Short Courses & Seminars
IMSA Distinguished Lecture Series
Boris Botvinnik, University of Oregon
Brunnian Links and Kontsevich Graph Complex
Wednesday, October 29, 2025
at 10:00am
Ungar 528-B
Abstract: Recently, Tadayuki Watanabe disproved the Smale Conjecture, which stated that the group Diff∂(D4) is homotopically trivial. He showed that there are nontrivial homotopy groups πqDiff∂(D4)⊗Q. The first key idea was to exhibit smooth D4 -bundles using trivalent graphs equipped with Hopf and Borromean links as their blueprints. The second key in the construction was the use of Kontsevich’s configuration space integral to detect the non-triviality of such bundles. These ideas were generalized by Watanabe for higher-dimensional disks Dd , d ≥ 4. I will explain some of the ideas and constructions of Watanabe’s work. In our very recent joint work, we used Brunnian links to construct a chain map from the Kontsevich graph complex to the rational singular chain complex of BDiff∂(D2k) when the dimension 2k is sufficiently large. Then we use again Kontsevich’s configuration space integral to detect the nontriviality of such homology elements In particular, we provide new constructions of non-trivial elements in the homotopy groups π8k−10(BDiff∂(D2k)) ⊗ Q (for k ≥ 17) which are derived from well known cycles in the graph complex.
Several Mysteries of Positive Scalar Curvature
Thursday, October 30, 2025
at 5:00pm
Ungar 528-B
Abstract: While Scalar Curvature might seem like a basic geometric invariant, it has actually become an important character in the mathematical world. In fact, Positive Scalar Curvature plays a key role in several areas of mathematics, including Riemannian geometry, geometric analysis (such as Ricci flow and conformal geometry), Index theory, Surgery theory, and Homotopy theory. It even gets involved in recent dramatic developments in the theory of Moduli Spaces of Manifolds and Cobordism Categories. In this lecture, I will attempt to pull back the curtain on some of this character’s most remarkable features. We’ll start with the backstory: the basics of Scalar Curvature, the Einstein-Hilbert functional, conformal metrics, and the Yamabe problem. Then I will show why and how Surgery theory and Index theory naturally intertwine with Positive Scalar Curvature.
At the end I will present some new results on the Space of Metrics with Positive Scalar Curvature.
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Thursday, November 13, 2025, 4pm & 5pm
Ungar Bldg, Room 528B
Ernesto Lupercio, CINVESTAV & Enrique Becerra, CINVESTAV
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Tuesday, November 11, 2025, 2pm
&
Wednesday, November 12, 2025, 11am
Ungar Bldg, Room 528B
&
Thursday, November 13, 2025, 11am
Leonardo Cavenaghi, CAMPINAS
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