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SUMMARY:Matthew Foreman (UC Irvine)
DTSTART;VALUE=DATE-TIME:20200410T230000Z
DTEND;VALUE=DATE-TIME:20200410T235000Z
DTSTAMP;VALUE=DATE-TIME:20240329T113706Z
UID:UCLALogicC/1
DESCRIPTION:Title: Attacking Classical Problems in Dynamical Systems with Descriptive Set
Theory\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;VALUE=DATE-TIME:20200522T230000Z
DTEND;VALUE=DATE-TIME:20200523T000000Z
DTSTAMP;VALUE=DATE-TIME:20240329T113706Z
UID:UCLALogicC/2
DESCRIPTION:Title: Removal and amalgamation\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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