**Download slides:** The troublesome reflection rule (TYPES 2015) [PDF].

A position is available for a PhD student at the University of Ljubljana in the general research area of modelling and reasoning about computational effects. The precise topic is somewhat flexible, and will be decided in discussion with the student. The PhD will be supervised by Alex Simpson who is Professor of Computer Science at the Faculty of Mathematics and Physics.

The position will be funded by the Effmath project (see project description). Full tuition & stipend will be provided.

The candidate should have a master’s (or equivalent) degree in either mathematics or computer science, with background knowledge relevant to the project area. The student will officially enrol in October 2015 at the University of Ljubljana. No knowledge of the Slovene language is required.

The candidates should contact Alex.Simpson@fmf.uni-lj.si by email as soon as possible. Please include a short CV and a statement of interest.

]]>The candidate should have as many of the following desiderata as possible, and at the very least a master’s degree (or an equivalent one):

- a master’s degree in mathematics, with good knowledge of computer science
- a master’s degree in computer science, with good knowledge of mathematics
- experience with functional programming
- experience with proof assistants
- familiarity with homotopy type theory

The student will officially enrol in October 2015 at the University of Ljubljana. No knowledge of Slovene is required. However, it is possible, and even desirable, to start with the actual work (and stipend) earlier, as soon as in the spring of 2015. The candidates should contact me by email as soon as possible. Please include a short CV and a statement of interest.

**Update 2015-03-28:** the position has been taken.

The zeroes GitHub repository contains the code.

There really is not much to show. The C program which computes the zeroes uses the GNU Scientific Library routines for zero finding and is just 105 lines long. It generates a PPM image which I then processed with ImageMagick and ffmpeg. The real work was in the image processing and the composition of movies. I wrote a helper Python program that lets me create floyovers the big image, and I became somewhat of an expert in the use of ImageMagick.

The code also contains a Python program for generating a variation of the picture in which roots of lower degrees are represented by big circles. I did not show any of this in the TEDx talk but it is available on Github.

Oh, and a piece of Mathematica code that generates the zeroes fits into a tweet.

The “Zeroes” Vimeo album contains the animations. The ones I showed in the TEDx talk are in Full HD (1920 by 1080). There is also a lower resolution animation of how zeroes move around when we change the coefficients. Here is one of the movies, but you really should watch it in Full HD to see all the details.

- The computed image zeroes.ppm.gz (125 MB) at 20000 by 17500 pixels is stored in the PPM format. The picture is dull gray, and is not meant to be viewed directly.
- The official image zeroes26.png (287 MB) at 20000 by 175000 pixels in orange tones. Beware, it can bring an image viewing program to its knees.
- I computed tons of closeups to generate the movies. Here are the beginnings of each animation available at Vimeo, and measly 1920 by 1080 pixels each (click on them).

**Abstract: **In constructive mathematics we often consider implications between non-constructive reasoning principles. For instance, it is well known that the Limited principle of omniscience implies that equality of real numbers is decidable. Most such reductions proceed by reducing an instance of the consequent to an instance of the antecedent. We may therefore define a notion of *instance reducibility*, which turns out to have a very rich structure. Even better, under Kleene’s function realizability interpretation instance reducibility corresponds to Weihrauch reducibility, while Kleene’s number realizability relates it to truth-table reducibility. We may also ask about a constructive treatment of other reducibilities in computability theory. I shall discuss how one can tackle Turing reducibility constructively via Kleene’s number realizability.

**Slides with talk notes: ** lc2014-slides-notes.pdf