研究ユニット

カイラル表現論ユニットは、量子場の理論に現れる対称性を研究します。より具体的には、無限次元リー代数、例えば Kac–Moody 代数や、より一般的には頂点代数の表現論に焦点を当てています。

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.

偏微分方程式の解析は豊富な内容を持つ数学の分野で、科学の様々な学問領域において幅広く応用されています。特に幾何学及び関連分野に現れる非線形偏微分方程式を考察することが重要です。幾何学的偏微分方程式ユニットでは、新たな解析手法を考案することによって、幾何学的発展方程式の解の振る舞いについて理解し、そしてサブリーマン多様体や距離空間などの一般的な幾何学的設定における非線形方程式の可解性問題を探究します。研究の動機として、材料科学、結晶成長，画像処理への応用が多く知られていて、最適制御やゲーム理論、機械学習などのテーマにも密接に関係しています。

幾何学的群論ユニットでは、無限群の大域的幾何構造を研究し、双曲空間や非正曲率空間、群の増大度や射影複体に焦点を当てています。

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. Areas of interest include the algebraic cryptanalysis of symmetric and asymmetric key algorithms; design and analysis of primitives for privacy-preserving cryptographic mechanisms; and the design and analysis of quantum-safe cryptographic constructions.

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.

代数学の中でも数学の多くの分野に関係する表現論は話題豊富な分野です。表現論と代数的組合せ論ユニットは、表現論の中でも対称群の研究をいくつかの方向に一般化する研究...

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...

距離空間上の解析ユニットは、多様な空間、特に先験的な平滑構造を持たない空間において解析的および幾何学的問題を考察します。本ユニットでは、サブリーマン多様体を含む...

重力、量子幾何と場の理論ユニットでは、時空の量子性について研究しています。一般相対性理論によって明らかになった重力の幾何学的な側面は、マトリックスモデル、テンソ...

量子マシンユニットでは、量子サブシステムが個々の能力を凌ぐ機能を集合的に発揮できる量子デバイスの実用化を目指して、理論と実験の両面から研究しています。