Dr. Annika Burmester is a postdoctoral researcher at Bielefeld University working at the interface of key areas of modern mathematics. She recently received a Klaus Tschira Boost Grant, which supports outstanding early-career researchers. Her work focuses, among other topics, on multiple zeta values and their connections to disciplines such as physics. In doing so, she contributes to Bielefeld University’s research focus area “Emergent Synergies in Mathematics,” which strengthens links between mathematical disciplines and related fields. International experience, including a research stay in Japan, has also shaped her academic path. In this interview, she discusses her current projects, interdisciplinary perspectives, and the new opportunities created by the grant.
© Sarah Jonek
You are one of the new fellows of the Klaus Tschira Boost Fund. If you had to describe your research in just a few sentences: what is it about, and why is it worth a closer look?
Dr Annika Burmester: In my research, I work on two mathematical topics that are important both in mathematics and in physics: so-called multiple zeta values and modular forms.
Multiple zeta values are special numbers that arise from infinite series—that is, calculations that in theory continue indefinitely. A simple example is the series1 + 1/2² + 1/3² +…. Such numbers appear surprisingly often in different mathematical and physical contexts.
Modular forms, on the other hand, are special mathematical functions with very strong symmetries. Roughly speaking, one can think of them as mathematical objects that contain a great deal of hidden information about numbers and their properties.
I investigate connections between these two areas. To do so, I study so-called q-analogues of multiple zeta values and multiple Eisenstein series. These are mathematical constructions that combine properties of both topics.
My goal is to uncover new shared patterns and structures. This helps us better understand the often very complicated relationships between multiple zeta values and to explain more precisely how they relate to the symmetries of modular forms.
You work on highly abstract mathematical topics that connect to many different fields. The research focus area “Emergent Synergies in Mathematics” (ESyMath) deliberately promotes such links between mathematical disciplines and related fields. Where do surprising bridges emerge in your research—for example to physics or other areas of mathematics?
Surprising connections arise where these abstract structures appear directly in the results of other research fields.
In high-energy physics, for example, multiple zeta values are used to calculate particle collisions. More precisely, they help in the evaluation of so-called Feynman integrals—mathematical formulas physicists use to describe how particles behave during collisions. Together with colleagues, I am also developing applications in stochastic analysis. Put simply, this field deals with random processes and random developments, as they occur in physics, biology, or financial mathematics. Here, we investigate how multiple zeta values are related to so-called signatures in rough path theory. This theory is a modern mathematical tool for better understanding and solving complicated stochastic differential equations.
Exciting synergies also arise in geometry: so-called multiple q-zeta values appear in topological invariants—mathematical quantities used to describe the structure of geometric spaces. More specifically, this concerns Hilbert schemes, special spaces studied in modern geometry. Multiple zeta values are also used to compute volumes and intersection numbers in certain geometric spaces. I am working on these questions together with fellow mathematicians at Bielefeld University.
In summary, my research fits well within the ESyMath focus area, since the underlying structures of multiple zeta values provide a unifying basis for various questions in mathematics and physics. In particular, surprising connections emerge with other topics in number theory as well as with geometry and stochastic analysis.
What new opportunities does the funding open up for you, and how does your international experience, for example in Japan, play a role?
The funding provides me with the freedom to advance my research more intensively. I benefit greatly from my experience in Japan, which is home to a world-leading research community on multiple zeta values. The exchange there helped me develop new perspectives on this theory, which I am now incorporating into my current projects.
The grant will also enable me to maintain close collaboration with Japanese mathematicians in the future, both through regular research visits to Japan and by hosting project partners at Bielefeld University.
