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SUMMARY:Matthew Foreman (UC Irvine)
DTSTART:20200410T230000Z
DTEND:20200410T235000Z
DTSTAMP:20260422T212900Z
UID:UCLALogicC/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UCLALogicC/1
 /">Attacking Classical Problems in Dynamical Systems with Descriptive Set 
 Theory</a>\nby Matthew Foreman (UC Irvine) as part of UCLA logic colloquiu
 m\n\n\nAbstract\nIn his classical 1932 paper\, von Neumann asked 3 questio
 ns: Can you classify the statistical behavior of differentiable systems? A
 re there systems where time-forward is not isomorphic to time-backward? Is
  every abstract statistical system isomorphic to a differentiable system? 
 These questions can be addressed with some surprising consequences by embe
 dding them in Polish Spaces. Indeed the tools answer other questions from 
 the 60's and 70's such as the existence of diffeomorphisms with arbitrary 
 Choquet simplexes of invariant measures. Moreover there are surprising ana
 logues to Hilbert's 10th problem.\n\nIn a different category\, building on
  work of Poincare\, Smale proposed classifying the qualitative behavior of
  differentiable systems on compact manifolds. His 1967 paper explicitly ar
 gued that the equivalence relation of "conjugacy up to homeomorphism" capt
 ures this notion and he proposes classifying it. Call this notion topologi
 cal equivalence. Very recent joint results with A. Gorodetski show:\n\n- T
 he equivalence relation $E _ 0$ is Borel reducible to topological equivale
 nce of diffeomorphisms of any smooth 2-manifold.\n\n- The equivalent relat
 ion of Graph Isomorphism is Borel reducible to topological equivalence of 
 diffeomorphisms of any smooth manifold of dimension 5 or above.\n\nAs coro
 llaries\, none of the classical numerical invariants such as entropy\, rat
 es of growth of periodic points and so forth\, can classify diffeomorphism
 s of 2-manifolds\, and there is no Borel classification at all of diffeomo
 rphisms of 5-manifolds.\n\nIn the same 1967 paper Smale asks (in different
  language) whether there is a generic class that can be classified. This i
 s still an open problem.\n
LOCATION:https://researchseminars.org/talk/UCLALogicC/1/
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SUMMARY:Henry Towsner (University of Pennsylvania)
DTSTART:20200522T230000Z
DTEND:20200523T000000Z
DTSTAMP:20260422T212900Z
UID:UCLALogicC/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UCLALogicC/2
 /">Removal and amalgamation</a>\nby Henry Towsner (University of Pennsylva
 nia) as part of UCLA logic colloquium\n\n\nAbstract\nThe key step in the p
 roof of the triangle removal lemma can be viewed as saying that we can ide
 ntify a small number of edges in a graph as being the "exceptional" edges\
 , and the remaining edges are sufficiently "representative of the neighbor
 hood around them" that\, if there are any triangles left\, there must have
  been many triangles. This can be viewed as a amalgamation problem in the 
 sense of model-theory: given types p(x\,y)\, q(x\,z)\, and r(y\,z)\, each 
 of which indicates that there is an edge between the vertices\, when are t
 he types p\,q\,r "large" in a way which guarantees that there are many (x\
 ,y\,z) extending each of these types?\n\nThe exceptional types can be char
 acterized as the non-Lebesgue points - that is\, the points which fail to 
 satisfy the Lebesgue density theorem in the right measure space. We give a
  way to generalize this to types of higher arity and use this to prove a n
 ew generalization\, an "ordered hypergraph removal lemma"\, extending the 
 recent ordered graph removal lemma of Alon\, Ben-Eliezer\, and Fischer.\n
LOCATION:https://researchseminars.org/talk/UCLALogicC/2/
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