PUMAS: PUre MAthematics for Statistical inference
Lead Research Organisation:
University of Oxford
Department Name: Mathematical Institute
Abstract
Algebraic geometry plays a prominent role in pure mathematics as illustrated by the 2018 Fields Medal awarded to the UK-based mathematician Prof Caucher Birkar. Algebraic Geometry is centered on the important problem of solving polynomial equations, which has captivated mathematicians for thousands of years. A polynomial equation is an equation that has multiple terms made up of numbers and variables. Such equations are ubiquitous in mathematical sciences and arise naturally in various contexts, from computational biology to natural language processing.
In recent years, modern algebraic geometry techniques and algorithms have been used in seemingly unrelated fields, such as statistics, evolutionary biology and integer programming. As a result, there has been an increasing emphasis on computational and algorithmic problems in algebraic geometry and the new field of Algebraic Statistics has emerged.
This research programme has a two-fold overarching aim: firstly, to apply powerful tools from algebraic geometry to advance statistical methods, and secondly, to generate new mathematics motivated by models in biology, networks and data science. The idea is that many statistical models correspond to geometrical objects and one can employ techniques from algebraic geometry to study their properties. For example:
*One advantage of using algebraic geometry is that the mathematical language of polynomials is appropriate to model complex processes accurately. For instance, when studying the evolution of species, it enables us to lift restricting modelling assumptions by allowing species to have evolved with different mutation rates or even different parts of a gene to have mutated with different rates.
*At the same time, passing from a statistical model to its geometric description and deriving the equations that describe it presents a new mathematical challenge in algebraic geometry. This challenge is connected to the statistical problem of whether categorical characteristics, for instance gender differences and mortality, are related.
*Finally, mathematical models that are used in Artificial Intelligence often present anomalies and are called "singular". For such singular models, a celebrated theorem in algebraic geometry for which Hironaka received the Fields medal in 1970, can be exploited to address the problem of choosing the best model that explains the data for which current statistical methods fail.
This research programme will contribute to the development of a toolkit of superior algebraic and geometric methods which end users of the statistical theory, such as biologists and computer scientists, can access and exploit.
In recent years, modern algebraic geometry techniques and algorithms have been used in seemingly unrelated fields, such as statistics, evolutionary biology and integer programming. As a result, there has been an increasing emphasis on computational and algorithmic problems in algebraic geometry and the new field of Algebraic Statistics has emerged.
This research programme has a two-fold overarching aim: firstly, to apply powerful tools from algebraic geometry to advance statistical methods, and secondly, to generate new mathematics motivated by models in biology, networks and data science. The idea is that many statistical models correspond to geometrical objects and one can employ techniques from algebraic geometry to study their properties. For example:
*One advantage of using algebraic geometry is that the mathematical language of polynomials is appropriate to model complex processes accurately. For instance, when studying the evolution of species, it enables us to lift restricting modelling assumptions by allowing species to have evolved with different mutation rates or even different parts of a gene to have mutated with different rates.
*At the same time, passing from a statistical model to its geometric description and deriving the equations that describe it presents a new mathematical challenge in algebraic geometry. This challenge is connected to the statistical problem of whether categorical characteristics, for instance gender differences and mortality, are related.
*Finally, mathematical models that are used in Artificial Intelligence often present anomalies and are called "singular". For such singular models, a celebrated theorem in algebraic geometry for which Hironaka received the Fields medal in 1970, can be exploited to address the problem of choosing the best model that explains the data for which current statistical methods fail.
This research programme will contribute to the development of a toolkit of superior algebraic and geometric methods which end users of the statistical theory, such as biologists and computer scientists, can access and exploit.
People |
ORCID iD |
| Dimitra Kosta (Principal Investigator / Fellow) |