Mathematics

The MB-OpenQS Unit theoretically explores hybrid many-body quantum systems composed of strongly interacting matter and light.

The Algebraic Combinatorics and Fundamental Physics Unit investigates new algebro-combinatorial and geometric structures underlying quantum field theory, focusing on scattering amplitudes, total positivity, amplituhedra, and cluster algebras.

Chiral representation theory unit investigates the symmetries arising in quantum field theories. More specifically, it focuses on the representation theory of infinite-dimensional Lie algebras such as Kac–Moody algebras, and more generally, on vertex algebras.

The Geometric Group Theory Unit studies the large-scale geometry of infinite groups, focusing on hyperbolic and non-positively curved spaces, growth, and projections.

The Information Theory, Probability and Statistics Unit performs theoretical research at the intersection of the fields described in the name with applications to various areas that include Estimation Theory, Computational Biology, Hypothesis Testing, etc.

The biological nonlinear dynamics data science unit investigates complex systems explicitly taking into account the role of time. We do this by instead of averaging occurrences using their statistics, we treat observations as frames of a movie and if patterns reoccur then we can use their behaviors in the past to predict their future. In most cases the systems that we study are part of complex networks of interactions and cover multiple scales. These include but are not limited to systems neuroscience, gene expression, posttranscriptional regulatory processes, to ecology, but also include societal and economic systems that have complex interdependencies. The processes that we are most interested in are those where the data has a particular geometry known as low dimensional manifolds. These are geometrical objects generated from embeddings of data that allows us to predict their future behaviors, investigate causal relationships, find if a system is becoming unstable, find early warning signs of critical transitions or catastrophes and more. Our computational approaches are based on tools that have their origin in the generalized Takens theorem, and are collectively known as empirical dynamic modeling (EDM). As a lab we are both a wet and dry lab where we design wet lab experiments that maximize the capabilities of our mathematical methods. The results from this data driven science approach then allows us to generate mechanistic hypotheses that can be again tested experimentally for empirical confirmation. This approach merges traditional hypothesis driven science and the more modern Data driven science approaches into a single virtuous cycle of discovery.

The Applied Cryptography Unit investigates the design and analysis of modern cryptographic primitives and schemes used to protect the confidentiality and integrity of data – at rest, being communicated or computed upon – both in the classical and the quantum settings. Particular areas of interest include the design and analysis of quantum / post-quantum cryptography schemes, the algebraic cryptanalysis of symmetric and asymmetric key algorithms, as well as the design and analysis of primitives for privacy-preserving cryptographic mechanisms.

Analysis of partial differential equations is a very rich mathematics subject, which is broadly applied in a large variety of fields of science. It is particularly important to study nonlinear PDEs that arise in geometry and many related areas. The Geometric Partial Differential Equations Unit aims to develop new analytic methods to understand behavior of solutions to various geometric evolutions and explore solvability of nonlinear equations in general geometric settings such as sub-Riemannian manifolds and metric spaces. Our research is motivated by numerous applications in material sciences, crystal growth, image processing and is also closely connected with topics in optimal control, game theory and machine learning, etc.

The Gravity, Quantum Geometry and Field Theory Unit’s research interest lies in revealing the quantum nature of spacetime. Geometrical aspects of gravity, manifested in General Relativity, h...

The mission of the Analysis and PDE unit is to reveal and analyze the mathematical principles reflecting natural phenomena expressed by partial differential equations and advance the boundar...

Analysis on Metric Spaces Unit explores analytic and geometric problems arising in diverse spaces, especially those with no priori smooth structures. Our research focuses on partial diff...

Our unit studies the representation theory of algebras via tools from algebraic combinatorics. Our study largely centres around the philosophy of decomposing representations.

We study in theory and experiments the engineering of quantum devices built from different subsystems that can collectively perform beyond the individual capabilities of their parts.