Events

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

Damaris Schindler, University of Göttingen

Density of rational points near manifolds

Given a bounded submanifold M in R^n, how many rational points with common bounded denominator are there in a small thickening of M? How does this counting function behave if we let the size of the denominator go to infinity? The study of the density of rational points near manifolds has seen significant progress in the last couple of years. In this talk I will explain why we might be interested in this question, focusing on applications in Diophantine approximation and the (quantitative) arithmetic of projective varieties.

MC 5403

Damaris Schindler, University of Göttingen

Quantitative weak approximation and quantitative strong approximation for certain quadratic forms

In this talk we discuss recent results on optimal quantitative weak approximation for certain projective quadrics over the rational numbers as well as quantitative results on strong approximation for quaternary quadratic forms over the integers. We will also discuss results on the growth of integral points on the three-dimensional punctured affine cone and strong approximation with Brauer-Manin obstruction for this quasi-affine variety. This is joint work with Zhizhong Huang and Alec Shute.

MC 5479