Distances to Lattice Points and Integer Linear Programming
My current research is in 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 well known 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). I aim to obtain optimal worstcase upper bounds for the distance between an optimal solution to the LP to some nearby feasible integer solution, which solves the ILP.
Publications & Preprints

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:

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:

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)
Responsibilities

International Journal for Mathematics Teaching and Learning  Reviewer

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