Neil R Hoffman

Office: 165
Phone: 61-3-8344-9710
Email: nhoffman(at)ms.unimelb.edu.au
Consultation hours: W 2-4, Th 10-11


About Me

I am very happy to be at the University of Melbourne, funded by Australian Research Council Discovery Grant DP130103694.
(The grant is held by Craig Hodgson, Hyam Rubinstein, and Stephan Tillmann (Sydney).)

For the academic year 2012-2013, I was a guest at the Max Planck Institute for Mathematics in Bonn. Before that, I had the pleasure of
visiting Boston College. Going back any further puts me back in graduate school at the University of Texas where I obtained by PhD.

For more complete information, have a look at my CV (updated January 2016) or here.



Research

I study low-dimensional topology. Broadly, speaking I study 3-dimensional objects. More concretely, I focus on classifying and understanding 3-dimensional spaces called 3-manifolds. These spaces all have the property that the neighbourhood of a point in the space ``looks like'' a 3-dimensional ball. Also, every 3-manifold can be decomposed into tetrahedra, and this gives a combinatorial description of that 3-manifold. In addition, to creating fast and rigourous classification schemes for 3-manifolds, I have particular interest in how 3-manifolds are related
to each other. Currently, I am looking at two ways in which this can happen. The first way
is if two manifolds are commensurable ie they share a common finite sheeted cover.
The second way is if two manifolds are related via a Dehn surgery.



Here is a list of my publications and preprints (yet to be published work has its arxiv listing).
A list of this work with short descriptions, questions, and slides is available here.

My Papers


In preparation.
12. (with Ken Baker and Joan Licata) Unifying unexpected exceptional Dehn fillings.

Preprints.
11. (with Stavros Garoufalidis, Craig Hodgson, and Hyam Rubinstein) The 3D-index and normal surfaces arxiv
10. (appendix to Ken Baker's paper) The Poincare homology sphere, lens space surgeries, and some knots with tunnel number two arxiv.
9. (with Jessica Purcell) Geometry of planar surfaces and exceptional fillings.arxiv
8. (with Nathan Sunukjian) Surfaces in 4-manifolds: smooth isotopy arxiv

Publications (in print and accepted).
7. (with Genevieve Walsh) The big Dehn surgery graph and the link of $S^3.$ Proc. AMS, Ser B (open access) $\dagger$ (Also, see ancillary files.)
6. (with Nathan Dunfield and Joan Licata) Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling, arxiv (Note: see ancillary files for the code.) Math Research Letters, Vol. 22, No. 6 (2015) 1679-1698.
5. (with Kazuhiro Ichihara, Masahide Kashiwagi, Hidetoshi Masai, Shin'ichi Oishi, and Akitoshi Takayasu) Verified computations for hyperbolic 3-manifolds. arxiv code Exp. Math. 25.1 (2016): 66-78.
4. (with Ken Baker and Brandy Guntel Doleshal) On manifolds with multiple cyclic fillings, and generalized Berge knots. Boletin de la Sociedad Matematica Mexicana, 20(2) 2014 pp 401-447. arxiv
3. On knot complements that decompose into regular ideal dodecahedra, Geom. Ded. Vol 173(1) (2014) arxiv The computation written as a magma file is here.
2. Small knot complements, exceptional surgeries and hidden symmetries, to appear in Alg & Geo. Top, arxiv
1. Commensurability classes containing three knot complements, Alg. & Geo. Top, Vol 10 (2010) pdf

$\dagger$ Detailed instructions on how to input the Kanenobu tangle into Damian Heard's ORB, can be found here.


Some of this work (1 & 2) made it into my thesis, which was advised by Alan Reid.

Finally, I had the wonderful oppurtunity to be apart of the SMALL program as an undergraduate and work with Frank Morgan.

Here is a publication that resulted from that project:

(with J. Corneli et al.) Double Bubbles in $S^3$ and $H^3$, J. of Geo. Anal. Vol 17, No 2 (2007) pdf ****Code available in the arxiv version


Supervised Research


(co-supervised with Craig Hodgson) Blake Dadd and Alex Duan, "Constructing infinitely many geometric triangulations of the figure eight knot complement". arxiv

(co-supervised with Craig Hodgson) Emma Kong and Curtis Mustgrave-Evans. "Maximal equal area cusp packings of punctured spheres".



Teaching and outreach


Current course: Linear Algebra M10007 (Spring) Course page (requires login).
Classes: MWF 10-11
Consulation hours: W 2-4, Th 10-11

Here is a list of courses I have enjoyed teaching (all seasons local to the hemisphere):
BC Spring 2012:
Linein Math Circle
UT Fall 2009:
Differential Calculus for Natural Science Students (408N)
UT Fall 2007-Spring 2008:
Pre-calculus
TA roles:
Calculus for engineers, Multivariable calculus, Algebraic topology (IBL), Point set topology (IBL)


BC - Boston College - Newton, MA Northern Hemisphere
UT - University of Texas - Austin, TX Northern Hemisphere
IBL - Inquiry based learning aka the Moore method, aka the Socratic method.


Miscellany

In 2011, I was a member of the Infinite Cardinals .
It turns out there are good softball math puns out there, for example Tufts' department is proud of their Vector Bases.. (See there are fantastic prizes for getting to the bottom of the page.)