Many prominent trends in modern algebra represent mixtures of what was classically labeled algebra with geometry, combinatorics, and topology. The flux moves in both directions: Longstanding algebraic problems are solved with techniques adapted from neighboring fields, and conversely. Consequently, this area offers broad exposure to mathematical ideas.
Here are a few specific lines pursued by UCSB's algebra faculty:
- The representation theory of finite dimensional algebras focuses on `linearized snapshots' of certain nonlinear objects, such as finite groups. In the past decades, this branch of algebra has been strongly linked to combinatorics and geometry through seminal work of Auslander and Kac, among others, showing that much of the desired structural information on the nonlinear objects and their linear snapshots is encoded in directed graphs. Other aspects of a well-rounded structure theory can, for instance, be accessed by way of `derived categories' (due to Grothendieck and Verdier), powerful tools in studying maps among sets with various types of additional structure that one wishes to explore.
- Representation theory has proved equally important in the realm of infinite dimensional algebras, where it has long been utilized to `linearize' the study of objects such as infinite groups or differential operators. It has led, in particular, to the study of algebras with built-in `multiplicative twists', algebras which, in the past two decades, have played key roles in a rapidly developing new field labelled `quantum groups'. This new field originated in theoretical physics -- in quantum inverse scattering theory and the search for solutions to the `quantum Yang-Baxter equation', to be a little more precise -- but rapidly built connections with areas of mathematics as seemingly disparate as knot theory, algebras of operators on Hilbert spaces, and special functions. The algebraic side of this field includes the ongoing development of `noncommutative counterparts' to classical algebraic geometry, such as `quantized' versions of the algebras of polynomial functions on algebraic spaces.
- Algebraic Geometry is the study of the solutions of polynomial equations. At first sight this task would seem modest, but in fact this problem is so hard that algebraic geometry draws upon many areas of mathematics such as algebra, differential geometry, topology, number theory, analysis and differential equations to attack this problem. First one considers the zero set as a geometric object, a variety. The modern approach to the problem of classifying varieties involves classifying all possible embeddings into projective space. It turns out that this problem is intimately related with the study of the topology of the curves that lie on the variety, especially the combinatorial structure of an associated cone.
- Kenneth Goodearl
- Birge Huisgen-Zimmermann
- Bill Jacob
- Jon McCammond
- David Morrison
- Adil Yaqub
Mathematical Analysis initially developed from arguments needed in the infinitesimal study of geometrical objects and physical motions. After the great success of the heroic age of differential and integral calculus, it was realized that, by focusing attention not on a single function in isolation, but rather on an appropriate class of functions, one could obtain a better understanding and an expansion of the classical theories. Thus, Functional Analysis was born about a century ago, with great benefits to several areas of modern mathematics and many remarkable applications to the natural sciences.
The faculty listed by our department as analysts are actually all functional analysts. They like to work in infinite dimensional spaces (of functions, operators, representations, dynamical systems). Among their main objectives are the solution of equations, minimization of functionals, and classification and study of algebras of operators. The numerous applications vary from classical problems of continuum mechanics and information theory to image analysis, knot theory and quantum computing.
Several research seminars devoted to such subjects are organized all year around. They are attended by many of our graduate students and attract many distinguished visitors.
- Charles A. Akemann
- Gustavo Ponce
- Mihai Putinar
Bjorn Birnir is interested in infinite-dimensional dynamical systems and their applications to partial differential equations.
Hector D. Ceniceros' research focuses on the investigation of free boundary problems in multi-phase flow and complex (polymeric) fluids through the design, analysis, and application of effective numerical methodologies. His specific research areas include: boundary integral methods, phase field and level set approaches, immersed boundary method, field-theoretic polymer models, adaptive and multi-scale methods, interfacial rheology, and liquid crystalline polymers.
Carlos Garcia-Cervera is interested in problems arising in Material Science and Condensed Matter Physics, and in particular, the dynamic and static properties of complex systems. His research interests are diverse, and include fields such as Ferromagnetism, Superconductivity, Liquid Crystals, and Polymeric Fluids. His approach to these problems is flexible in the techniques employed, which include modelling, asymptotic analysis, and numerical simulation.
Paul Atzberger is interested in problems arising in Soft Matter Physics and Biology. A particular aspect is the role of stochastic effects. Specific application areas include the study of Membrane Bilayers, Complex Fluids, Molecular Motor Proteins, and Osmotic Phenomena. His research draws on techniques from stochastic analysis, asymptotics, and numerical analysis. An emphasis is also placed on scientific computing and the development of general numerical methods.
- Paul Atzberger
- Bjorn Birnir
- Maria Isabel Bueno Cachadina
- Hector Ceniceros
- Carlos Garcia-Cervera
- Xu Yang
The Geometry Group of the Mathematics Department at UCSB has Differential Geometry as its core part, and includes two important related fields: Mathematical Physics, and part of Algebraic Geometry in the department.
The core part, Differential Geometry, covers Riemannian Geometry, Global Analysis and Geometric Analysis. A central topic in Riemannian geometry is the interplay between curvature and topology of Riemannian manifolds and spaces. A well-known example is the classical Bonnet-Myers theorem which states that a complete Riemannian manifold of uniformly positive Ricci curvature must be compact and have a finite fundamental group. Global analysis, on the other hand, studies analytic structures on manifolds and explores their relations with geometric and topological invariants. For example, the celebrated Atiyah-Singer index theorem establishes the relation between the index of elliptic operators-an analytic quantity, and characteristic classes of the underlying manifold which are topological invariants. Finally, geometric analysis combines geometric tools with analytic tools such as PDE, geometric measure theory and functional analysis in geometric contexts to study geometric and topological problems which are often nonlinear. An important example is Hamilton's Ricci flow. Recently, spectacular results in geometry and topology were achieved by employing the Ricci flow. These include Perelman's seminal work on the Poincare Conjecture and the Geometrization Conjecture for 3-manifolds. The research of the Geometry Group covers diverse topics in Riemannian geometry, Global analysis and Geometric Analysis, such as manifolds with lower bounds on the Ricci curvature, minimal surfaces in Riemannian manifolds, Einstein manifolds, the index theory and the eta invariants, Ricci flow, pseudo-holomorphic curves in symplectic geometry, and Seiberg-Witten invariants in the theory of the topology of 4-dimensional manifolds.
The research of the Geometry Group in Mathematical Physics covers various topics such as knot and link homologies, gauge theory, Chern-Simons theory, Calabi-Yau spaces, D-branes, mirror symmetry, the positive mass theorem in general relativity, and constant mean curvature foliations on asymptotically flat manifolds.
The research of the Geometry Group in Algebraic Geometry covers various topics such as mirror symmetry, Calabi-Yau spaces, the minimal models, moduli spaces, and the Kahler-Ricci flow.
Interactions between the various directions of Riemannian Geometry, Global Analysis, Geometric Analysis, Mathemtical Physics and Algebraic Geometry play an important role in the research of the Geometric Group. Interactions with other groups of the Mathematics Department, the Physics Department and KITP play an equally important role.
- Xianzhe Dai
- Doug Moore
- David Morrison
- Guofang Wei
- Xin Zhou
- Rick Rugang Ye
Number theory abounds in problems that are easy to state, yet difficult to solve. An example is "Fermat's Last Theorem," stated by Pierre de Fermat about 350 years ago. Finding a proof of this theorem resisted the efforts of many mathematicians who developed new techniques in number theory, for example with the theory of elliptic curves over finite fields. A proof of Fermat's Last Theorem was finally presented by Andrew Wiles in 1995 in a landmark paper in the Annals of Mathematics.
Another famous problem from number theory is the Riemann hypothesis. This problem asks for properties of the Riemann zeta function, a function which plays a fundamental role in the distribution of prime numbers. Although it is over one hundred years old the Riemann hypothesis is still unresolved; in fact, the Clay Mathematics Institute has offered a prize of one million dollars for its solution.
Yet another famous open problem from number theory is the Goldbach conjecture which states that every even positive integer is a sum of two primes. Understanding this conjecture requires nothing more than high school mathematics, yet it has resisted the efforts of countless mathematicians.
- Adebisi Agboola
- Jeffrey Stopple
- Yitang Zhang
Our understanding of the fundamental processes of the natural word is based to a large extent on partial differential equations (PDE). For example, the Einstein equations describe the geometry of space-time and its interaction with matter. The dynamics of fluids and elastic solids are governed by partial differential equations that go back to Euler and Cauchy. Electro-magnetic waves including the propagation of light in various media are modeled by Maxwell's equations. More recently, PDE's are gaining importance in the social and life sciences. The Black-Sholes PDE underpins the theory of option pricing in financial mathematics. Reaction-diffusion models are used in neurophysics and population dynamics.
The analysis of PDE's has given rise to new mathematical ideas. For example, the study of the equation thermal diffusion lead to the discovery of Fourier series and ultimately the field of Fourier analysis. Hilbert's investigations into eigenvalue problems arising in the PDE's for vibrating strings and Schrodinger's equation in quantum mechanics evolved into to modern theory of functional analysis and operator theory.
Nonlinear PDE is major research area at UCSB. Bjorn Birnir specializes in application of dynamical systems tools to infinite dimensional systems. Gustavo Ponce is an expert in modern techniques of harmonic analysis applied to nonlinear dispersive equations. Tom Sideris studies nonlinear waves with a particular interest in hydro- and elasto-dynamics. Denis Laubutin studies geometric measure theory in connection with PDE's arising in differential geometry. Close connections exist between the groups in PDE and Applied Mathematics.
- Bjorn Birnir
- Denis Labutin
- Gustavo Ponce
- Thomas Sideris
Geometric topology is often split into low dimensional (4 or less) and high dimensional. This split is based upon the techniques employed, the kinds of question that can be answered, and the state of knowledge. There were enormous advances in high dimensional topology during the 60’s including the solution of the high dimensional Poincare conjecture, and a good understanding of how differentiability enters into the picture, for example through the existence of exotic smooth structures on spheres.
Today a considerable effort is being made to better understand manifolds of dimensions 3 and 4. The techniques, conjectures and outlooks in these two areas are very different, although there have also been hints of various unifying themes. In the 80’s it was discovered by Donaldson, Freedman and Casson that Euclidean space has exotic smooth structures only in 4 dimensions.
The theory of 3 dimensional manifolds was revolutionized in the late 70’s by Thurston’s Geometrization Conjecture. There are eight geometries (homogeneous Riemannian metrics) which (appear to) play a similar role in 3 dimensions to the three constant curvature geometries in two dimensions. Some problems in 3-dimensions are best studied through combinatorial and topological techniques using surfaces and their generalizations. Many problems in knot theory are of this type.
There are many connections to number theory, Riemannian geometry, geometric group theory and dynamical systems to name only a few.
- Stephen Bigelow
- Daryl Cooper
- Darren Long
- Jon McCammond
- Ken Millett
- Marty Scharlemann
- RTG in Topology and Geometry
- Applied Mathematics Research Group
- Computational Science and Engineering
- String Theory Group
- Complex Fluids Design Consortium
- Center for Mathematical Inquiry
- Center for Control, Dynamical Systems and Computation
- California Nanosystems Institute
- Center for Complex & Nonlinear Science
- UCSB Math Circle