 The thesis, old interests as well as publication and preprints and seminar page or the last chapter in my probability book illustrate earlier work in ergodic and spectral theory. Having always done what is now called "experimental mathematics" (initially got hooked with additive number theory), I have a passion for mathematical problems in computer science, with discrete structures, geometry or probability. Always having been fascinated by almost periodicity, (like packings, fluids, operators, walks, matrices,cellular automata), or Dirichlet series. A particular question on Birkhoff sums or the Birkhoff sum of the cotangent function. (Some notes (43 pages PDF) for a talk). A couple of years ago, playing with polyhedra led to a paper in graph theory which continues to interest me and fits well into quantum calculus. Teaching got me interested also in pedagogical questions, especially in web pedagogy and the use of technology in teaching. I love movies, especially if they contain math. In that context, a central interest is calculus and especially quantum calculus (which has a dedicated blog). About the history: here [PDF] is a course developed and ran first in the spring of 2010 and is now taught for the 9th time.

Some work highlights:
• A generalized Cauchy-Binet formula gives the coefficients of the characteristic polynomial of a product of two arbitrary mxn matrices article (preprint), slides [PDF]).
• A graph theoretical Lefshetz fixed point theorem: the fixed point index sum of a graph automorphism agrees with the cohomologically defined Lefshetz number. article.
• Illustrating mathematics and proofs using 3D printers, with Elizabeth Slavkovsky. In book, Updated preprint, talk (2013), Slides (2016).
• Some analytic continuation results for almost periodic Dirichlet series with John Lesieutre. Journal Project Page.
• Some spectral theory in ergodic theory. Example: weakly mixing, ergodic shift invariant measures are generic for a Zd shift. PDF.
• Weakly mixing is abundant PDF: a Hamiltonian system with a KAM torus can be perturbed to have a weakly mixing measure. Unresolved: is there an open set of aperiodic Hamiltonian systems in dim larger than 3 without weakly or strongly mixing invariant measure? Most probably "No": either chaos (a horse shoe already gives a mixing measure) or then KAM integrability i.e. near a periodic orbit. The paper shows that in the later case a perturbation produces weak mixing measure. Many ergodic theoretical questions are open for flows on tori: in this paper with Bert Hof we showed that for 2-tori the spectral measure are 0-dimensional by linking first in full generality Hausdorff dimension of spectral measures with speed of cyclic approximation in ergodic theory.
• Work on the cohomology of higher dimensional group actions started in my thesis (updated Feb 2000 and a Nov 2000). Realizing stochastic processes like random magnetic fields lead to higher dimensional cohomological problems in ergodic theory. A IID gauge field on the edges of a Z2 lattice for example does not produce IID magnetic curl F=dA in general. In order to apply a theorem of Feldman and Moore, I prove the equivalence of three cohomologies leading to ergodic analogue of the de Rham theorem equating singular and de Rham cohomology (generalizing a result of Jeraume De Pauw in 2D). The generalization to ergodic equivalence relations allows for example to generate IID random magnetic fields on a Penrose lattice.
• Almost periodic cellular automata: with Bert Hof a cool way to evolve infinite cellular automata using finite data points. Unlike with periodic boundary conditions, with aperiodic boundary conditions, there is no recurrence in general and complexity can grow indefinitely. It can also done for fluids (Vlasov systems) or Riemannian metrics (see with Evan Reed).
• The paper gives I think the first example of a caustic in differential geometry which is nowhere differentiable. No example with non-integer Hausdorff dimension is known yet.
• The paper [PDF] with Bert Hof and Barry Simon got recently some attention.
• An ergodic theoretical approach of sphere packings: in a class of almost periodic packings, periodic packings maximize the packing density. HTML, PDF,and some code.
• For an aperiodic ergodic sequence of SL(2,R)-valued random variables, the class with positive Lyapunov exponent are dense. PDF. (This 1992 paper was the first density result of this type). [It was rescanned 2019 as the original the ETDS archive scan has problems.]
Here is some more unpublished work (most of it is not even submitted once as writing down ideas currently has priority). This is ordered according my personal current preferences:
• Newer in graph geometry is about the expectation of simplex cardinality in complexes, then a new approach to Dehn-Sommerville via Gauss-Bonnet (see slides and bloging (about D-S) or about Gauss-Bonnet.
• The unimodularity theorem scrambled a bit priorities recently as this is too remarkable to not pursue a bit more. The connection Laplacian L for graphs or more generally for simplicial complexes is always invertible. Green functions g(x,y), where g is the inverse of L are all bounded leading to a bounded potential theory. There is some local topology, energy Dehn Sommerville stuff or arithmetic of graphs involved. The most important result so far is the energy theorem: it tells that for an arbitrary simplicial complex, the sum of the green function entries g(x,y) is the Euler characteristic. This result can be extended to the strong ring generated by simplicial complexes and has fantastic spectral properties also. The connection spectra multiply as the connection Laplacians tensor.
• This is also related to Wu characteristic and more general connection calculus. Here is a handout explaining the corresponding cohomology (which is almost as simple as simplicial cohomology) but which allows to distinguish topological spaces like Stiefel-Whitney classes. In this handout, the cohomology was not yet finalized yet. Here and here (4K downloadable) are some slides explaining the Wu characteristic. Bowen-Lanford Zeta Functions are interesting in connection calculus: a Handout [PDF] from a math table.
• I'm interested in basic structures of calculus and especially quantum calculus (see also the Video) Here is a writeup, which explains a bit how easy calculus is on a graph. In order to work, we have to see calculus structures on finite discrete structures. Some examples of results illustrating how porting results is Gauss-Bonnet, Poincare Hopf, Brouwer-Lefshetz, Jordan-Brouwer, McKean Singer or Lysternik-Schnirelman. A notion of dimension for graphs proposed here was explored here in detail. A notion of continuity and homeomorphism for graphs,a Cartesian product for graphs, a definition of sphere (I now call Evako sphere) and level surfaces in graph theory.
• A limit theorem for Barycentric subdivision (see also an earlier write-up) is exciting not only because it is a central limit type theorem but because the limits need still to be investigated. This is a follow up on a renormalization story for random Jacobi matrices, where I had searched for higher dimensional versions. While the Euclidean discrete lattice is too confining the Barycentric subdivision of an arbitrary graph makes things more natural. In the case a graph has no triangle, the limiting operator is almost periodic over the dyadic integers. The underlying translation is the von Neumann Kakutani system and the density of states the equilibrium measure on a Julia set.
• An integrable evolution equation in geometry as well as a generalization to the complex shows that any exterior derivative both in Riemannian geometry or in graph theory allows an isospectral deformation in form of a Lax pair. I had worked on integrable systems as grad student originally with the hope to use it as a tool to prove an entropy estimation problem in Hamiltonian dynamics which eventually crashed after an attempt to beat a deadline. The deformation system is part of the Noether symmetry of any geometry and not only natural, it is also exciting from the mathematical and physics point of view. The deformation always produces an expansion of space with an inflationary start: for small graphs, we can even compute the evolution explicitly. When allowing the deformation to become complex, we obtain asymptotically a wave evolution.
• The limiting structure of the Birkhoff sum of the Cot function is I believe the first example, where one can compute explicitly the Birkhoff sum of an aperiodic dynamical system for arbitrary large sums because we know everything about it. The random walk converges asymptotically to a self similar graph which is given analytically even so highly non-smooth. While it appears first that the cot function is special, it is quite universal: we could adding general random variables with infinite variance. The Cauchy distribution is the "high risk" version of the Gaussian distribution. This paper especially answers a question worked on with Tangerman. The mathematics is closely related also to the analysis of the Zeta function for circular graphs, where I proved a "Baby Riemann Hypothesis" (no relation with the real Riemann hypothesis, which is the case of the circle).
• After some summer work with Jenny Nitishinskaya lead to Coloring graphs using topology and Graphs with Eulerian unit spheres (see also curvature). The idea is simple and illustrated here and here: embed a planar graph into a 3 dimensional Eulerian sphere to 4 color it. I currently believe this works. some notes here: to prove the 4 color theorem one has only to be able to 4-color any discrete 2-sphere (this follows from work of Cayley and Whitney). Now write the 2-sphere as the boundary of a three ball. Now make edge refinements with edges in the interior of the ball to render the ball Eulerian in the inside (we essentially construct an Eulerian 3 sphere in which the 2-sphere is embedded. It is a basic idea going back to Fiske that an Eulerian 3 sphere is 4 colorable. Of course, any embedded 2 sphere is then also 4 colorable. To make the 3 ball Eulerian, we have to cut edges in an organized way. There appears currently no difficulty but it still needs to be programmed and tested.
• A fascinating class of networks we call orbital networks were found with Montasser Ghachem. Some write-ups: Part 1, part 2, and part 3. Unfortunately, due to other commitments we both had to move on to other projects. For now, we have just to admire the immense beauty of these graphs.
• I love elementary number theory since early on. These are topics which also produce nice teaching material. Here are examples: Additive number theory (theory of partitions) (highschool), Encounters with Goldbach (college), Sphere packings (1995), Multivariable Chinese remainder theorem (2005-2012), Diophantine equations (2006), Perfect numbers (2007), Hunting Euler bricks (2009). A prime project page (2016) and a graph on which a Morse function shows that counting is topology.

Oliver Knill, Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA. SciCenter 432 Tel: (617) 495 5549, Email: knill@math.harvard.edu Quantum calculus blog, Twitter, Youtube, Vimeo, Linkedin, Scholar Harvard, Academia, Harvard Academia, Google Scholar, ResearchGate, Slashdot, Ello, Webcam, Fall 2019 office hours: Tentative: Mon-Fri 11:30-12:30 AM and by appointment.