Dr Bogdan Grechuk

I am an Associate Professor at School of Computing and Mathematics Sciences, University of Leicester. 

I fell in love with mathematics as a child, taking part in various mathematics competitions, including the International Mathematical Olympiad where I won gold and silver medals. I have got a Ph.D. from Moscow Institute of Physics and Technology (MIPT, Russia) in 2006, and then another Ph.D. from Stevens Institute of Technology, USA, in 2009. After one year in the University of Edinburgh, I have joined Leicester in January 2011.  

My research vision is to develop mathematical methods that will transform and improve the world of financial markets, medicine, and industry, and transform the way people do mathematics.

I started my research career working on a new mathematical theory of high-gain safe investment strategies. The theory allows stakeholders to increase profit from investments without suffering from increased risks. 

In Machine Learning (ML), which I joined only in 2018, I made a significant contribution to a theory explaining efficiency and generality of novel error-correction mechanisms in AI systems. The importance, value,  and impact of my work is evidenced by that the error-correction mechanisms to which my results apply already underpin 2 US patents, UoA10  REF2021 Impact Case Study, and were used in Ingenic T01 chip featuring in smart cameras (Hive camera). The results, evaluated as 4* by external review, are published in leading journals and presented at a leading conference in this area (IJCNN, 2019). 

In 2020, I initiated and won, as a PI, a 60K Institute and Faculty of Actuaries (IFoA) tender on modelling mortality risks in people with diabetes. In 2020, as a Co-I, I contributed to successful award of 2 KTP projects with the overall value about 600K. There I lead key work on modelling, managing and understanding risk, which is at the heart of these projects.

In addition to my main research interests in Financial mathematics and machine learning I am interested in

Exposition of general mathematics: I have wrote two book about great recent mathematical theorems.

Automated theorem proving. The idea that mathematical theorems and proofs should be verified automatically by a computer has been around for at least 45 years. The progress however is extremely slow. Only a tiny fraction of all theorems in mathematics are formalised. I believe that the main reason for this is that writing down mathematics using a special language is a challenging and difficult task. My long-term aim is to help to create a program for formalisation statements of mathematical theorems and paper abstracts with fully natural language interface. 

Diophantine equations. The idea is to arrange all Diophantine equation by size and solve them systematically starting from the smallest ones.

Monographs

B. Grechuk, Landscape of 21st Century Mathematics, Springer, Cham, 2021, https://doi.org/10.1007/978-3-030-80627-9

B. Grechuk, Theorems of the 21st Century, Springer, Cham, 2019, https://doi.org/10.1007/978-3-030-19096-5

Selected journal publications

Grechuk B. On the smallest open Diophantine equations. ACM SIGACT News 2022, 53(1), pp 36-57

Gorban, A., Grechuk, B., Mirkes, E., Stasenko, S., Tyukin, I. High-dimensional separability for one-and few-shot learning. Entropy 2021, 23(8), 1090

B. Grechuk, A.N. Gorban, I.Y. Tyukin, General stochastic separation theorems with optimal bounds. Neural Networks Vol. 138, 2021, pp 33-56

Gorban, A. N., Golubkov, A., Grechuk, B., Mirkes, E. M., Tyukin, I. Y. Correction of AI systems by linear discriminants: Probabilistic foundations. Information Sciences, Vol. 466, 2018, pp 303-322

Grechuk, B., Zabarankin, M., Sensitivity Analysis in Applications with Deviation, Risk, Regret, and Error Measures. SIAM Journal on Optimization, Vol. 27, Issue 4, 2017, pp 2481-2507 

Grechuk, B., Molyboha, A., Zabarankin, M., ""Mean-Deviation Analysis in the Theory of Choice"". ""Risk Analysis"", Vol. 32, No. 8, August 2012, pp. 1277-1292

Grechuk, B., Molyboha, A., Zabarankin, M., ""Chebyshev Inequalities with Law Invariant Deviation Measures,"" Probability in the Engineering and Informational Sciences, Vol. 24, No. 1, 2010, pp. 145-170

Grechuk, B., Molyboha, A., Zabarankin, M., ""Maximum Entropy Principle with General Deviation Measures,"" Mathematics of Operations Research, Vol. 34, No. 2, 2009, pp. 445-467

Topic 1. Risk evaluation in finance and portfolio optimization. See my publications for details.

Topic 2. Application of machine learning to automated theorem proving.

The idea that mathematical theorems and proofs should be verified automatically by a computer has been around for at least 45 years. The progress, however, is extremely slow. Only a tiny fraction of all theorems in mathematics are formalised. We believe that the main reason for this is that writing down mathematics using a special language is a challenging and difficult task. This project proposes to create a program for formalisation statements of mathematical theorems and paper abstracts with fully natural language interface. The program will be AI-powered, and learn continuously from human experts. 

Topic 3. Diophantine equations.

This project is to arrange all Diophantine equation by size and solve them systematically, starting from the smallest ones. Try to solve as many equations as you can, and leave the smallest equations you cannot solve as open questions. Maybe, one of the equations from your list will remain open for centuries, and got the name XXX last equation, where XXX is your surname.

I currently teach

MA2404/MA7404 Markov Processes, Semester 1 

MA2510 Investigations in Mathematics, Semester 1

MA3073/MA4073/MA7073 Financial Risk, Semester 2

MA7514 Models and Mortality (Distance Learning)

and supervise various projects including

MA7002 Financial Mathematics Project, Summer

MMath project, Semesters 1 and 2

MA3516 Undergraduate projects

I would be happy to talk about my books

B. Grechuk, Landscape of 21st Century Mathematics, Springer, Cham, 2021

B. Grechuk, Theorems of the 21st Century, Springer, Cham, 2019, https://doi.org/10.1007/978-3-030-19096-5

that offer a detailed cross section of contemporary mathematics. Important results of the 21st century are motivated and formulated, providing an overview of recent progress in the discipline. The theorems presented in these books have been selected among recent achievements whose statements can be fully appreciated without extensive background. Grouped by subject, the selected theorems represent all major areas of mathematics: number theory, combinatorics, analysis, algebra, geometry and topology, probability and statistics, algorithms and complexity, and logic and set theory. The presentation is self-contained with context, background and necessary definitions provided for each theorem. Rigorous yet accessible, these books presents an array of breathtaking recent advances in mathematics. It is written for everyone with a background in mathematics, from inquisitive university students to mathematicians curious about recent achievements in areas beyond their own.

2009, Prize for excellence in graduate research, Department of Mathematical Sciences, Stevens Institute of Technology.
2002, Gold Medal, International Mathematical Olympiad for students, Poland
1998, Silver Medal, 39th International Mathematical Olympiad for high school students, Taipei
1997, Gold Medal, 38th International Mathematical Olympiad for high school students, Argentina

Practical stochastic separation theorems for product distributions B Grechuk 2019 International Joint Conference on Neural Networks (IJCNN), 1-8 2019 

Kernel Stochastic Separation Theorems and Separability Characterizations of Kernel Classifiers IY Tyukin, AN Gorban, B Grechuk, S Green 2019 International Joint Conference on Neural Networks (IJCNN), 1-6 2019

Workshop: Risk Management Approaches in Engineering Applications 2018, University of Florida, October 2018. Presentation: “Decision Making under Catastrophic Risk”

APMOD 2014 conference, University of Warwick, April 2014, Presentation “Inverse Portfolio Problem with Mean-Deviation Model”

INFORMS 2011 Annual Meeting, Charlotte, NC, Nov. 2011. Presentation ""Risk Averse Decision Making under Catastrophic Risk""

INFORMS 2011 Annual Meeting, Charlotte, NC, Nov. 2011. Presentation ""Cooperative Games with General Deviation Measures""

INFORMS 2008 Annual Meeting, Washington, DC, Oct. 2008. Presentation ""Probability Inequalities with General Deviation Measures""

INFORMS 2007 Annual Meeting, Seattle, WA, Nov. 2007. Presentation ""Maximum Entropy Principle with General Deviation Measures""

In addition to my main research interests in Financial mathematics and machine learning, I am interested in

Exposition of general mathematics: I have wrote two book about great recent mathematical theorems.

Automated theorem proving. The idea that mathematical theorems and proofs should be verified automatically by a computer has been around for at least 45 years. The progress, however, is extremely slow. Only a tiny fraction of all theorems in mathematics are formalised. I believe that the main reason for this is that writing down mathematics using a special language is a challenging and difficult task. My long-term aim is to help to create a program for formalisation statements of mathematical theorems and paper abstracts with fully natural language interface. 

Diophantine equations. The idea is to arrange all Diophantine equation by size and solve them systematically, starting from the smallest ones. 

HEA Fellow April 2013 University of Leicester (Recognition reference PR057705)