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
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
Jules Ribolzi, University of Waterloo
On Two Model-Theoretic Approaches to Complex Analytic Geometry
There is a first-order multi-sorted structure for compact complex spaces which satisfies important model-theoretic properties (quantifier elimination, elimination of imaginaries, finiteness of Morley rank,…). We call this theory $CCM$. On the other hand, any compact complex manifold is definable in the O-minimal structure $\mathbb{R}_{an}$. In this talk, we will discuss the relation between these two structures (and also their elementary extensions).
MC 5417
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
Elisabeth Werner, Case Western Reserve University
The $L_p$-Floating Area and Isoperimetric Inequalities on the Sphere
Euclidean convex bodies in spaces of constant positive curvature. We introduce the family of $L_p$-floatingareas for spherical convex bodies, as an analog to $L_p$-affine surface area measures from Euclidean geometry.We investigate a duality formula, monotonicity and isoperimetric inequalities for this new family of curvaturemeasures on spherical convex bodies. Based on joint works with Florian Besau.
MC 5417
Nathaniel Bannister, Carnegie Mellon University
Condensed Sets and the Solovay Model
We exhibit a geometric morphism from the Grothendieck topos representing the Solovay model to the κ-pyknotic sets of Barwick--Haine and Clausen--Scholze. We then use the properties of this morphism andautomatic continuity in the Solovay model to outline a proof of Clausen--Scholze's resolution of the Whiteheadproblem for discrete condensed abelian groups. Joint work with Dianthe Basak.
MC 5417
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
Siyuan Lu, McMaster University
Interior C^2 estimate for Hessian quotient equation
In this talk, we will first review the history of interior C^2 estimates for fully nonlinear equations. As a matter offact, very few equations admit this property, not even the Monge-Ampère equation in dimension three or above.We will then present our recent work on interior C^2 estimate for Hessian quotient equation. We will discuss themain idea behind the proof. If time permits, we will also discuss the Pogorelov-type interior C^2 estimate forHessian quotient equation and its applications.
MC 5417
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