Meet Dr Sacha Mangerel from our Department of Mathematical Sciences

Tell us a bit about yourself and your research interests?

I was born in Toronto, Canada. My father studied Physics at university, and discussions with him helped spark in me a taste for the sciences.

Throughout my childhood and teenage years I enjoyed challenging myself with logic puzzles, riddles and learning languages. Eventually my interests gravitated towards the mathematics, where every new problem provides an opportunity to test myself.

My main area of interest in research is called “Analytic Number Theory”. The overarching philosophy of this area of maths is that techniques from analysis (especially the theory of complex functions and Fourier series), which involve continuous, approximate objects, can be used to study problems in number theory, involving counting discrete objects, in a surprisingly fruitful way.

At a broad level, I am interested in understanding how the multiplicative and additive structures of integers interact with one another. This can be studied in many concrete forms by looking at the behaviour of multiplicative functions – functions defined on the integers whose values are determined by the prime factorisations of their arguments.

When did you join us and what is your role at Durham University?

I joined the Mathematical Sciences department at Durham in September 2021 as an assistant professor of pure mathematics. I am currently a member of both the Analysis and Algebra and Number Theory groups within the department.

I split my time between active research and teaching. For the last couple of years I have taught 2^nd year modules in number theory and algebra, but have previously taught a postgraduate module more closely related to my research area.

What are the projects/research you are currently working on?

One of my key areas of interest within analytic number theory is about “rigidity phenomena” for multiplicative functions. We have well-understood heuristics governing how multiplicative functions should behave, but we are often unable to give formal proofs that these heuristics hold. I am interested in testing these heuristics by looking at multiplicative functions that fail to follow them in some extreme way. A typical problem of this kind is as follows: if a multiplicative function is known only to satisfy a particular strong property in this vein, must this property then (nearly completely) determine the function?

As an example, I am currently working on classifying all multiplicative functions taking integer values, such that the gaps between their consecutive values is uniformly bounded from above. It is expected that any such function looks, more or less, like the linear (and multiplicative) function f(n) = n.

In a different direction, there is an historically significant line of inquiry on certain periodic multiplicative functions called Dirichlet characters. These are rather strange objects that interact predictably with both additive and multiplicative structure. Many open problems exist about their behaviour, specifically the distribution of their values. I am keenly interested in studying a certain collection of ‘typical’ characters, which has as yet not been looked at sufficiently in the literature.

Finally, I am interested in classical problems about covering integers by binary quadratic forms. It has been known for more than three centuries precisely which integers can be written as a sum of two perfect squares; in a precise sense, very few of them are. One can ask about variations on this representation, e.g., which integers can be written as sums of the form Ax² + By², with prescribed constraints on the coefficients A and B. With specific applications in mind, I would like to know how many different such forms one requires in order to ensure that every integer N can be represented in a form of this kind.

What is interesting about your projects/research?

I find that these research problems tend to blend more than one area of maths together. In studying the first two of the problems described above, for instance, I have found ideas from a subject called Additive Combinatorics – focused on how the structure of sets of integers can be gleaned from studying how often sums of its elements coincide – to be quite useful. In these and other such investigations, finding unexpected connections is exciting, and motivates me to learn about other areas of maths previously not well known to me. A great deal of my research tastes have been influenced by finding precisely these links, and I feel it has helped me better understand my area.

What is the scientific and societal relevance of your research?

There is a long-standing open problem in number theory called the ‘twin prime’ conjecture. It posits that there are infinitely prime numbers p for which p+2 is also a prime number. This basis for this problem concerns whether one can identify precisely the multiplicative structure (i.e. prime factorisation) of two integers, p and p+2, which are related by addition. We expect that for such a pair of additively-related integers their multiplicative structures are independent, and that the property of being prime can be shared by both. My research seeks to make inroads on this and related problems within a broader scope.

It is not always possible to see works of pure mathematics as relevant within society at large. To paraphrase G.H. Hardy, studying maths of this kind is more akin to art than to engineering, and that there is beauty both in understanding fundamental truths and in rigorously expressing some of these phenomena as theorems through sometimes clever and elegant proofs.

What are your plans for future research/study?

In general I am open-minded about the types of problems that I work on, and as such welcome new problems to look at, both of my own finding as well as from collaborators. My research plans are therefore not typically set in stone. I would like to continue investigating problems within the scope described earlier, but hope to continue to find ways of branching off from these problems into areas as yet unknown to me.

In terms of developing myself as a researcher, I have observed that several new developments in my area have grown out of a familiarity with areas such as ergodic theory and graph theory, and I plan to become better acquainted with these subjects over the next several years.

I would also like to help grow the analytic number theory group in Durham by continuing to attract doctoral students and early career researchers, and helping them develop. In addition, I would like to promote the influence of number theory research in the northeast, and am hoping to continue building connections with mathematicians in the north of England and in Scotland in the near future.

Any interesting hobbies/passion outside of work?

I am an avid road cyclist and runner, and compete in duathlons from time to time.

I benefit a lot from being in nature, and regularly go for walks. I’ve recently spent a lot of time outdoors birdwatching with my partner.

In my more idle time, I like to feed my addiction for cryptic crosswords, and derive a lot of enjoyment out of cooking.