Activities This Week

One easy way of representing knot is via a knot diagram. However, inferring properties of the knot from its diagram and deciding when two diagrams represent the same knot are quite difficult problems. Surprisingly, when the diagram is sufficiently “twisty” then some structure starts to emerge. I will discuss two results of this nature: hyperbolicity of highly twisted knot diagrams and uniqueness of highly twisted plat diagrams.

Based on joint works with Yoav Moriah, Tali Pinsky and Jessica Purcell. All relevant notions will be explained in the talk.

In this talk, I will discuss the irreducibility problem of Severi varieties on toric surfaces. The classical Severi varieties were introduced by Severi almost 100 years ago in the context of Severi’s attempt to provide an algebraic treatment of the irreducibility problem of the moduli spaces of curves. Although, the irreducibility of the moduli spaces was achieved algebraically by Deligne and Mumford in 1969 using completely different techniques, Severi varieties remained in the focus of study of many algebraic geometers including Harris, Fulton, Zariski and others.

In this talk I will present the main result of my M.Sc. Thesis providing a complete description of the irreducible components of the genus-one Severi varieties on toric surfaces.

This work was done under the supervision of Professor Ilya Tyomkin.

In recent years, the study of measure preserving and stationary actions of Lie groups and hyperbolic groups have produced many geometric consequences. This talk will continue the tradition. We will show that stationary actions of hyperbolic groups have large critical exponent, namely exponential growth rate more than half of entropy divided the drift of the random walk.

This can be used to prove an interesting geometric result: if the bottom of the spectrum of the Laplacian on a hyperbolic n manifold M is equal to that of its universal cover (or equivalently the fundamental group has exponential growth rate at most (n-1)/2) then M has points with arbitrary large injectivity radius.

This is (in some sense the optimal) rank 1 analogue of a recent result of Fraczyk-Gelander which asserts that any infinite volume higher rank locally symmetric space has points with arbitrary large injectivity radius.

This is joint work with Arie Levit.