Functional Analysis stems from the studies of differential and integral equations, it began at the dawn of the 20th century, and is a branch under Analytics. It is mainly focused on the mapping in infinite dimensional spaces and the variety of Topology and Geometric structures. It analyzes mathematical problems through incorporating the perspectives and methodologies of analytics, algebra and geometry.
The main research content of Functional Analysis includes Spectral theory, Banach algebra, the topological linear space theory and the generalized function theory. It is an extremely adaptive field, and it is because of its adaptivity that propelled and enriched the evolvement of fields such as Partial Differential Equations, Probability Theory, Harmonic Analysis, and etc. Nowadays, it is not only widely used in mathematics, but has also become an indispensable tool in studies related to physics, chemistry and engineering theory.
Number theory is one of the oldest branches (arithmetic and geometry) of mathematics. Classical number theory is devoted primarily to the study of the properties of integers, like the distribution of primes and integer solutions of polynomial equations. This kind of problem is used to elementary in form (like Goldbach's conjecture and Fermat's last theorem). However, the related researches involve a lot of deep theory in mathematics. The studies in this area not only promote the development of mathematics but also inspire plenty of new methods and new topics.
The studies of modern number theory, according to the tool they used, can be roughly classified into two parts: algebraic number theory and analytic number theory. On the other hand, in other mathematic areas, like dynamic systems, geometry, representation theory and even computational science, they all have huge connections with number theory. As Carl Friedrich Gauss said, “Mathematics is the queen of sciences and number theory is the queen of mathematics.”
Professor Yi-Fan Yang(Number Theory, modular forms)
Number theory is one of the oldest branches of mathematics. It is mainly concerned with properties and relations among rational numbers, integers, and their generalizations. For example, Fermat’s last theorem, which asserts that if is an integer greater than 2 and is an integer solution of the equation , then one of the three integers must be 0, is one of the most famous problems in number theory. In order to solve problems in number theory, mathematicians used ideas and tools from other areas of mathematics, such as algebra, geometry, differential equations, complex analysis, and discrete mathematics, to name a few. This in turn stimulates the development of other areas of mathematics.
In the past, number theory has been considered as a branch of pure mathematics, but in the past three decades, this has changed and mathematicians have found applications of number theory in cryptography and coding theory. For example, the RSA cryptosystem and the elliptic curve cryptosystem, which are commonly used in e-commerce and other security-related situations nowadays, are both based on theories from number theory.
Our research is mainly focused on construction and applications of modular forms and related problems about modular curves and Shimura curves. In short, a modular form is an analytic function with many symmetries. Naturally, we would expect that a function satisfying such special requirement must have lots of interesting properties. Indeed, mathematicians have found that a modular form carries a lot of arithmetic information and can be used to solve problems in number theory. For example, the proof of Fermat’s Last Theorem given by Andrew Wiles is in fact a proof of a certain connection between arithmetic properties of elliptic curves and those of modular forms. Nowadays, it is well-established that modular forms lies at the cross road of representation theory, arithmetic geometry, string theory, quantum field theory, and many others. Even though modular forms have been studied for one and half a centuries, there are still many open problems that need to be solved. Also, their higher-rank generalizations remain significant challenges to mathematicians.
Professor Michael Fuchs (Analytic Combinatorics, Discrete Probability Theory, Analysis of Algorithms, Mathematical Biology, Metric Number Theory)
I am interested in metric number theory, more precisely, Diophantine approximation over the field of real numbers and over fields with positive characteristic and metric theory of continued fractions.
In recent years, I have mainly worked on Diophantine approximation over function fields. We have studied the classical problem of approximating irrationals by rationals in this context and have obtained many results such as $0$-$1$ laws, strong laws of large numbers with error terms, laws of iterated logarithm, central limit theorems and invariance principles. I am currently working on extending these results to inhomogeneous and/or simultaneous Diophantine approximation.
Professor Yi-Jung Hsu
My primary research interests include the following major parts: ：
Professor Kuo-Zhong Wang (Operator Theory, Matrix Analysis, Numerical Ranges)
The main work of our research on the numerical range is to analysis theoretically the correlation between numerical ranges, functional analysis and matrix analysis
The numerical range and numerical radius are very useful for studying linear operators acting on Hilbert spaces or Banach spaces. For instance, it is known that the close set of the numerical range always contains its spectrum, and many geometrical properties of the numerical range correlate with its spectrum. The spectrum and the numerical range are useful tools for studying operators and matrices. In this respect we're now approaching the research about product numerical ranges and it is used in the study of quantum information science. On the other hand, we have also been interest in studying the numerical range of a special matrix such as a partial isometry matrix or a stochastic matrix.
A generalization of the numerical range has been applied widely in many fields. The classical numerical range is also generalized to valuable different types which play important rolls in many fields. For instance, the higher-rank numerical range is applied in quantum physics, the C-numerical radii is applied in unitary similarity invariant norms, the joint numerical range is applied in the joint spectrum and joint spectral norm. I’m interested in these questions, and I’m looking forward to cooperating with experts in this field.
Professor Ming-Hsuan Kang (Operator Theory, Matrix Analysis, Numerical Ranges)
My research interests include algebraic groups , representation theory and its application. My collaborates and I have a series works about zeta functions of complexes, which are classifying spaces of uniform lattices in the algebraic groups. We are also interested in the spectral behavior of these zeta functions, like the distributions of zero and poles.
Besides, we are also interested in the philosophy of the field with one element, which reduces the problems on p-adic algebraic groups to the problems on affine Weyl groups. We also have some works on Generalized Poincare series on affine Weyl groups and Iwahori-Hecke algebras.
On the other hands, we are also very interested in the applications of pure mathematics in other disciplines. For instance, we apply the theory of extremal lattices and representation theory to spherical Monte Carlo methods. Moreover, we also use representation theory to study the spectral properties of toroidal fullerenes.
Professor Shiah-Sen Wang（Variational Methods., Geometric Measure Theory）
In the theory of calculus of variations, we are mainly interested study the partial regularities and singular structures of the minimizing mappings of energy functionals under various conditions originated from geometry and physics and their relations with the geometric and topological structures of the target spaces.
Quantitative Geometric Measure Theory: For some subsets of Euclidean space, we look for suitable Hausdorff measures so that these subsets are appreciable with respect to these measures. Using these measures, we can decompose the subset into the union of subsets of different dimensions, study the geometric properties of each subset, for example, smoothness of the set and its curvatures and investigate how these subsets are patched back to the original subset.
- Real Analysis (Junior/Senior/Graduate)
- Matrix Analysis (Junior/Senior/Graduate)
- Partial Differential Equations (Graduate)
- Functional Analysis (Graduate)
- Advanced Algebra (Sophomore)
- Algebraic Number Theory (Graduate)
- Analytic Number Theory (Graduate)
- Representation Theory (Graduate)
- Elliptic Curve Theory (Graduate)
- Algebraic Geometry (Graduate)
- Modular Forms Theory (Graduate)
- General Topology (Junior/Senior)
- Geometry (Junior)
- Introduction to Analysis (Junior/Senior)
- Introduction to Partial Differential Equations (Sophomore)
- Elementary Number Theory (Junior/Senior/Graduate)