Applied and Computational Mathematics
The Applied and Computational Mathematics group combines expertise from various disciplines such as analysis, modeling and scientific computing. Our primary goal is the development of mathematical and computational methods in our pursuit to target real-world problems in science, engineering, medicine and other fields. We integrate different mathematical techniques from PDEs, numerical analysis, functional analysis, optimization, machine learning and data science, among others. As a result, our interests intersect with those of other research groups such as ‘Analysis and PDEs’, ‘Mathematical Medicine and Biology’, and ‘Mathematics for AI and Big Data’. The focus on real-world problems naturally involves our members in interdisciplinary collaboration with other experts in various scientific areas.
Algebra and Algebraic Geometry
Traditionally, algebra studies the structures of sets equipped with operations, such as sum or product, and algebraic geometry studies the solution spaces of equations formed by such algebraic relations. In the mid-20th century, algebra and algebraic geometry, along with several other disciplines, experienced important revolutionary moments. Arguably, the most impactful changes were instigated by Alexander Grothendieck’s new languages and tools such as schemes, stacks, sheaves, together with developments in commutative algebra and homological algebra. This resulted in a total revision of the foundations of algebraic geometry, which opened up entirely new classes of objects to study, and eventually enabled researchers to establish connections between modern algebraic geometry and diverse mathematical disciplines that previously were considered disjoint. Examples include representation theory, differential geometry, complex geometry, number theory and arithmetic geometry, algebraic topology, homotopy theory, category theory, non-archimedean analysis and rigid geometry, to name a few, and the connections to algebraic geometry keep getting stronger and mutually beneficial.
Through these developments, the terminology and methodology of algebraic geometry is now accepted as part of protocols in communications with neighboring disciplines. As a result, it has become ever more important to acquire some fluency in the language of algebraic geometry. This is comparable to how the general jargon of analysis is now part of the standard glossary in partial differential equations, harmonic analysis and part of applied mathematics, for instance.
Our department has various faculty members specializing in diverse topics ranging from classical algebra and algebraic geometry to newer territories such as representation theory or arithmetic geometry, etc. Students can benefit from the chances to learn the modern languages and foundations of algebraic geometry, and pursue diverse opportunities to engage in cutting-edge research programs.
Analysis and PDEs
Analysis is an area studying quantitative properties of functions and operators acting on them. Since the discovery of calculus, physics describes natural phenomena via functional equations involving differentiation and integration, so called, differential equations. Analysis is developed as the corresponding mathematical theory, and has become a central topic in mathematics. A distinct feature is a mathematical rigor of analytic proof in describing qualitative and quantitative properties of phenomena. This distinguishes it from other sciences dealing with natural phenomena or differential equations. The most extensive and central topic in this area is differential equations, and it also includes related topics, such as dynamical systems, mathematical physics, harmonic and complex analysis, and functional analysis. In our department, several major topics are being studied. Research interests include the calculus of variations, partial differential equations describing fluid mechanics or wave phenomena, inverse problems, mathematical modeling, and mathematical physics.
Discrete Mathematics is a relatively young area of mathematics which explores discrete structures such as graphs, permutations, partitions, and partially ordered sets. Since today’s computers operate in “discrete” steps and store information in “discrete” bits, there is a strong connection to theoretical computer science, and for this reason, the field of discrete mathematics has been attracting a lot of attention recently.
In discrete mathematics, we investigate problems on discrete structures such as the following: problems on counting certain patterns on discrete structures, problems on unavoidable substructures in large discrete structures, problems on intersection patterns of geometric objects, problems on extremal structures of discrete structures with a certain property, and problems on finding efficient algorithms to solve certain problems.
Solving these problems often requires techniques from other areas of mathematics, such as algebra, topology, representation theory, and probability theory. Methods developed by combining ideas from several areas of mathematics for solving problems in discrete mathematics are frequently used in various research areas including number theory, topology, geometry, and theoretical computer science.
Two main areas of financial mathematics are option pricing theory and portfolio theory: The former studies pricing of financial derivatives including options on underlying assets such as stocks and bonds, and the latter finds ways to manage portfolios efficiently. The beginning of option theory was the publication of two papers in 1973 by Black and Scholes, and Merton. The idea of efficient portfolio was proposed by Markowitz. Two Noble prizes in economics were given to the founders. To study financial mathematics, we need to understand financial concepts such as risk, hedging, interest rate and no arbitrage principle. Necessary mathematical prerequisites are linear algebra, ordinary differential equations, partial differential equations, Lebesgue integral, probability theory, numerical analysis, statistical ideas, Fourier analysis and complex analysis. Among these, the most essential tool is Ito integral which may be regarded as a combination of probability theory and Lebesgue integral. In Ito integral the integrator is Brownian motion, and the most fundamental fact is Ito’s lemma in which the square of the increment of Brownian motion is regarded as being equal to the increment in time. To solve problems in financial mathematics, sometimes we need to employ computational methods in addition to theoretical investigations: Two main tools for option pricing are numerical analysis of partial differential equations and Monte Carlo method for estimating expectation.
Students with sufficient knowledge in financial mathematics can continue their study in academia after graduation, or find jobs in financial industry including banks, stock trading companies, insurance companies, and financial technology companies.
Geometry and Topology
Geometry and Topology deal with all problems about classifying spaces based on their structure and shape, hence they affect virtually every branch of mathematics. They are intertwined closely these days as seen in the notable development toward Poincare conjecture and geometrization conjecture. Research of the members of our Geometry and Topology group are focused on low dimensional topology and complex geometry. The study of low dimensional manifolds (dimensions less than five) has great significance to theoretical physics and has many applications. For one thing, many interesting phenomena happen in these dimensions. For instance, every three-dimensional manifold is triangulable whereas non-triangulable manifolds exist for each dimension greater than three. Also, a well-known yet surprising fact is that the only dimension where the Euclidean space allows an exotic smooth structure is dimension four. Our faculty have worked on many important topics in the field including geometric structures, geometric group theory, foliation theory, knot theory, gauge theory, and Heegaard Floer theory. On the geometry side, our faculty focuses on pluripotential theory and complex geometry. A real valued function of several complex variables is plurisubharmonic if its restriction to any complex line is subharmonic. In a way this is the convexity notion in the complex setting. Pluripotential theory studies properties of such functions. It has found many applications in complex geometry and algebraic geometry recently. It gives the background to construct a singular Kähler-Einstein metric on a Kähler manifold.
Mathematical Biology and Biomedical Mathematics
If the 20th century was the century of physics, the 21st century will be the century of biology. Currently, biology is the field that produces the most doctorates in the United States, and one in six students in mathematics and half in statistics are receiving degrees in biology-related research. However, unlike physical phenomena that move according to laws, biological phenomena are so diverse that there seems to be no universal laws. In particular, our body is composed of about 100 trillion various cells, and each cell is composed of about 100 trillion different molecules. It is impossible to understand such a complex system only with our intuition. Mathematical biology aims to find the basic principles of complex life phenomena. For this, we simplify the complex systems to their core principles and project them to the virtual and digital space, allowing for virtual experiments. This requires the application and development of mathematical theories in diverse areas. Typically, differential equations such as ODE, PDE, and SDE are used to describe biological systems. Theories from algebra, topology, and combinatorics are also used. One of the beauties of mathematical biology is that it can use various fields of mathematics to solve a given life science puzzle, rather than one specific field of mathematics. In addition, one can contribute to people’s health and happiness through mathematics, as the research results lead to understanding the cause and spread of diseases, and the development of treatments.
Mathematics for AI and Big Data
Artificial intelligence (AI) is the study of systems capable of mimicking learning and problem solving abilities of human beings. The availability of big data has made recent remarkable achievements in AI possible by extracting meaningful information from it. So AI and big data are considered as the key elements of the forth industrial revolution that we are facing now. However, in spite of remarkable achievements in AI and big data, it is still challenging to understand the structure of complex data and how to use it for reliable and explainable AI systems. To address such questions and lay a foundation for the development of artificial general intelligence (AGI), mathematics for AI and big data is required, which includes mathematical analysis, geometry, topology, algebra, probability and statistics. With the help of mathematics for AI and big data, ongoing research focuses on topics such as mathematical principles in AI, reliable and explainable AI, statistical learning and optimization for AI and big data within a reasonable time span and with available computing power, and convergence research with information technology (IT), biological technology (BT), medical imaging, and social sciences.
Number theory has come from the study of the properties of numbers. In particular, algebraic number theory has been developed from exploring the properties of numbers through solving polynomial equations. More specifically, it studies the arithmetic properties and invariants of a number field, which is obtained from the roots of polynomials, or studies “the properties of numbers” through the algebraic and arithmetic structures and properties of various algebraic varieties defined over number fields. Most typically, algebraic varieties such as elliptic curves, modular curves, and abelian varieties are mainly studied. In this process, various tools in the area of algebra including Galois theory and algebraic geometry are used. It also studies objects over finite fields and p-adic fields as a tool to access objects over number fields, and also deals with problems on the objects over function fields that are structurally analogous to number fields.
In the 19th century, Riemann proved important results on the distribution of prime numbers through the analytic properties of the zeta function. This inspired the study of the Riemann zeta function and its generalizations (so-called L-functions), which has been established as an important subject of number theory. Recently, as Fermat’s Last Theorem has been proved by the modularity of elliptic curves over the rational numbers, which is a result of correlating the L-functions of elliptic curves over the rational numbers and modular forms, the arithmetic properties of modular forms and the more general automorphic forms have been very actively studied to date.
In modern number theory, the properties of numbers are studied through entities appearing in various areas, and number theory is also applied to cryptography and coding theory beyond the realm of pure mathematics. Modern research in number theory employs not just tools from “classical” number theory, but also those borrowed from other areas in mathematics such as group theory, commutative algebra, algebraic geometry, combinatorics, representation theory, Lie theory, analysis, dynamics, topology and hyperbolic geometry. The research group in our department is studying problems mostly in an algebraic and geometric approach.
Probability theory started out of interest in predicting the outcome of a game, and it has established itself as a branch of mathematics based on a strict understanding of random variables through a measure-theoretic method about 100 years ago. In modern times, research related to probability theory is being conducted in various fields of mathematics, and probability theory is widely used to understand natural or social phenomena with random characteristics in various fields other than mathematics such as physics, life sciences, and economics. Recently, probability theory has also been applied in fields that need to deal with large amounts of data, such as artificial intelligence research. The main research subjects in probability theory include random variables, probability distributions, and stochastic processes. Major research tools used in probability theory include martingales and Markov chains, and the most basic yet the most representative research results of probability theory are the law of large numbers and the central limit theorem.
Probabilistic models considered in probability theory are very diverse, including various kinds of stochastic differential equations, random graphs, random matrices, and spin glass models. In addition, the results obtained in probability theory are applied to queuing theory, probability control theory, optimization theory. These results are recently used for theoretical understanding of technologies used in machine learning, such as reinforcement learning algorithms. The importance of probability theory continues to grow, and its applications are expected to expand further. Notable future research topics for probability theory and its application are research on distributions that appear universally in various probabilistic models, the so-called universality, and research on reliable and explainable artificial intelligence based on mathematical understanding.
Data science is an interdisciplinary field that uses scientific methods, processes, algorithms and systems to extract knowledge and insights from structured and unstructured data, and apply knowledge and actionable insights from data in a broad range of scientific domains. Statistics, as a key component of data science, is a study that develops mathematical and statistical methodologies to collect, analyze, and interpret various types of data based on probability theory and applies them to various fields of science. In the era of the 4th industrial revolution and big data, the form of data has become increasingly complex, and the dimension and the size of data are rapidly increasing. Therefore, research of Statistics Group focuses on developing statistical methodologies to analyze ultra-high-dimensional, high-throughput, large-scale, and unstructured data, developing algorithms to implement them, and pioneering new statistical theories in line with the rapidly changing data environment. Specific topics of methodological research include function estimation, nonparametric inference, statistical machine learning, statistical computation, time series and spatial statistics, Bayesian modeling, multivariate statistics, and sampling design. At the same time, research also focuses on the application of novel statistical methodologies that leads to expand the knowledge in different fields of science such as basic science, life and medical science, engineering, social science, economics, public health, and environmental science.