Seminar

William Dan, University of Waterloo

Solovay Reducibility

Having discussed the relationship between Solovay reducibility and the newly introduced reducibilities, K-reducibility and C-reducibility, we turn back to study its relationship with previously discussed reducibilities, Turing reducibility and wtt-reducibility. Then, if time permits, we will completely finish sections 9.1 and 9.2 by discussing a final characterization of Solovay reducibility and going beyond random left-c.e. reals to look at random left-d.c.e. reals.

MC 5403

Fateme Peimany, University of Waterloo

Strongly minimal groups in CCM

We continue our study of the structure of groups definable in CCM, now in our second session on this topic, withthe goal of proving that every strongly minimal group is either a complex torus or a (commutative) linearalgebraic group.

MC 5479

Spencer Kelly, University of Waterloo

Constructing a Slice Theorem in Infinite Dimensions

The slice theorem is a powerful tool for understanding proper group actions on manifolds; however it does nothold on infinite dimensional manifolds, nor does there exist a general infinite dimensional extension of it.However, on specific infinite dimensional manifolds, working on a case-by-case basis, we have been able toconstruct analogues of the slice theorem. In this talk, we will investigate one of these cases, namely the space ofconnections on a bundle over a compact Riemannian manifold, acted on by the gauge group.

MC 5403

William Dan, University of Waterloo

Solovay Reducibility and Relative Randomness

Having completed our characterization of left-c.e. random reals, we return to the concept of Solovay reducibilityto study it more deeply. We will see that beyond the characterizations we have seen so far, Solovay reducibilitycan be viewed as a measure of relative randomness, and connect this perspective back to the Kucera-Slamantheorem. We will also relate it to the reducibilities we have studied previously, and give a final, possiblysimplest, characterization of Solovay reducibility. This seminar follows sections 9.1 and 9.2 from the Downeyand Hirschfeldt book.

MC 5403

Fateme Peimany, University of Waterloo

Definable groups in CCM

We continue to study the structure of groups definable in CCM, toward showing that every strongly minimal group is either a complex torus or a (commutative) linear algebraic group.

MC 5479

Viktor Majewski, University of Waterloo

Filling Holes in the Spin(7)-Teichmüller Space and String Cohomology

In this talk, I apply the analytic results from the first talk to study the boundary of the Spin(7) Teichmüller space.Using compactness results for Ricci-flat metrics together with known examples of Spin(7) manifolds, it is knownthat Spin(7) orbifolds with SU(N) isotropy arise as boundary points of the moduli space. Building on theresolution scheme for Spin(7) orbifolds that I discussed in 2024, and which I will briefly review, we show howthis boundary can be removed by requiring Spin(7) orbifolds to encode information about their resolutions. Inthis way, the Teichmüller space is enlarged to include orbifold limits together with their compatible resolutions,thereby filling in the boundary. Finally, we explain how this perspective is related to a Spin(7) analogue of thecrepant resolution conjecture from string cohomology, providing a geometric interpretation of the obstructioncomplex discussed in the linear gluing analysis in the first talk.

MC 5403

Fateme Peimany, University of Waterloo

Model Theory Working Seminar: Definable groups in CCM

We continue to study the structure of groups definable in CCM, toward showing that every strongly minimal group is either a complex torus or a (commutative) linear algebraic group.

MC 5479

Amanda Petcu, University of Waterloo

Some results on hypersymplectic structures

A conjecture of Simon Donaldson is that on a compact 4-manifold X^4 one can flow from a hypersymplectic structure to a hyperkahler structure while remaining in the same cohomology class. To this end the hypersymplectic flow was introduced by Fine-Yao. In this thesis the notion of a positive triple on X^4 is used to define a hypersymplectic and hyperkahler structure. Given a closed positive triple one can define either a closed G2 structure or a coclosed G2 structure on T^3 x X^4. The coclosed G2 structure is evolved under the G2 Laplacian coflow. This descends to a flow of the positive triple on X^4, which is again the Fine-Yao hypersymplectic flow. In the second part of this thesis we let X^4 = R^4 \0 with a particular cohomogeneity one action. A hypersymplectic structure invariant under this action is introduced. The Riemann and Ricci curvature tensors are computed and we verify in a particular case that this hypersymplectic structure can be transformed to a hyperkahler structure. The notion of a soliton for the hypersymplectic flow in this particular case is introduced and it is found that steady solitons give rise to hypersymplectic structures that can be transformed to hyperkahler structures. Some other soliton solutions are also discussed.

MC 5403

Amanda Maria Petcu, University of Waterloo

Some results on hypersymplectic structures

A conjecture of Simon Donaldson is that on a compact 4-manifold X^4 one can flow from a hypersymplecticstructure to a hyperkahler structure while remaining in the same cohomology class. To this end thehypersymplectic flow was introduced by Fine-Yao. In this thesis the notion of a positive triple on X^4 is used todefine a hypersymplectic and hyperkahler structure. Given a closed positive triple one can define either a closedG2 structure or a coclosed G2 structure on T^3 x X^4. The coclosed G2 structure is evolved under the G2Laplacian coflow. This descends to a flow of the positive triple on X^4, which is again the Fine-Yaohypersymplectic flow. In the second part of this thesis we let X^4 = R^4 \ {0} with a particular cohomogeneityone action. A hypersymplectic structure invariant under this action is introduced. The Riemann and Riccicurvature tensors are computed and we verify in a particular case that this hypersymplectic structure can betransformed to a hyperkahler structure. The notion of a soliton for the hypersymplectic flow in this particularcase is introduced and it is found that steady solitons give rise to hypersymplectic structures that can betransformed to hyperkahler structures. Some other soliton solutions are also discussed.

MC 5479

Rahim Moosa, University of Waterloo

Definable groups in CCM

I will continue totalk about “Strongly minimal groups in the theory of compact complex maniflds".

MC 5479