Post-Doctoral Research Visit F/M Multipoint evaluation and spatial subdivisions
Contract type : Fixed-term contract
Level of qualifications required : PhD or equivalent
Fonction : Post-Doctoral Research Visit
Context
The INRIA team Gamble is part of the INRIA Nancy Grand Est research center and also of the department Algorithms, Computation, Geometry and Image of the LORIA laboratory of Université de Lorraine.
Classical computational geometry usually deals with linear objects in a Euclidean setting and when other situations happen, curved objects are typically linearized and non-Euclidean spaces are locally approximated by Euclidean spaces. The goals of the Gamble team are to address such limitations of classical computational geometry.
Assignment
The goal of this postdoc is to study questions that are relevant both to discrete and computational geometry and to computer algebra, by combining combining classical methods and recent developments from both fields.
As an example, consider the typical problem, underlying several questions in computational geometry and computer algebra, is semi-algebraic range counting: given a set P of n points and a set F of polynomials, count the number of pair (f,p) ∈ F × P such that f(p) > 0. Consider for instance the case where the points are in ℝ2 and the polynomials are bivariate. On one hand, recent results in computational geometry show that if F consists of n polynomials of constant degrees, then it is possible to do it in O(n4/3) operations [1]. On the other hand, results in computer algebra based on a completely different set of tools show that if F is reduced to a single polynomial of degree n1/2, then the problem can be solved in O(n1.334) operations [2]. Is it possible to combine these two sets of tools to obtain better bounds? Or could we obtain lower bounds for those problems, in the spirit of the recent lower bounds obtained for selialgebraic range reporting problems [3]?
State of the art
The main tools from computational geometry that we will consider are cuttings [4, Ch. 4 and 6] and their algebraic generalization [5]. Initially, cuttings were introduced to allow divide-and-conquer algorithms in geometry. For example, given n lines in the plane and a parameter 0 < ε < 1, a ε−cutting is a triangulation of ℝ2 such that each triangle contains at most εn lines.
In computer algebra, the main tools used for fast bivariate polynomial evaluation are based on fast matrix multiplication [6, Ch. 12]. This tool was used to compute the modular composition of polynomials [7], and later extended to evaluate a bivariate polynomial on multiplie points [2].
An important data structure underlying these approaches is the biclique partition. Given a subset S of a product of sets F × P, a biclique partition of S is a union of pairwise distinct subsets Si ⊂ S of the form Fi × Pi ⊂ F × P such that S = ∪iSi. This kind of data structure appears notably in several geometric incidence and algebraic rigidity problems. Such problems include bounding the number of line intersections, or bounding the number of intersection of an algebraic surface with a Cartesian product [8].
Main activities
The candidate will study the recent results on semi-algebraic-relational queries to try and narrow the gaps between upper and lower bounds. They will try to improve asymptotic bounds on related problems in computational geometry and computer algebra, and to compare and combine the methods from both fields.
Depending on the interests of the candidate, the postdoc can open up in a number of directions. For example, these problems are naturally studied in the Real RAM model of computation, but a comparison to the word-RAM model is relevant (in particular, it would be interesting to inspect the frontier between the classes NP and ETR, if any). Other possiblities include, but are not limited to, investigating from a computer algebra standpoint problems in dicrete and computational geometry where polynomials play a key role (e.g. the various phenomena of algebraic rigidity that arise in incidence geometry, semi-algebraic graphs arising from geometry, certain LP-type optimization problems, etc.).
Skills
Academic background: Ph.D. in computer science or mathematics.
Required knowledge and skills: algorithms in discrete and computational geometry.
Benefits package
- Subsidized meals
- Partial reimbursement of public transport costs
- Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
- Possibility of teleworking (after 6 months of employment) and flexible organization of working hours
- Professional equipment available (videoconferencing, loan of computer equipment, etc.)
- Social, cultural and sports events and activities
- Access to vocational training
- Social security coverage
Remuneration
2788 € gross/month
General Information
- Theme/Domain : Algorithmics, Computer Algebra and Cryptology
- Town/city : Villers lès Nancy
- Inria Center : Centre Inria de l'Université de Lorraine
- Starting date : 2024-09-01
- Duration of contract : 2 years
- Deadline to apply : 2024-04-28
Warning : you must enter your e-mail address in order to save your application to Inria. Applications must be submitted online on the Inria website. Processing of applications sent from other channels is not guaranteed.
Instruction to apply
Defence Security :
This position is likely to be situated in a restricted area (ZRR), as defined in Decree No. 2011-1425 relating to the protection of national scientific and technical potential (PPST).Authorisation to enter an area is granted by the director of the unit, following a favourable Ministerial decision, as defined in the decree of 3 July 2012 relating to the PPST. An unfavourable Ministerial decision in respect of a position situated in a ZRR would result in the cancellation of the appointment.
Recruitment Policy :
As part of its diversity policy, all Inria positions are accessible to people with disabilities.
Contacts
- Inria Team : GAMBLE
-
Recruiter :
Moroz Guillaume / Guillaume.Moroz@inria.fr
About Inria
Inria is the French national research institute dedicated to digital science and technology. It employs 2,600 people. Its 200 agile project teams, generally run jointly with academic partners, include more than 3,500 scientists and engineers working to meet the challenges of digital technology, often at the interface with other disciplines. The Institute also employs numerous talents in over forty different professions. 900 research support staff contribute to the preparation and development of scientific and entrepreneurial projects that have a worldwide impact.