Faculty of Science URSA Project: Tropical Quantifier Elimination with Real Implications

This project is one of a number that are in competition for funding from the University of Bath URSA competition, for entry in September 2025.

Primary Supervisor: Dr. Ali K. Uncu 

Secondary Supervisor: Hebron & Medlock Professor of Information Technology Prof Dr James H. Davenport 

Overview of the Research:

Many computational problems can be phrased as systems of polynomial relations with quantifiers. For example: do we have two rainy days for every sunny day in the UK? Another, less tongue-in-cheek, example: does there exist a safe flight path from A to B with all those other flights already up in the air? 

Eliminating the quantifiers — the “for every” and “there exist” above — from such a quantified question is an important step in solving these problems. Real (in the sense of real numbers rather than complex numbers) quantifier elimination (QE) is highly useful and is present everywhere from autonomous vehicles to optimization research. However, the computational complexity of QE is high when a system is studied over the real numbers. This is due to the degrees of polynomials getting higher as the necessary computations proceed. Working in a different algebra, where we replace the operation of multiplication with taking maximum, would keep the degrees low. Such systems are called tropical algebras. Recently breakthroughs showed that tropical algebra structures admit QE. It is also known that a lot of information can be carried over between real systems and their tropical counterparts. We want to take advantage of these ideas by developing new QE procedures and investigating the relationship between tropical QE and real QE.  

Join us if you would like to be the one to bridge the tropical and the real worlds for QE problems. You will develop and implement brand new procedures that will streamline complex mathematical models, optimize computation, and enhance our understanding in many mathematical fields such as optimization, combinatorics, analysis and algebra. You will be a part of a highly collaborate team of experts that tackles real-world problems and contribute to cutting-edge research that connects theory with practical applications.  

Research Environment 

The successful applicant will experience a unique and intellectually stimulating research environment within the Department of Computer Science and the Department of Mathematical Sciences, undertake a specific programme of specialist research skill development, and contribute to advancing a strategic research agenda. The successful applicant will be a part of the Mathematical Foundations of Computation group of the Computer Science department, one of UK’s leading centres in computational mathematics. It is an expanding group with wide variety of research interests that will enhance the applicant’s knowledge. The successful applicant will enjoy a collaborative research environment within the group and the departments. They will have full access to University of Bath resources, mentoring, and all offered in-house training. 

Candidate Requirements:

Applicants should hold, or expect to receive, a First Class or good Upper Second Class UK Honours degree (or the equivalent) in a relevant subject. A master’s level qualification would also be advantageous.

Non-UK applicants must meet the programme’s English language requirement by the application deadline.

Applicants without a Master’s level degree in computer science, mathematics, physics, or a strongly correlated field, would have to provide strong justification (and evidence) that they would be able to handle the mathematics and programming necessary to complete a PhD in this field. 

·      Being self-motivated and enthusiastic about doing high-quality research. 

·      Experience in programming and familiarity with computer algebra softwares (e.g. Maple, Mathematica, SageMath) is preferred. 

·      Experiences of presenting or preparing scientific manuscripts in journals is advantageous. 

Enquiries and Applications:

Informal enquiries are encouraged and should be directed to Dr. Ali K. Uncu at [email protected]   

Formal applications should be submitted via the University of Bath’s online application form for a PhD in Computer Science prior to the closing date of this advert.

PLEASE BE AWARE: We encourage early applications and will be interviewing on a rolling basis. We may close the advert early if we find a suitable before the closing date. We therefore recommend that you contact the lead supervisor prior to applying and submit your formal application as early as possible.

IMPORTANT:

When completing the application form:

1.      In the Funding your studies section, select ‘University of Bath URSA’ as the studentship for which you are applying.

2.      In the Your PhD project section, quote the project title of this project and the name of the lead supervisor in the appropriate boxes. 

Failure to complete these two steps will cause delays in processing your application and may cause you to miss the deadline.

More information about applying for a PhD at Bath may be found on our website.

Equality, Diversity and Inclusion:

We value a diverse research environment and aim to be an inclusive university, where difference is celebrated and respected. We welcome and encourage applications from under-represented groups.

If you have circumstances that you feel we should be aware of that have affected your educational attainment, then please feel free to tell us about it in your application form. The best way to do this is a short paragraph at the end of your personal statement.