Distances to Lattice Points and Integer Linear Programming
I received my PhD in mathematics from Cardiff University at the start of 2022 under the supervision of Professor Iskander Aliev. My PhD thesis entitled "distances to lattice points in rational polyhedra" is available below. My current research lies within the field of integer optimisation. This mathematical discipline looks at optimisation or feasibility problems where some or all of the variables are restricted to integers. Because integer optimisation is concerned with discrete values, it is unsurprising that it is widely used to model problems in many real world scenarios including operational research tasks such as scheduling and manufacturing.
I currently focus on 'integer linear programming' (ILP) problems. These problems have the property that both the constraints and objective functions are linear. It is wellknown that in general, finding a solution to an ILP is NPcomplete. Despite this, one can obtain an approximation within polynomial time by solving the problem's related 'linear program' (LP). One of my aims is to under certain assumptions refine current worstcase upper bounds for the distance between an optimal solution to the LP to some nearby feasible integer solution, which solves the ILP. Further, I am actively interested in how the righthand side of an ILP impacts on the feasibility of the problem and the sparsity of the underlying solutions.
For those who are interested, my current Erdös Number is 3 (meaning I am a coauthor of a coauthor of a coauthor of Erdös). In particular, the connection to Erdös is A. Williams => I. Aliev => P. Gruber => P. Erdös.
Publications & Preprints

Insights to Weighted Sum Sampling Approaches for MultiCriteria Decision Making Problems
A. Williams and Y. Cai (2024)

Markov Chain Monte Carlo Methods for Dynamic Student Assessment
A. Williams (2024)

Identifying Job Satisfaction Parameters among the Employees in Higher Educational Institutions: A Mathematical Model
M. Bhatia and A. Williams (2024)
[WSEAS Transactions on Business and Economics] ; [arXiv.org]

Considering a Classical Upper Bound on the Frobenius Number
A. Williams and D. Haijima (2023)

Selection of Criteria Using MCDM Techniques  An Application in Renewable Energy
M. Bhatia and A. Williams (2023)
[American Journal of Operations Management and Information Systems] ; [arXiv.org]

Target Setting: A Case Study Looking at How Greater Collaboration Can Impact the Effectiveness of Target Setting
A. Williams (2022)
[WSEAS Transactions on Business and Economics]

Finding an Optimal Proximity Bound in a Very Special Scenario
A. Williams (2022)
[WSEAS Transactions on Mathematics]

Distances to Lattice Points in Rational Polyhedra (PhD Thesis)
A. Williams (2022)

Intelligent Security Control Based on the New Criterion of Edwards and Montgomery Curves, Isogenous of These Curves Supersingularity
R. Skuratovskii, A. Williams and Y. Osadchyy (2021)
[Proceedings of International Conference on Advanced Computing Applications]

DistanceSparsity Transference for Vertices of Corner Polyhedra
I. Aliev, M. Celaya, M. Henk and A. Williams (2021)
[SIAM Journal on Optimization] ; [arXiv.org]

Irreducible bases and subgroups of a wreath product in applying to diffeomorphism groups acting on the Möbius band
R. Skuratovskii and A. Williams (2020)
[Rendiconti del Circolo Matematico di Palermo Series 2 (Springer)]

Some Approach to Key Exchange Protocol Based on NonCommutative Groups
R. Skuratovskii and A. Williams (2020)
[International Journal of Mathematical Models and Methods in Applied Sciences] ; [arXiv.org]

A Solution of the Inverse Problem to Doubling of Twisted Edwards Curve Point over Finite Field
R. Skuratovskii and A. Williams (2019)
[Wydawnictwo Naukowe Akademii TechnicznoHumanistycznej w BielskuBiałej]

Minimal Generating Set and a Structure of the Wreath Product of Groups, and the Fundamental Group of the Orbit Morse Function
R. Skuratovskii and A. Williams (2019)
[Вісник Донецького національного університету. Серія А: Природничі науки] ; [arXiv.org]
Conferences & Workshops
Contributed Talk:

International Meeting on Numerical Semigroups (IMNS)  Universidad de Cádiz (2024)

33rd European Conference on Operational Research  Technical University of Denmark (2024)

75th British Mathematical Colloquium (Number Theory Workshop)  University of Manchester (2024)

Sharing Best Practices in Teaching (LSE ECR Network's Peer Learning Café)  LSE (2023)

Seminar on Combinatorics, Games and Optimisation  LSE (2023)

6th European Conference on Electrical Engineering and Computer Science  Bern, Switzerland (2022)

PhD Seminar on Combinatorics, Games and Optimisation  LSE (2022)

World Conference on Information Systems for Business Management  Bangkok, Thailand (2022)

7th International Conference on Mathematics and Computers in Sciences and Industry  Athens, Greece (2022)

British Early Career Mathematicians' Colloquium  University of Birmingham (2021)

Young Researchers in Mathematics  University of Bristol (2021)

International Conference of Young Mathematicians  Online Kiev, Ukraine (2021)

British Early Career Mathematicians' Colloquium  University of Birmingham (2020)

Algebraic and Geometric Methods of Analysis  Odessa, Ukraine (2020)

Intellectual Systems and Information Technologies  Odessa, Ukraine (2019)

British Combinatorial Conference  University of Birmingham (2019)

SIAMUKIE National Student Chapter Conference  University of Manchester (2019)

International Conference of Young Mathematicians  Institute of Mathematics NAS of Ukraine (2019)

Welsh Mathematics Colloquium  Gregynog Hall (2019)

British Young Mathematicians Colloquium  University of Birmingham (2019)
Contributed Poster:

SIAMUKIE National Student Chapter Conference  University of Manchester (2019)

SIAMIMA Student Chapter Conference  Oxford University (2019)
Attended:

Induction Course for New Lecturers in the Mathematical Sciences  University of Cambridge (2022)

NATCOR Multi Criteria Decision Making  University of Portsmouth (2022)

IPCO and Summer School  London School of Economics (2020)

SCIP Online Workshop 2020  University of Exeter (2020)

Mixed Integer Programming Workshop 2020  Online Edition (2020)

LMS Computer Science Colloquium “Mathematics of Security”  London (2019)

NATCOR Combinatorial Optimisation  University of Southampton (2019)

NATCOR Stochastic Modelling  Lancaster University (2019)

Computational Optimisation in Action  Edinburgh University (2018)

NATCOR Convex Optimisation  Edinburgh University (2018)

Young Researchers in Mathematics  Southampton University (2018)
Society Membership

Society for Industrial and Applied Mathematics (Discrete Mathematics, Linear Algebra, Optimisation)

South Wales Operational Research Discussion Society (SWORDS)
Other Responsibilities

Editorial Board Member of the American Journal of Operations Management and information Systems

Joint organiser of the Combinatorics, Games and Optimisation (CGO) Seminar  London School of Economics

Member of the inaugural Early Career Research (ECR) Steering Group  London School of Economics

Undergraduate Academic Mentor  London School of Economics

International Journal for Mathematics Teaching and Learning  Reviewer

Canadian Journal of Science, Mathematics and Technology Education (Springer)  Reviewer