Research Areas
Introduction to Research Topics in the AI Graduate School
Solving Mathematical Challenges and Exploring New Theories Using AI
This research uses AI as a powerful tool to solve longstanding problems in pure mathematics or to discover new mathematical concepts and theories.
Number Theory, Algebra, and Algebraic Geometry
Research on explicit computation and pattern recognition of the Breuil–Mézard conjecture using AI (Wansu Kim, Bohae Lim).
Research on elliptic curves over function fields related to the Cohen–Lenstra and BKLPR conjectures (Wansu Kim, Bohae Lim).
Search for counterexamples to a weak version of Kaplansky’s conjecture (Hyung-Ryeol Baek).
Topology
Exploration of major problems in low-dimensional topology using AI (Jung Hwan Park).
Study on torsion elements and the existence of left-invariant order in (hyperbolic) groups (Hyung-Ryeol Baek).
Investigation of properties of algebraic integers appearing as entropy in pseudo-Anosov dynamical systems (Hyung-Ryeol Baek).
Other Mathematical Theories
Proposing new mathematical conjectures in low-dimensional topology using AI (Jung Hwan Park).
Research on reducing computational complexity in topological data analysis (Woojin Kim).
Detection of changes in time-series data and generation of mathematical conjectures using topology (Hoseung Song, Woojin Kim).
Research on detecting and quantifying abrupt changes in topological structures in data that vary over time and space.
This involves detecting topological changes using persistent homology, proposing new conjectures, and exploring ways to reduce computational cost with AI.
Research on AI Models for Enhancing Mathematical Reasoning
This research aims to enhance mathematical reasoning capabilities and develop theoretical and experimental methods to achieve this.
Large Language Models (LLMs) and Deep Learning
Theoretical and experimental research to improve reasoning ability in LLMs (Donghwan Kim).
Enhancing mathematical reasoning in LLMs through learning latent causal variables (Wooseok Ha).
Research on applying diffusion models to generative AI (Donghan Kim, Jinwoo Shin).
Research on quantifying uncertainty in LLMs (Wooseok Ha).
Research on data generation methods for training LLMs in mathematical theorem proving (Hyung-Ryeol Baek, Juho Lee).
Optimization and Learning Algorithms
Design of efficient training methods and faster optimization algorithms for LLMs (Donghwan Kim, Cheolhee Yoon).
Mathematical analysis of phenomena occurring during neural network training (Cheolhee Yoon).
AI-based Hyperparameter Control: Research on automatically controlling hyperparameters using AI to optimize deep learning model performance (Jaekyung Kim, Won Jang).
Study on optimization dynamics of pre-trained neural networks and LLM fine-tuning under distribution shifts (Wooseok Ha, Cheolhee Yoon).
Convergence and Implicit Bias in Neural Network Training in the Mean-field Limit (Beomjun Choi, Cheolhee Yoon).
Research analyzing optimization dynamics of neural network training via mean-field theory.
This includes proving convergence between real training algorithms and mean-field limit PDEs, analyzing long-term dynamics, and studying implicit bias.
AI-based Mathematical Data Analysis and Software Development
This research combines mathematics and AI to analyze data and develop software and platforms for this purpose.
Data Analysis Platforms
Development of data analysis software from a researcher’s perspective (Woojin Kim).
Research on using AI for topological data analysis (Woojin Kim).
Development of statistical machine learning methods for analyzing unstructured and complex data (Cheolwoo Park).
Applications and Others
Research on iterated function systems (IFS) for fractal image compression (Minju Lee).
Research on AI-based synthetic data generation and privacy protection techniques (Cheolwoo Park).
Research on measuring inefficiencies and optimal design through transportation network modeling (Hyung-Ryeol Baek, Changhyun Kwon).
Application of Physics-Informed Neural Networks (PINNs) to biological complex systems (Jaekyung Kim).
Prediction of sleep-wake rhythms based on change-point detection in wearable time-series data (Hoseung Song, Jaekyung Kim).
Research analyzing wearable time-series data to predict phases, state transitions, and anomalies in sleep-wake rhythms.
This combines nonparametric change-point detection with AI models to detect changes in personal sleep patterns and predict sleep onset, wake time, and fatigue risk.
AI Theory Based on Probability and Statistical Mechanics
This research uses probability theory and statistical mechanics to deeply analyze the learning process and operational principles of AI.
Stochastic Processes and Dynamics
Analysis of neural network training using stochastic differential equations (SDEs) (Cheolhee Yoon, Ilchul Moon).
Study of mixing in Markov chains (Kyungsik Nam).
Research on matching problems using optimal transport (Kyungsik Nam).
Research on conjectures regarding free energy fluctuations in the Sherrington–Kirkpatrick model (Jiwoon Lee).
Statistical Analysis
Statistical inference on random graphs (Kyungsik Nam).
Research on change-point detection and A/B testing (Hoseung Song).
Research on spatial clustering methodologies (Hoseung Song).
Analysis of computational efficiency and in-context learning performance using foundation model quantization and randomized algorithms (Wooseok Ha, Insu Han).
AI-Enhanced Statistical Inference: Research on improving the accuracy and efficiency of statistical inference in complex data using AI (Jaekyung Kim, Won Jang).