a num
 ber of conjectures concerning cluster algebras. 
LOCATION:CMS\, MR12
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