Weimin Chen, Ph.D. - US grants

Proposal: DMS-9971454

Principal Investigator: Weimin Chen

Abstract: This project will center on three topics in symplectic topology and geometry. In the first part the principal investigator (joint with Y. Ruan) will establish the Gromov-Witten invariants for symplectic orbifolds. This work aims at computing the Gromov-Witten invariants of a symplectic manifold by decomposing it into two pieces. Such an operation will in general introduce orbifold singularities, so the establishment of Gromov-Witten invariants for symplectic manifolds with orbifold singularities will lay the foundation for a program of computing Gromov-Witten invariants, which has great potential applications in symplectic topology as well as other related fields such as birational geometry. Other motivations come from mirror symmetry and string theory of theoretical physics, in which there is a demand to consider spaces with controlled singularities such as orbifold singularities. In the second part of this project the principal investigator will continue his work on the 3-dimensional Reeb dynamical systems by exploiting a potential connection between Seiberg-Witten Floer homology and the contact homology. The last part concerns symplectic structures on smooth 4-manifolds with vanishing second homotopy group. Currently very less is known about them.

In recent years, one has witnessed some great interactions between different branches of mathematics in the area of geometry and topology, with the input of ideas from theoretical physics. This project seeks to exploit the intimate interplay between these different fields to investigate some very interesting or fundamental questions in such areas as quantum cohomology, birational geometry, the existence of periodic orbits of Reeb dynamics on a 3-dimensional space, and symplectic 4-manifolds. The principal investigator believes that a successful outcome of this project will make a significant advancement of knowledge in the areas listed above, which are important not only within mathematics but also in real life problems. For example, the importance of Reeb dynamics is seen in the following example. The motion of a satellite in the presence of the gravitational forces of the sun, the planets and the moon is described mathematically as a Reeb dynamics. The relevant part of this project aims at solving the 20-year-old Weinstein conjecture for Reeb dynamical systems on a 3-dimensinal space, which is one of the most important conjectures in the field.

DMS-0304956
Weimin Chen

The Principal Investigator plans to continue his work in gauge
theory and symplectic, low-dimensional topology. More concretely,
he plans to study the following problems: (1) general properties of
finite groups of symplectomorphisms of a symplectic 4-manifold,
(2) whether a symplectic circle action on a 6-dimensional manifold
is Hamiltonian if its fixed-point set is non-empty and finite, and
(3) general, global restrictions on the occurrence of quotient singularities
in an algebraic surface. A unifying theme in these studies is a theory,
yet to be developed in this project, on the Seiberg-Witten and Gromov
invariants of a symplectic 4-orbifold, which will be built on and
substantially extend the related fundamental work of Taubes on
symplectic 4-manifolds.

A 4-dimensional orbifold is a space which locally looks like the space-time
we all live in, except that there is a certain degree of ambiguity caused by
a finite symmetry occured locally. Many examples of this kind of space can be
found easily and naturally in Mathematics (e.g. topology and geometry) as well
as models in Theoretical Physics (e.g. orbifold string theory). The proposed
research seeks to understand a fundamental duality in a 4-dimensional orbifold
between a certain type of fields that are distributed all over the space and a certain
type of worldsheet that is wiped out by a collection of strings in the space which
is condensed in a 2-dimensional subspace. When there is no such
ambiguity of finite symmetries, such a duality is well-understood, and
is a fundamental piece of
mathematics.

DMS-0603932
Weimin Chen

Gromov's pseudoholomorphic curve theory has been at the center of many great advances in mathematics in the past twenty years. This project seeks to apply Gromov's ideas and techniques of pseudoholomorphic curves in the context of group actions on manifolds, which technically amounts to studying pseudoholomorphic curves in the quotient space of the group action. Such a quotient space, called an orbifold, may have singularities in general, which correspond to the fixed points of the group action. Particularly, this project proposes to systematically study a certain class of smooth finite group actions on four-dimensional manifolds, which includes finite order automorphisms of nonsingular algebraic surfaces, a subject which has been long studied in algebraic geometry. The project also seeks to classify a certain type of circle actions on the five-dimensional sphere involved in the study of Einstein metrics on the sphere, which is a subject of central importance in Riemannian geometry.

The importance of symmetry in mathematics has been long recognized. A
crucial issue in the study of symmetry is to understand the structure
of the set of points in the space which are fixed under the symmetry.
The central new idea in this project is the observation that when studying symmetries of a four-dimensional space (for instance, the universe in which we live), one can often extract useful information about the fixed points of a symmetry by studying a certain type of rigid two-dimensional subspaces (like a soap bubble) in the four-dimensional space.

This project is to provide funding for the international conference ``Transformation Groups in Topology and Geometry'', which will take place at the University of Massachusetts in Amherst, July 14-17, 2008. The participants of this conference come from many countries. The conference plans to cover three subjects. The first is symmetries of 3-dimensional manifolds. The topics to be discussed include Thurston's Conjecture of geometrization of 3-manifolds
with symmetries whose singular set is of dimension at least 1, geometrization (after Perelman) of 3-manifolds with finite
fundamental groups, and symmetries of 3-manifolds related to knots and links.
A second topic is group actions on 4-manifolds. The topics to be discussed include locally linear, topological actions on 4-manifolds, smooth and symplectic symmetries of 4-manifolds, and gauge theoretic techniques and their roles in studying symmetries of 4-manifolds.
A third subject is transformation groups on higher dimensional manifolds.
The topics to be discussed include homotopy theory in transformation groups, group actions and surgery theory, group actions in symplectic geometry, and group actions on algebraic varieties. Besides the standard conference talks (30 minutes long) there will be invited (1-hour long) plenary talks delivered by leading researchers in the corresponding areas. These talks will give an overview of a given research area and suggest possible new research directions.

Symmetry is a fundamental phenomenon in mathematics (as well as physics),
which is a unifying theme of a wide range of research areas.
The main purpose of this conference is to bring together researchers from
quite diverse areas of topology and geometry whose research is related to or
involves transformation groups. This conference will be especially useful to young researchers and graduate students. For it will give them a chance to obtain a more global and comprehensive view of various research areas which only on a surface look to be disjoint and unrelated.

This collaborative project will study the topology of smooth 4-dimensional manifolds, in connection with well-known problems in low-dimensional topology. We will focus on the construction of new smooth manifolds with symplectic structures, including Stein manifolds and symplectic fillings of certain contact 3-manifolds. Recent advances in techniques based on knot surgery and Luttinger surgery for creating exotic manifolds with small Euler characteristic will be coupled with computations of gauge-theoretic and symplectic invariants. We will make use of 4-dimensional handlebody techniques in these constructions, with an organizing principle being the search for 'corks' and 'plugs' as a technique for changing the smooth structure. Techniques of gauge theory and symplectic geometry will be used to investigate the classification of symplectic 4-manifolds and their symmetry groups.

The physical world of space and time is a 4-dimensional space whose local structure is well understood but whose large-scale (or topological) properties remain mysterious. This Focused Research Group will explore the global topology of 4-dimensional spaces, with a goal of understanding what kinds of spaces (called 4-dimensional manifolds) can exist as mathematical objects, and what the properties of such manifolds are. Of particular interest will be the problem of existence and uniqueness of symplectic structures, as well as that of determining the symmetries of a given manifold. The group will investigate how subtle changes in the smooth structure of a manifold can be achieved by gluing together pieces of different manifolds. Such changes will be detected by combining expertise from several disciplines, including powerful techniques derived from gauge theories of mathematical physics.

The "Northeast Conferences on Geometry and Topology of 4-manifolds" series will have two meetings in 2015-2016, both at the University of Massachusetts, Amherst. The first conference, entitled "Geometry and Topology of Symplectic 4-Manifolds," will meet between April 24-26, 2015, and the second, entitled "Floer Homologies, Gauge Theory, and Topology of 4-Manifolds," will meet in fall 2016. The conference series focuses on the geometry and topology of 4-manifolds, regarded as a melting pot for various research areas, such as low dimensional topology, contact, symplectic, complex and differential geometry, geometric analysis, and mathematical physics. The organizers plan to bring together leading experts and rising young researchers from across the country as speakers for each meeting and to support participation of interested researchers and graduate students, especially from the institutions in the Northeast. An objective of the conference series is to fertilize new research directions by increasing interaction and collaboration among the wealth of geometers and topologists in Massachusetts, New York, New Jersey and Connecticut, particularly among graduate students, junior researchers, and their more senior colleagues, while providing all with a panorama of the subject area through a variety of talks and discussion sessions. The series will help build a regional network and support ties among graduate students, faculty, and other researchers throughout the Northeast.

The themes for both conferences are chosen from the most active and dynamic fields in recent years. Taubes' work connecting Seiberg-Witten gauge theory and symplectic topology has had great impact on both symplectic geometry and 4-dimensional smooth topology, while work of Donaldson and Gompf has led to many new constructions and applications of Lefschetz fibrations. The advances led to deep understanding of the internal structure of symplectic 4-manifolds. Remarkably, this led to the complete smooth and symplectic classification of symplectic 4-manifolds of negative Kodaira dimension. These will be the main topics of the first conference. On the other hand, gauge theory has had an enormous impact on the study of smooth 4-manifolds, starting with the seminal work of Donaldson in the 1980's and the introduction of Seiberg-Witten theory some 10 years later. Coupled with new methods for building 4-manifolds, this has led to the constructions of exotic smooth manifolds in many homotopy types, and to an understanding of the internal structure of smooth 4-manifolds. The deepest results of recent years have been based on the use of gauge theoretic invariants of 3-manifolds known as Floer homology theories, which give rise in turn to invariants of 4-manifolds with boundary. The second conference will focus on these invariants for all smooth 4-manifolds.

Web link for the Spring 2015 conference: http://people.math.umass.edu/~baykur/Symplectic4manifoldsConference.html