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Let G be a finite group. For any element g in G, the centralizer of g is denoted by
C_G(g) = {h in G : gh=hg}
We associate a graph PC_G with a group G (called the non-prime centralizer graph of G) as follows: let P be the set of elements of G in which their orders are prime power and take P as the vertex set of PC_G; and two vertices x and y are adjacent if
C_G(x) is not equal to C_G(y)
where C_G(x) is the centralizer of x in G. In this project, we study the graph theoretical properties of PC_G and its induced sub-graph on P_G, where P_G is a prime graph which is a famous graph that many prominent researchers work on it.
Here we try to find the clique number of this graph and the influence of this graph on the prime graph.
Public-key cryptography is fundamentally based on algebraic structures. RSA relies on arithmetic in the multiplicative group of integers modulo N, while elliptic curve cryptography (ECC) is built on the group law of algebraic curves over finite fields. With the advent of quantum computing, both classes of cryptosystems face fundamental challenges, as quantum algorithms can exploit their underlying algebraic structure.
This project studies quantum cryptography through the lens of algebraic geometry and quantum algorithms, with a concrete path from mathematical foundations to FPGA-based cryptographic hardware implementation. Rather than focusing on abstract quantum complexity alone, the project emphasizes how algebraic constraints, curve geometry, and group structure shape both quantum algorithms and real hardware designs.
The project is inspired by FPGA-based cryptographic architectures such as Reconfigurable Quantum Crypto Processor using FPGA, but is reorganized to highlight the role of algebraic curves, quotient structures, and reconfigurable hardware design as a coherent research theme.
Fault-tolerant quantum computation requires robust quantum error correction (QEC) to protect quantum information from decoherence and noise. Among many proposed QEC schemes, surface codes and quantum low-density parity-check (Q-LDPC) codes play a central role due to their geometric structure, locality, and scalability.
This project studies surface codes and Q-LDPC codes from a unified perspective, emphasizing their topological, combinatorial, and algebraic structures, as well as their implications for hardware implementation and decoding. The goal is to guide students from mathematical foundations to research-level questions in quantum error correction, with an optional FPGA-based implementation track.
In law, economics, and ethics, courageous behavior is usually considered a virtue. In fact, Aristotle considered it the primary virtue that produces all others. This research project seeks to develop new mathematical methods to identify and quantify courage. It also seeks to more definitively determine whether courage can be trained or is instead inborn.
In statistics and machine learning, divergences such as the Kullback-Leibler divergence, Hellinger distance, chi-squared divergence, and total variation distance are widely used to measure dissimilarity between probability distributions. Each of these belongs to the broader family of f-divergences, and each induces a different geometric structure on the underlying statistical manifold. A classical result in information geometry states that in the small-distance limit, all f-divergences converge to the Fisher-Rao metric, but their global behavior can differ substantially.
In this project, students will fit parametric models (e.g., Gaussian, exponential family) to real-world datasets and systematically compare the performance of several f-divergences as distance measures for k-nearest neighbor classification. On the theoretical side, students will study the geometric properties of each divergence — whether it defines a true metric, its symmetry, and its relationship to the Fisher information — and investigate whether these properties correlate with classification performance. The goal is to bridge the gap between the abstract geometry of divergences and their practical behavior in data analysis.
The subject of this project are prime graphs of finite groups with a focus on groups that are very close to solvable groups. There are several open questions on these graphs which are purely combinatorial in nature. In the realm of solvable groups, an important notion here is the one of minimal prime graphs. A simple graph is called a minimal prime graph if it is connected, has at least two vertices, and its complement is triangle-free and has chromatic number 3, and removing any edge results in a graph whose complement has a triangle or chromatic number 4.
An interesting notion in this context is the one of superminimal prime graphs which in a sense have as few edges as possible. Finding new families of such graphs would be interesting, as would be answering many questions, such as: What can be said about minimal prime graphs that have a vertex of degree 3?
Logistic regression is a powerful method for estimating the probability that a particular event will occur. It is commonly used to distinguish between two or more categories. A simple example is email filtering, where logistic regression classifies messages as either spam or legitimate and routes them to the appropriate inbox.
In recent years, logistic regression has become an essential technique in the rapidly growing field of machine learning. Compared with more complex models such as neural networks, it is generally easier to implement, interpret, and train, especially for data collection and analysis tasks. For this reason, it is often regarded as a foundational method in machine learning. Its applications span many areas, including the social sciences, credit scoring, medicine, text classification, hotel booking prediction, gaming, and school admission rate analysis, among others. Logistic regression also provides a conceptual foundation for modern deep learning algorithms.
Despite its advantages, solving logistic regression efficiently can be challenging. One of the most widely used solvers is implemented in scikit-learn, which relies on Newton-type methods. However, Newton methods require computing the Hessian of the objective function, making them computationally expensive and often slow for large-scale problems. Therefore, there is a strong need for the development of faster and more efficient solution techniques.
In this summer project, we propose to develop an efficient solver for logistic regression based on conjugate gradient methods.
Prerequisites:
Greedy algorithms play an important role in modern machine learning, particularly in high-dimensional approximation, sparse modeling, and neural network training. Methods such as the Orthogonal Greedy Algorithm (OGA) construct models incrementally by selecting features (or neurons) that provide the greatest immediate improvement at each step.
Specifically, OGA builds an approximation by iteratively choosing atoms from a predefined dictionary—these atoms can be interpreted as candidate features or basis functions—that are most strongly correlated with the current residual error. After each selection, the algorithm performs an orthogonal projection onto the subspace spanned by the chosen atoms, ensuring an optimal update within the selected feature set. This forward stagewise strategy closely relates to sparse learning, feature selection, boosting methods, and shallow neural network training.
In recent years, greedy methods have been used to design neural network solvers for partial differential equations and other supervised learning tasks, offering interpretable and computationally efficient alternatives to fully trained deep models.
In this project, we will develop a preconditioned version of OGA tailored to machine learning applications. The goal is to significantly improve convergence speed, scalability, and numerical stability, making the method more suitable for large-scale learning problems.
We will study a partially ordered structure called the Bruhat order, which organizes permutations and related symmetry objects. While lower and upper intervals in this order are well understood and closely related to Schubert varieties, middle intervals are much less studied despite their connection to Richardson varieties. The project focuses on the combinatorics of these middle intervals by viewing them as graphs via cover relations. Motivated by geometric considerations, we will investigate a conjecture asking whether any connected middle interval of length n admits the deletion of a low-degree vertex (degree at most n) while preserving connectivity and not increasing remaining vertex degrees.
Prerequisites: Some familiarity with group theory would be helpful
Generative AI systems such as ChatGPT and Gemini have become increasingly popular in recent years. Recent developments have also made it possible for AI to generate synthetic data that is similar to the original datasets, and it is anticipated that by 2030, AI-generated data may account for more than 80% of all data. Although large amounts of synthetic data could improve the performance of many statistical methods due to increased sample size, it remains unclear whether synthetic data can be used to draw reliable conclusions. In this project, we will propose several measures to evaluate the similarity between synthetic and real datasets. We will investigate the scenarios under which synthetic data can more accurately reflect real data based on the proposed measures, and we will also compare synthetic data generated by different AI models.
The summer math camp project will focus on analyzing various types of DNA microarray and/or sequencing datasets to identify genetic and epigenetic patterns associated with cancer. This project is ideal for students who are interested in tackling challenging problems in genetics using statistical, computational, and bioinformatics approaches. Students are expected to be self-motivated and willing to learn foundational concepts in genetics, statistics, bioinformatics, and programming (including R and Unix/Linux) in order to conduct rigorous data analysis. Additionally, students who are interested are encouraged to continue working on the project after the math camp concludes, with the goal of completing the study and submitting a manuscript to a peer-reviewed journal.
The box-ball system is a combinatorial dynamical system that models a lot of interesting phenomena in physics (Toda lattice, KdV equations, etc). Hiro and Dr. Makiko Sasada (she’s a speaker for the 2026 ICM) have many new, unanswered questions about a two-dimensional version of the box-ball system. Hiro would love to gather a group of students who enjoy working with others! If you know basic programming, in any language, great, but programming expertise is absolutely not required. (Programming will be useful for generating examples and conjectures; but proofs will not require any programming skills.) There is also no knowledge of physics or calculus needed; we will mainly focus on the box-ball system, rather than the physics applications.
Online mentoring
We say that two non-empty subsets A and B of a group G are non-commuting subsets if xy != yx for every x in A and y in B. We associate a graph S[G] with a group G (called the non-power set graph of G) as follows: let P(G) be the power set of G and take P(G) as the vertex set of S[G]; and two vertices are adjacent if they are non-commuting subsets. In this project, we study the influence of this graph S[G] on the structure of group G.
Among other results, we conjecture that if G and H are two finite groups such that S[G] ~ S[H], then |G|=|H|.
