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On a given Riemann surface, we may talk about various projective structures that are compatible with the given complex structure. These structures are equivalent to what are known as $sl_2$ opers, or in terms of Faltings’ paper that we mainly follow, as permissible connections. Then there are several interesting questions we may ask about the monodromies of these objects, and the one we focus on today is whether the monodromy group is conjugate into $PSL_2(\mathbb R)$ in $PSL_2(\mathbb C)$ (“the real monodromy property”).
The dual Garside left canonical form is used to solve the conjugacy problem in braid theory, introduced by Xu (3-braids), Kang-Ko-Lee (4-braids) and Birman-Ko-Lee (n-braids). The fractional Dehn twist coefficient (FDTC) is a measurement of the boundary twist of a surface homeomorphism, introduced by Honda-Kazez-Matic. In contact geometry, it is a powerful tool to detect tightness/overtwistedness of a given contact structure. In this talk, I will give applications of the left canonical form to the computation of the FDTC of 4-braids.
Knot concordance, in both the smooth and topological categories, is a well-studied equivalence relation on knots. We will introduce the smooth, topological, and algebraic concordance groups, and discuss several invariants that help answer questions about the structures of these groups and the relationships between them. We will also discuss the ways facts about knot concordance can be used to construct exotic pairs of 4-manifolds.
We discuss a new problem by Dmitri Burago and Anton Petrunin. Consider a $C^2$-smooth surface embedded in $\RR^3$ with principal curvatures $\leq 1$. Does such a surface necessarily enclose a volume at least that of the unit ball? We address results from two opposite perspectives: On the one hand, we can construct a counterexample by deforming Lagunov's fishbowl (a fattened Bing's house); on the other hand, there are positive results with additional conditions. Many problems remain open.
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