Departmental PhD Thesis Exam – Boris Lishak

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Monday, May 2, 1016
11:10 a.m.
BA6183

PhD Candidate:  Boris Lishak
Supervisor:  Alex Nabutovsky
Thesis title:  Balanced Presentations of the Trivial Group and 4-dimensional Geometry

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Abstract:

We construct a sequence of balanced presentations of the trivial group with two generators and two relators with the following property: The minimal number of relations required to demonstrate that a generator represents the trivial element grows faster than the tower of exponentials of any fixed height of the length of the finite presentation.

We prove that 1) There exist infinitely many non-trivial codimension one “thick” knots in $\mathbb{R}^5$; 2) For each closed four-dimensional smooth manifold $M$ and for each sufficiently small positive $\epsilon$ the set of isometry classes of Riemannian metrics with volume equal to $1$ and injectivity radius greater than $\epsilon$ is disconnected; and 3) For each closed four-dimensional $PL$-manifold $M$ and any $m$ there exist arbitrarily large values of $N$ such that some two triangulations of $M$ with $<N$ simplices cannot be connected by any sequence of $<\exp_m(N)$ bistellar transformations, where $\exp_m(N)=\exp(\exp(\ldots \exp (N)))$ ($m$ times).

We construct families of trivial $2$-knots $K_i$ in $\mathbb{R}^4$ such that the maximal complexity of $2$-knots in any isotopy connecting $K_i$ with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of $K_i$. Here we can either construct $K_i$ as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots $K_i$, consider isotopies through PL knots, and measure the complexity of a PL-knot as the minimal number of flat $2$-simplices in its triangulation.

For any $m$ we produce an exponential number of balanced presentations of the trivial group with four generators and four relations of length $N$ such that the minimal number of Andrews-Curtis transformations needed to connect any two of the presentations is at least $\exp_m(N)$.

A copy of the thesis can be found here: ut-thesis