Abstract:

A recurring phenomenon in algebraic geometry is that certain structures remain stable under a surprisingly broad class of changes of the underlying objects. Such rigidity phenomena have a striking formal resemblance to homotopy invariance in topology: in both cases, certain information becomes insensitive to a class of transformations that may substantially alter the object itself.

In this talk I will discuss this analogy from the perspective of algebraic K-theory and its connections to homotopy theory. We will consider rigidity phenomena for local rings and their consequences for algebraic K-theory, including symplectic and Grothendieck–Witt theories. Particular emphasis will be placed on the conceptual relationship between rigidity and $\mathbb A^1$-homotopy invariance, and on the possibility that these phenomena reflect a common underlying homotopy-theoretic principle.

The final part of the talk will turn to topology.  I will discuss several ways in which algebraic rigidity phenomena may have topological counterparts or consequences.  These include questions about the passage from algebraic to topological K-theory, possible interpretations in stable homotopy theory, and the extent to which rigidity techniques might provide new information about classical topological invariants.  The aim is to suggest a broader perspective in which rigidity serves as a bridge between algebraic geometry and topology, and to identify directions in which this analogy may lead to new results.

Abstract (William Easton):

In the late 1960s, Kirby gave a remarkable proof of the Stable Homeomorphism Theorem (and hence the Annulus Theorem) which laid the foundations for much of the theory of topological manifolds that was to follow. This theorem gives a very powerful decomposition of homeomorphisms on R^n which allows one to show, for example, that all orientation preserving maps are isotopic to the identity. The proof consists of a series of neat tricks that allow bootstrapping from simpler cases. I will give the key arguments in the proof, omitting only some surgery-theoretic inputs and plausible lemmas.

Abstract (Ribhu Hooja):

In this talk, we will introduce bordered Heegaard Floer homology as a tool to compute concordance invariants of satellite knots. Satellite operations are a well-studied class of operations on knots that descend to the knot concordance group, and the cut-and-paste methods of bordered Heegaard Floer homology are particularly suited to describing how they affect the Heegaard Floer homology of knots. We will survey prior work and explain new results on (1, 1)-patterns generalizing the Mazur pattern. This is joint work done in the SHUR 2026 summer REU alongside Sean Kim, Filip Rupchin, Shreya Sinha, and Allison Tsypin.

Abstract: In 2019, Budney and Gabai introduced a concrete way to construct diffeomorphisms of 4-manifolds called barbell diffeomorphisms. We will review this construction and use barbells to construct interesting knotting in S^4 and S^5, including knotted splitting spheres and knotted solid tori in S^4, and knotted 3-spheres in S^5. This is partially joint work with Seungwon Kim and Gheehyun Nahm.

Abstract: TBD

Abstract: TBD

Abstract: Spheres of revolution having just one equator carry many closed geodesics, which can be classified according to their winding number around the axis of revolution and the number of intersections with the equator. This leads to a natural definition of marked length spectrum. I will discuss rigidity and flexibility phenomena which are associated to this notion. This talk is based on a recent joint work with Marco Mazzucchelli.