SGGTC this Friday: Agustin Moreno and Mariano Echeverria
Oleg Lazarev
Dear all,
This Friday we have two speakers. Agustin Moreno from Augsburg will give a talk at 10:30 am in Math 520. Mariano Echeverria from Rutgers University will give a talk at 1:00 pm in Math 407. We will meet at 11:35 in the lobby and go out for lunch.
Best, Oleg
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Agustin Moreno: Bourgeois contact structures: tightness, fillability and applications.
Abstract: Starting from a contact manifold and a supporting open book decomposition, an explicit construction by Bourgeois provides a contact structure in the product of the original manifold with the two-torus. In this talk, we will discuss recent results concerning rigidity and fillability properties of these contact manifolds. For instance, it turns out that Bourgeois contact structures are, in dimension 5, always tight, independent on the rigid/flexible classification of the original contact manifold. Moreover, Bourgeois manifolds associated to suitable monodromies provide new examples of weakly but not strongly fillable contact 5-manifolds. We also present the following application in any dimension: the standard contact structure in the unit cotangent bundle of the n-torus, which is a Bourgeois manifold, admits a unique aspherical filling up to diffeomorphism. This is joint work with Jonathan Bowden and Fabio Gironella.
Mariano Echeverria: A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori
Abstract: Starting from a contact manifold and a supporting open book decomposition, an explicit construction by Bourgeois provides a contact structure in the product of the original manifold with the two-torus. In this talk, we will discuss recent results concerning rigidity and fillability properties of these contact manifolds. For instance, it turns out that Bourgeois contact structures are, in dimension 5, always tight, independent on the rigid/flexible classification of the original contact manifold. Moreover, Bourgeois manifolds associated to suitable monodromies provide new examples of weakly but not strongly fillable contact 5-manifolds. We also present the following application in any dimension: the standard contact structure in the unit cotangent bundle of the n-torus, which is a Bourgeois manifold, admits a unique aspherical filling up to diffeomorphism. This is joint work with Jonathan Bowden and Fabio Gironella.
Mariano Echeverria: A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori
Abstract: Given a knot K inside an integer homology sphere Y , the Casson-Lin-Herald invariant can be interpreted as a signed count of conjugacy classes of irreducible representations of the knot complement into SU(2) which map the meridian of the knot to a fixed conjugacy class. It has the interesting feature that it determines the Tristram-Levine signature of the knot associated to the conjugacy class chosen.
Turning things around, given a 4-manifold X with the integral homology of S1 × S3, and an embedded torus which is homologically non trivial, we define a signed count of conjugacy classes of irreducible representations of the torus complement into SU(2) which satisfy an analogous fixed conjugacy class condition to the one mentioned above for the knot case. Our count recovers the Casson-Lin-Herald invariant of the knot in the product case, thus it can be regarded as implicitly defining a Tristram-Levine signature for tori.
This count can also be considered as a singular Furuta-Ohta invariant, and it is a special case of a larger family of Donaldson invariants which we also define. In particular, when (X, T ) is obtained from a self-concordance of a knot (Y,K) satisfying an admissibility condition, these Donaldson invariants are related to the Lefschetz number of an Instanton Floer homology for knots which we construct. Moreover, from these Floer groups we obtain Frøyshov invariants for knots which allows us to assign a Frøyshov invariant to an embedded torus whenever it arises from such a self-concordance.
Abstract: Given a knot K inside an integer homology sphere Y , the Casson-Lin-Herald invariant can be interpreted as a signed count of conjugacy classes of irreducible representations of the knot complement into SU(2) which map the meridian of the knot to a fixed conjugacy class. It has the interesting feature that it determines the Tristram-Levine signature of the knot associated to the conjugacy class chosen.
Turning things around, given a 4-manifold X with the integral homology of S1 × S3, and an embedded torus which is homologically non trivial, we define a signed count of conjugacy classes of irreducible representations of the torus complement into SU(2) which satisfy an analogous fixed conjugacy class condition to the one mentioned above for the knot case. Our count recovers the Casson-Lin-Herald invariant of the knot in the product case, thus it can be regarded as implicitly defining a Tristram-Levine signature for tori.
This count can also be considered as a singular Furuta-Ohta invariant, and it is a special case of a larger family of Donaldson invariants which we also define. In particular, when (X, T ) is obtained from a self-concordance of a knot (Y,K) satisfying an admissibility condition, these Donaldson invariants are related to the Lefschetz number of an Instanton Floer homology for knots which we construct. Moreover, from these Floer groups we obtain Frøyshov invariants for knots which allows us to assign a Frøyshov invariant to an embedded torus whenever it arises from such a self-concordance.