Page 356 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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GEOMETRIC ANALYSIS AND LOW-DIMENSIONAL TOPOLOGY (MS-59)
The Witten Conjecture for homology S1 × S3
Nikolai Saveliev, saveliev@math.miami.edu
University of Miami, United States
The Witten conjecture (1994) poses that the Seiberg–Witten invariants contain all of the topo-
logical information of the Donaldson polynomials. The natural domain of this conjecture com-
prises closed simply connected oriented smooth 4-manifolds with b+ > 1, where the Seiberg–
Witten invariants are obtained by a straightforward count of irreducible solutions to the Seiberg–
Witten equations. The Seiberg-Witten invariants have also been extended to manifolds with
b+ = 1 using wall-crossing formulas. In our work with Mrowka and Ruberman (2009) we
defined the Seiberg–Witten invariant for a class of manifolds X with b+ = 0 having homol-
ogy of S1 × S3. The usual count of irreducible solutions in this case depends on metric and
perturbation but we succeeded in countering this dependence by a correction term to obtain a
diffeomorphism invariant of X. In the spirit of the Witten conjecture, we conjectured that the
degree zero Donaldson polynomial of X can be expressed in terms of this invariant. I will
describe the special cases in which the conjecture has been verified, together with some appli-
cations. This is a joint project with Jianfeng Lin and Daniel Ruberman.
Free boundary minimal surfaces in the unit ball
Mario Schulz, m.schulz@qmul.ac.uk
Queen Mary University of London, Germany
Coauthors: Alessandro Carlotto, Giada Franz
Free boundary minimal surfaces arise naturally in partitioning problems for convex bodies, in
capillarity problems for fluids and in the study of extremal metrics for Steklov eigenvalues
on manifolds with boundary. The theory has been developed in various interesting directions,
yet many fundamental questions remain open. One of the most basic ones can be phrased as
follows: Can a surface of any given topology be realised as an embedded free boundary minimal
surface in the 3-dimensional Euclidean unit ball? We will answer this question affirmatively for
surfaces with connected boundary and arbitrary genus.
Disoriented homology of surfaces and branched covers of the 4-ball
Sašo Strle, saso.strle@fmf.uni-lj.si
University of Ljubljana, Slovenia
Coauthor: Brendan Owens
An often used construction in low-dimensional topology is to associate to a properly embedded
surface F ⊂ B4 the branched double cover X of B4 with branch set F . If the surface is obtained
by pushing the interior of an embedded surface in S3 into the interior of B4, a classical result of
Gordon and Litherland states that H2(X; Z) is isomorphic to H1(F ; Z) and that the intersection
pairing of X may be described in terms of a pairing on H1(F ; Z) which is determined by the
embedded surface in S3.
We generalize this result to any surface F by defining a non-standard homology theory
DH∗(F ) that depends on a description of F in S3. This homology captures the homological
354
The Witten Conjecture for homology S1 × S3
Nikolai Saveliev, saveliev@math.miami.edu
University of Miami, United States
The Witten conjecture (1994) poses that the Seiberg–Witten invariants contain all of the topo-
logical information of the Donaldson polynomials. The natural domain of this conjecture com-
prises closed simply connected oriented smooth 4-manifolds with b+ > 1, where the Seiberg–
Witten invariants are obtained by a straightforward count of irreducible solutions to the Seiberg–
Witten equations. The Seiberg-Witten invariants have also been extended to manifolds with
b+ = 1 using wall-crossing formulas. In our work with Mrowka and Ruberman (2009) we
defined the Seiberg–Witten invariant for a class of manifolds X with b+ = 0 having homol-
ogy of S1 × S3. The usual count of irreducible solutions in this case depends on metric and
perturbation but we succeeded in countering this dependence by a correction term to obtain a
diffeomorphism invariant of X. In the spirit of the Witten conjecture, we conjectured that the
degree zero Donaldson polynomial of X can be expressed in terms of this invariant. I will
describe the special cases in which the conjecture has been verified, together with some appli-
cations. This is a joint project with Jianfeng Lin and Daniel Ruberman.
Free boundary minimal surfaces in the unit ball
Mario Schulz, m.schulz@qmul.ac.uk
Queen Mary University of London, Germany
Coauthors: Alessandro Carlotto, Giada Franz
Free boundary minimal surfaces arise naturally in partitioning problems for convex bodies, in
capillarity problems for fluids and in the study of extremal metrics for Steklov eigenvalues
on manifolds with boundary. The theory has been developed in various interesting directions,
yet many fundamental questions remain open. One of the most basic ones can be phrased as
follows: Can a surface of any given topology be realised as an embedded free boundary minimal
surface in the 3-dimensional Euclidean unit ball? We will answer this question affirmatively for
surfaces with connected boundary and arbitrary genus.
Disoriented homology of surfaces and branched covers of the 4-ball
Sašo Strle, saso.strle@fmf.uni-lj.si
University of Ljubljana, Slovenia
Coauthor: Brendan Owens
An often used construction in low-dimensional topology is to associate to a properly embedded
surface F ⊂ B4 the branched double cover X of B4 with branch set F . If the surface is obtained
by pushing the interior of an embedded surface in S3 into the interior of B4, a classical result of
Gordon and Litherland states that H2(X; Z) is isomorphic to H1(F ; Z) and that the intersection
pairing of X may be described in terms of a pairing on H1(F ; Z) which is determined by the
embedded surface in S3.
We generalize this result to any surface F by defining a non-standard homology theory
DH∗(F ) that depends on a description of F in S3. This homology captures the homological
354
