## Research Training Groups

In July 2012, the
National Science Foundation has
awarded
a
**Grometry and Topology RTG (Research Training Group)** grant to our department.

The University of Texas Geometry/Topology Research Training Grant (RTG) is a vertically integrated program to enhance the training of undergraduates, graduate students, and postdoctoral associates at the University of Texas and, through publication on the web, beyond. The thirteen faculty members will build on the success of our previous RTG grants and also initiate new activities. The latter include a Directed Reading Program pairing undergraduate students and graduate mentors; a new Journal Seminar for all graduate students in geometry and topology; a new series of First Cut Lectures on a variety of core topics, to be disseminated on the web; a new yearly Undergraduate Mathematics Conference; and a new series of Graduate Winter Schools for graduate students and postdocs. All RTG activities are designed around a unified perspective on geometry and topology. They include a healthy mix of ideas from low dimensional topology, algebraic geometry, differential geometry, global linear and nonlinear analysis, geometric group theory, geometric representation theory, and homotopy theory; often concepts and inspiration from theoretical physics play an important role as well. The RTG training activities are designed to promote this unified view of geometry and topology and, more broadly, of mathematics and science to young mathematicians.

Mathematics, the study of Number and Space, lies at the foundation of science and technology. Geometry, the study of Space, has long had connections with and applications to the real world. Topology investigates, very roughly, global geometric properties of spaces which do not rely on local structure. There is great interplay between Geometry and Topology-there is not a sharp boundary between them. In the ancient world measurement was introduced to facilitate commerce; more advanced concepts, such as trigonometry, were introduced to understand the stars. In our modern world advanced ideas in Geometry underlie abstract theories in physics and also have more immediate practical implications, for example in navigation, computer graphics, etc. The large scale structures studied in topology appear in biological and physical science; there are new applications to data analysis and signal processing as well. Effective training of the next generation of Geometers and Topologists is vital for the continued development of these ideas, both within mathematics and for external applications. The University of Texas Geometry/Topology Research Training Grant promotes a unified view of the field. An open attitude towards ideas from all directions is essential for success with the challenges facing mathematics and science today. Through local activities in Austin, and via dissemination on the web, the RTG will enhance training at all levels.

### Previous RTG awards

In June 2007, the National Science Foundation has awarded THREE Research Training Group (RTG) grants to our department.

**Program in Applied and Computational Analysis**(NSF award)The analysis program is hosted jointly by the Math Dept and ICES. The goal is the creation of a model program in applied and computational mathematics that includes enhanced undergraduate research experience, reduced time to graduation, a stronger background in applicable mathematics for graduate students, and the highest level of postdoctoral mentoring. The analysis program runs a series of summer workshops for select college students.

- 2008 Summer School: Multiscale Modeling and Analysis
- 2009 Summer School: Financial Mathematics
- 2010 Summer School: Long Term Behavior of Nonlinear Systems
- 2011 Summer School:
**Analysis, PDEs and Mathematical Physics**

**Unified Approach to Training in Geometry**(NSF award)The geometry program is a vertically integrated program to enhance the training of undergraduates, graduate students, and postdoctoral associates at the University of Texas. The RTG training activities are designed to promote a unified view of geometry and, more broadly, of mathematics and science to young mathematicians.

**Program in low-dimensional topology and its applications**(NSF award)The topology program is focused on training in low-dimensional topology and its applications.