Research Units

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.

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...

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.

Our group focuses on unveiling lots of mysteries surrounding astrophysical explosive phenomena such as supernovae (SNe) and gamma-ray bursts (GRBs). SNe and GRBs are the most powerful explosions in the universe, yet very little is known about their explosion mechanisms. These astrophysical big bangs fascinate us with their unknown physics and puzzling astronomical phenomena such as gravitational waves, neutrinos, nucleosynthesis, non-equilibrium ionization, ultra-high-energy cosmic rays. Through our theoretical and computational approaches, we strive to reveal the complete pictures of these explosions and provide the-state-of-the-art physical interpretations for current, cutting-edge observations and useful predictions for future observations by next-generation astronomical observatories.

The Biological Complexity Unit studies how biophysical systems, ranging from subcellular circuits to cellular populations, can function despite being subject to random fluctuations.

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.

We seek the principles governing the behavior of whole organisms, integrating physics, biology and computational approaches to understand life's most complex and fascinating phenomena.

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.

Collective Dynamics and Quantum Transport Unit explores dynamical and transport phenomena in various quantum matters and their spintronic, electronic, and quantum-information applications.

The CSE Unit analyzes the dynamics of complex adaptive systems, focusing on behavioral dynamics shaping social systems, eco-evolutionary dynamics shaping ecosystems, and their interactions.

The Droplet and Soft Matter Unit explores soft matter, droplet, and interfacial physics to do fundamental science that is both beautiful and useful, especially in water-health-energy nexus.

We study topics of electron correlation in physics and materials science, such as emergent phenomena at quantum phase transitions, spin frustration and excitation, and Fermi surface evolution.

The Experimental Quantum Information Physics Unit carries out experimental studies on highly controllable quantum systems. A particular research emphasis is put on the development of ion tra...

Utilizing cutting-edge time-resolved techniques, including ultrafast PEEM and ARPES, the Femtosecond Spectroscopy Unit explores extreme light-matter interaction on the nanometer and femtosecond scale.

The Future-Proof Cryptography Unit studies Multi-Party Computation, Fully Homomorphic Encryption, Verifiable Computation and Cryptanalysis and how these are affected by quantum computers.

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

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.