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SUMMARY:How Much Does Homology Really Know? - Professor Thomas Nikolaus (U
 niversität Münster)
DTSTART:20260428T150000Z
DTEND:20260428T160000Z
UID:TALK237184@talks.cam.ac.uk
CONTACT:HoD Secretary\, DPMMS
DESCRIPTION:Homology is one of the most fundamental tools in topology: it 
 assigns to every space  X a collection of groups H_*(X) that capture aspec
 ts of its shape. But there is a well-known limitation - different spaces c
 an share exactly the same homology while being fundamentally different fro
 m a topological point of view.\n\n\nSo what is missing?\n\n\nIn this talk\
 , we explore how much more information is hidden just beneath the surface.
  Instead of looking only at homology\, we return to the richer object it c
 omes from: the singular chains of a space. These chains carry additional s
 tructure that records how pieces of the space interact with one another - 
 structure that disappears when passing to homology\, but is responsible fo
 r subtle phenomena familiar to topologists.\n\n\nThe surprising message is
  that this extra structure is powerful enough to recover the entire space.
  More precisely\, when singular cochains are viewed with all of their high
 er-order structure (as an E_\\infty-coalgebra)\, they form a complete inva
 riant: from this data alone\, one can reconstruct the homotopy type of the
  space in a functorial way.\n\n\nThis result builds on and extends ideas f
 rom rational homotopy theory and deep work of Mandell\, Yuan\, and Bachman
 n–Hahn. At its core lies a classification of particularly well-behaved (
 “perfect”) E_\\infty-coalgebras\, which makes this reconstruction poss
 ible.\n\n\nThis is joint work with F. Riedel.\n\nA wine reception in the C
 entral Core will follow the lecture.
LOCATION:MR2\, CMS
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