PhD Position F/M Effective algebraic invariants for multi-parameter persistence

June 05, 2023
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2023-06310 - PhD Position F/M Effective algebraic invariants for multi- parameter persistence

Contract type : Fixed-term contract

Level of qualifications required : Graduate degree or equivalent

Fonction : PhD Position

About the research centre or Inria department

The Inria Saclay-Île-de-France Research Centre was established in 2008. It has developed as part of the Saclay site in partnership with Paris-Saclay University and with the Institut Polytechnique de Paris .

The centre has 39 project teams, 27 of which operate jointly with Paris- Saclay University and the Institut Polytechnique de Paris; Its activities occupy over 600 people, scientists and research and innovation support staff, including 44 different nationalities.


Topological Data Analysis (TDA) is among the recent trends in exploratory data analysis, and it has aroused a steadily growing interest within the last two decades. Its main contribution has been to provide new types of data descriptors that encode information about the topology of data.These descriptors are of an essentially different nature compared to classical descriptors encoding geometric or statistical quantities.

Moreover, they are invariant under reparametrization of the data (so no need for coordinates, just distances or (dis-)similarity measures), they are provably stable with respect to small perturbations of the data, and they can be defined and computed in very general settings and under very mild assumptions on the structure of the data. With such properties, they have found applications in a wide range of scientific areas.

One of the main current challenges in TDA is to improve the sensitivity and discrimination power of its descriptors. Traditionally, these descriptors are defined from a 1-parameter algebraic construction, which is to take the homology of the sublevel sets of a real-valued function; a natural approach to improve their sensitivity is to consider a multi-parameter construction using vector-valued functions. This changes everything at the algebra level, where the objects under consideration, called persistence modules, are now no longer indexed over the real line R, but over more general posets, typically Rd. While R-indexed modules have a simple algebraic structure, being decomposable as direct sums of thin modules supported on intervals, Rd-indexed modules can have arbitrarily complicated direct summands, whose classification is basically impossible as the underlying indexing poset is of wild type. In view of this negative fact, in the recent years the TDA community has put substantial effort into defining and studying incomplete invariants for persistence modules.

Main activities

As these invariants are incomplete, a central question is how much of the algebraic structure of the modules they are able to capture, and consequently, how they can be interpreted. Another question is about the trade-off between improving the computational complexity of computing and storing these invariants, versus giving up on some of their sensitivity. A last question is to find ways of converting these invariants into meaningful, stable, possibly learnable features for Machine Learning, as has been previously done for invariants of 1-parameter persistence modules. The goal of this Ph.D. is to address these three questions for the existing invariants of multi-parameter persistence modules, and possibly for new invariants to come. These questions form the three parts of the work program, whose details can be found in the full description (see link below).

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A solid background in applied algebraic topology and theoretical computer science.

Meanwhile, some knowledge and experience in data science.

For code development, good programming skills in C++ and Python.

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
  • 1st and 2nd year : 2051€ gross/month
  • 3rd year : 2158€ gross/month
  • General Information
  • Theme/Domain : Optimization and control of dynamic systems
  • Town/city : Palaiseau
  • Inria Center : Centre Inria de Saclay
  • Starting date : 2023-09-01
  • Duration of contract : 3 years
  • Deadline to apply : 2023-06-05
  • Contacts
  • Inria Team : GEOMERIX
  • PhD Supervisor : Oudot Steve / [email protected]
  • 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.

    Instruction to apply

    To apply, please provide :

  • CV
  • Cover letter
  • Letter(s) of recommendation, where applicable
  • 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.

    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.

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