## Pure Mathematics and Statistics

We carry out research in Algebra, Number Theory, and Logic. We study algebraic structures and their applications to problems arising in algebraic geometry, number theory, and combinatorics. Our work in Logic focuses on intuitionism, including implications for mathematical physics. Our main Statitsics research is on Regime switching models with applications in financial and economic time series. We are also interested in empirical analysis, especially volatility forecasting performance and its application in risk management. Currently, we carry our research on how to facilitate high frequency data to volatility estimation and forecasting. For more information on PhD opportunities, please see here.

The branch of Mathematics now known as Galois theory arose from a natural concrete question: given a polynomial in one variable f(x) (for example: f(x)=x^2+5x+6 , or f(x)=x^5+x+1 ), can we determine the solutions of the equation f(x)=0 (the roots of the polynomial) in terms of the coefficients? This answer to this question for quadratic polynomials is encapsulated in the familiar quadratic formula, and renaissance Mathematicians discovered similar formulae for cubic and quartic polynomials. However, they were then able to prove that there is no such formula for general polynomials of degree 5 or higher. The 19th century French Mathematician Evariste Galois studied certain symmetries present in the roots of polynomials, and established a criterion for determining whether the roots of a particular polynomial could be expressed in terms of its coefficients. The techniques he pioneered became known as Galois theory in his honour.

As the techniques of Galois theory evolved, concrete equations were replaced with abstract Mathematical objects called field extensions. The modern perspective on these problems is that if a particular field extension satisfies certain hypotheses then Galois theory yields important information about its structure and properties. Much of my work has focussed on a generalization of Galois theory known as Hopf-Galois theory. This approach was developed in the 1960s, and generalizes Galois theory in two ways: it applies to a much broader class of field extensions than Galois theory, and it can provide multiple perspectives on a given field extension. This raises the possibility that a particular perspective might be more or less suitable depending upon the precise question that one wishes to address.

Hopf-Galois theory was developed to study problems in algebra and number theory. However, in 2016 an unexpected connection emerged between Hopf-Galois theory and theoretical physics via a new abstract object called a skew brace. This connection has already been fruitful in both directions: results from Hopf-Galois theory have been translated into the language of skew braces, and vice versa.

*Dr. Paul Truman presenting at a conference in 2019*

**Here are some selected publications by Paul Truman: **

Koch, A., & Truman, P. J. (2020). Opposite skew left braces and applications. Journal of Algebra, 546, 218-235.https://doi.org/10.1016/j.jalgebra.2019.10.033

A new construction in the theory of skew left braces. One application of the results is the creation of new solutions to the Yang-Baxter equation in theoretical physics; another (via the results of the paper “Normality and short exact sequences of Hopf-Galois structures”) is to the theory of Hopf-Galois structures, which occur in algebra and number theory.

Truman, P. J. (2020). Hopf-Galois module structure of tamely ramified radical extensions of prime degree. Journal of Pure and Applied Algebra, 224(5), 106231.https://doi.org/10.1016/j.jpaa.2019.106231

One of the advantages of Hopf-Galois theory over classical Galois theory is that it applies in situations where the classical theory does not. This paper is the first to apply this principle to certain problems in algebraic number theory, specifically the structure of rings of algebraic integers in non-normal extensions of global fields.

Koch, A., Kohl, T., Truman, P. J., & Underwood, R. (2019). Isomorphism problems for Hopf–Galois structures on separable field extensions. Journal of Pure and Applied Algebra, 223(5), 2230-2245.https://doi.org/10.1016/j.jpaa.2018.07.014

This paper instigated the study of new structural questions for Hopf algebras occurring in Hopf-Galois theory. We provided answers and classifications in many cases, and gave examples of pathological behaviour. The paper marked a new direction in Hopf-Galois theory, which has been pursued by several authors.

Koch, A., Kohl, T., Truman, P. J., & Underwood, R. (2019). Normality and short exact sequences of Hopf-Galois structures. Communications in Algebra, 47(5), 2086-2101. https://doi.org/10.1080/00927872.2018.1529237

The centerpiece of classical Galois theory is the Fundamental Theorem of Galois theory. This paper generalizes important parts of that theorem to the context of Hopf-Galois theory, which paves the way for generalization of the many applications and consequences of that theorem. Some of these have now been explored by the authors, their collaborators and their students.

**Collaborators:**

Prof Lindsay Childs, University at Albany, USA

Prof Cornelius Greither, University of Munich, Germany

Prof Kevin Keating, University of Florida, USA

Prof Alan Koch, Agnes Scott College, USA

Dr. Timothy Kohl, University of Boston, USA

Prof Robert Underwood, Auburn University at Montgomery, USA

Intuitionism is a philosophy of mathematics, founded by L.E.J. Brouwer in 1907, which views mathematics as a creation of human mathematicians, and as constrained by human limitations. This is very different from the usual platonistic view of mathematics as existing timelessly, independent of human beings. Intuitionistic mathematics appears to have a subjective, time-bound nature, which is very disconcerting to conventionally trained mathematicians. In a recent paper [1] I argue that the difference between these two viewpoints is less than it seems. I examine Brouwer's apparently most subjective concepts, his 'weak counterexamples' and his 'theory of the creative subject', and show that the mathematical work that they do can be understood in a way that is not at all subjective or time-dependent.

**Regime switching modelling and risk management (**Dr Jie Cheng**)**

Traditional regime-switching models are widely used for volatility modelling in empirical economics and finance research for their ability to identify and account for the impact of latent regimes or states on the behaviour of the interested variables. Recently, there are several different ways of modelling endogenous regime changes. By different constructions, the resulting state transition probabilities are time-varying and dependent on the lagged values of the observed time series.

By considering regime switching chains, we proposed the general idea of using Reducible Diffusions (RDs) with nonlinear time-varying transformations for modeling financial and economic variables. Our application suggests that from an empirical point of view time-varying transformations are highly relevant and statistically significant. We expect that the proposed models can describe more truthfully the dynamic time-varying behavior of economic and financial variables and potentially improve out-of-sample forecasts significantly [Bu, R., J. Cheng, and K. Hadri. (2016). Reducible Diffusions with Time-Varying Transformations with Application to Short-Term Interest Rates. Economic Modelling 52: 266–277.].

We examine model specification in regime-switching continuous-time diffusions for modeling S&P 500 Volatility Index (VIX) [Bu, R., Cheng, J., & Hadri, K. (2017). Specification analysis in regime-switching Continuous-time diffusion models for market volatility. Studies in Nonlinear Dynamics and Econometrics, 21(1), 65–80.]. Our investigation is carried out under two nonlinear diffusion frameworks, the NLDCEV and the CIRCEV frameworks, and our focus is on the nonlinearity in regime-dependent drift and diffusion terms, the switching components, and the endogeneity in regime changes. While we find strong evidence of regime-switching effects, models with a switching diffusion term capture the VIX dynamics considerably better than models with only a switching drift, confirming the presence and importance of volatility regimes. Strong evidence of nonlinear endogeneity in regime changes is also detected.

We generalize the latent-factor-driven endogenous regime-switching Gaussian model of Chang, et al., Journal of Econometrics, 2017, 196, 127–143 by allowing the state-dependent conditional distributions to be non-Gaussian. Our setup is more general and promises substantially broader relevance and applicability to empirical studies. We provide evidence to justify our generalization by a simulation study and a real data application. Our simulation results confirm that when the state-dependent dynamics are misspecified, the bias of model parameter estimates, the power of the likelihood ratio test against endogenous regime changes, and the quality of the extracted latent factor all deteriorate quite considerably. [Bu, R., J. Cheng, and Jawadi. Fredj. (2020). A latent-factor-driven endogenous regime-switching non-Gaussian model: Evidence from simulation and application. International Journal of Finance & Economics, DOI: 10.1002/ijfe.2192].

Current research is focussed in understanding bivariate time series models driven by either threshold/smooth transition chains or (exo and endo) regime switching models with latent vector autoregressive factors, which allows for (un)synchronized switching and endogenous feedback. This will allow us to consider dynamics of several financial factors and the interactions among them. Several projects are based on this idea.

Another key topic we are interested in is risk management. It is vital for clearinghouses to employ appropriate market instruments (Margin, capital requirement and price limits) in order to strike a delicate balance between increasing futures price stability, not impairing price discovery, and facilitating futures growth. A framework is proposed that is rooted in extreme value theory to study the performance of margin, capital requirement and price limits and their interactions in the presence of clearing firms’ risk preferences. By applying the concept of self-enforcing contracts, we incorporate clearing firms’ risk attitudes into the framework. The efficacy of these market instruments under three risk measures (i.e. VaRs, ESs and SRMs) is studied, while the latter two risk measures present two approaches to gauge potential losses in the form of capital requirement [Cheng, J., Hong, Y., & Tao J (2015) How do risk attitudes of clearing firms matter for managing default exposure in futures markets? The European Journal of Finance, 22(10)]. The results cast new light on the economic rationale of price limits, which is examined now during the Coronavirus outbreak.

**Collaborators: **Dr Ruijun Bu, Liverpool University, UK

Prof Fredj Jawadi, University of Lille, France

**Current PhD Students**

- George Prestidge (supervisor: Dr Paul Truman)
- Menli Tirkishova (supervisor: Dr Jie Cheng)
- Isabel Martin-Lyons (Supervisor: Paul Truman)

**Recent PhD Students**

- James Appleby (supervisor: Dr Peter Fletcher)
- Stuart Taylor (supervisor: Dr Paul Truman)

- Hopf-Galois theory and skew braces (Paul Truman)
- Mathematical logic (Peter Fletcher)
- Regime switching modelling and risk management (Jie Cheng)