This is a list of questions and topics I would like to research. If you find any of them interesting and would like to collaborate, drop me an email.
Working papers
These papers need coauthors in order to realize their full potential:

Fairly Dividing a Cake after Some Parts were Burnt in the Oven: Given a cake and n agents with different valuations of its pieces, where the valuations can be positive or negative, does there always exist a connected envyfree division? The paper proves that the answer is yes when n=3. Breaking news: Frédéric Meunier and Shira Zerbib have just proved that the answer is yes when n=4 or n is a prime number. It is still open whether the answer is yes for all n. The proofs combine combinatorics and algebraic topology.

How to redivide a cake fairly? and PriceofFairness: Combines computational geometry, combinatorics, economics and social choice. I am looking for experts in any of these fields in order to help verify and improve the results.
Reasearch projects

Experiments in fair division. I have some simulation results on fair division of land, and an online game for experiments with humans on fair cakecutting. I need partners to advance.

Implementing fair division algorithms in realestate projects. Currently, apartments are divided by lottery and/or protection… There are better ways and I would like to check how they can be implemented.
Open Questions
I would very much like to know the answer to the following questions.

Two agents with different incomes have to divide a set of indivisible items. Does a competitive equilibrium exist for almost all incomes? This question was raised by Babaioff and Nisan and TalgamCohen. I solved some related open problems but the case of two additive agents is still open.

How many cuts are required for cakecutting when the agents have different entitlements? Mainly a combinatorics question. May be related to measure theory and topology. My MathOverflow question has some preliminary results.

Is it possible to find in bounded time an envyfree and proportional cakedivision with connected pieces? Mainly an algorithmic question. The Wikipedia page I wrote on envyfree cakecutting gives some background. Our wastemakeshaste paper presents the status as of 8/2016.

Given a collection of points in the plane, what is the largest possible collection of pairwisedisjoint wetsquares? Mainly a geometric question. This draft presents the question formally and gives some directions. Our fairandsquare papers (extended abstract and full preprint) present the economic motivation.