May 3, 2022
1:40PM - 2:45PM
066 University Hall or zoom
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2022-05-03 13:40:00
2022-05-03 14:45:00
Topology, Model Theory, and Hypergraph Regularity
Speaker: Henry Towsner (University of Pennsylvania)
Speaker's URL: https://www.sas.upenn.edu/~htowsner/
Title: Topology, Model Theory, and Hypergraph Regularity
Abstract: Szemeredi's Regularity Lemma tells us that binary relations can be viewed as a combination of a "roughly unary" part and a "quasirandom" part. When the binary relation is tame in the model theoretic sense - specifically, when it has finite VC dimension (that is, is NIP) - the quasirandom part disappears. When the relation is stable, a stronger theorem holds in which the "roughly unary" part has a particularly nice structure. We discuss a "topological" perspective on these results inspired by the Lebesgue differentiation theorem, showing that we can prove these results by looking at the behavior of "points of density".
We then discuss the generalization to hypergraphs - that is, to k-ary relations - where the same topological perspective gives us a short proof that a k-ary relation has no quasirandom part exactly when the relation satisfies the high arity generalization of finite VC dimension, finite VCk-1 dimension.
066 University Hall or zoom
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2022-05-03 13:40:00
2022-05-03 14:45:00
Topology, Model Theory, and Hypergraph Regularity
Speaker: Henry Towsner (University of Pennsylvania)
Speaker's URL: https://www.sas.upenn.edu/~htowsner/
Title: Topology, Model Theory, and Hypergraph Regularity
Abstract: Szemeredi's Regularity Lemma tells us that binary relations can be viewed as a combination of a "roughly unary" part and a "quasirandom" part. When the binary relation is tame in the model theoretic sense - specifically, when it has finite VC dimension (that is, is NIP) - the quasirandom part disappears. When the relation is stable, a stronger theorem holds in which the "roughly unary" part has a particularly nice structure. We discuss a "topological" perspective on these results inspired by the Lebesgue differentiation theorem, showing that we can prove these results by looking at the behavior of "points of density".
We then discuss the generalization to hypergraphs - that is, to k-ary relations - where the same topological perspective gives us a short proof that a k-ary relation has no quasirandom part exactly when the relation satisfies the high arity generalization of finite VC dimension, finite VCk-1 dimension.
066 University Hall or zoom
Department of Mathematics
math@osu.edu
America/New_York
public
Speaker: Henry Towsner (University of Pennsylvania)
Speaker's URL: https://www.sas.upenn.edu/~htowsner/
Title: Topology, Model Theory, and Hypergraph Regularity
Abstract: Szemeredi's Regularity Lemma tells us that binary relations can be viewed as a combination of a "roughly unary" part and a "quasirandom" part. When the binary relation is tame in the model theoretic sense - specifically, when it has finite VC dimension (that is, is NIP) - the quasirandom part disappears. When the relation is stable, a stronger theorem holds in which the "roughly unary" part has a particularly nice structure. We discuss a "topological" perspective on these results inspired by the Lebesgue differentiation theorem, showing that we can prove these results by looking at the behavior of "points of density".
We then discuss the generalization to hypergraphs - that is, to k-ary relations - where the same topological perspective gives us a short proof that a k-ary relation has no quasirandom part exactly when the relation satisfies the high arity generalization of finite VC dimension, finite VCk-1 dimension.
