# University Lecture

## November 17, 2015: Michael Barnsley (Australian National Unviersity)

## 6:00pm – 7:00pm in Laurel Hall 102

## Fractal Transformations

**Abstract: **Fractal transformations are a newly discovered way of changing smooth objects into rough ones, and vice-versa. They have exciting potential applications to modeling teapots, turbulent flows, and high resolution medical data. This lecture will explain, in simple terms, how fractal transformations can be built and explored (easily), using the “Chaos Game”.

**Event Details: **Professor Barnsley will deliver a public lecture for the University of Connecticut community. Reception with light refreshments to follow.

**About the Speaker: **Michael Barnsley, Professor of Mathematics at Australian National University, is the highly cited author of about two hundred research papers, and of several popular books, including “Fractals Everywhere”, “SuperFractals”, “The Science of Fractal Images” and “Fractal Image Compression”. He holds degrees in Mathematics and Theoretical Chemistry, and worked in universities and laboratories in the US, France, and Australia. In addition to being a prolific researcher and author, Michael Barnsley is a successful entrepreneur, with several patents related to fractal image compression.

# Colloquia Series

## October 1, 2015: Fabrice Baudoin (Purdue)

**Stochastic Analysis on Riemannian Foliations**

**Abstract:** We develop a Malliavin calculus on the horizontal path space over a foliated Riemannian manifold. This calculus is used to prove logarithmic Sobolev inequalities under curvature conditions and concentration inequalities for the horizontal Brownian motion of the foliation.

**Event Details: **Thursday, October 1, 2015; 4:00 – 5:00pm; Math Sciences Building (Gant Science Complex), Room 109A

**About the Speaker: **Fabrice Baudoin is a Professor at the Department of Mathematics of Purdue University. His area of research includes topics and techniques from stochastic analysis and geometry, namely, he studies diffusion processes, Riemannian and sub-Riemannian geometry, geometric functional inequalities as well as rough paths theory, Malliavin Calculus and Gaussian prcesses. His PhD thesis advisors were Marc Yor and Huyen Pham. Fabrice is the author of three books and numerous research articles. In addition, he is blogging on math and related issues.

## October 8, 2015: Bruce Kleiner (NYU)

### Spaces satisfying Poincare inequalities

**Abstract: **The lecture will be concerned with PI spaces—metric measure spaces that are doubling and satisfy a Poincare inequality. These were introduced by Heinonen-Koskela more than 20 years ago, and have been applied in differential geometry, geometric group theory, random walks, quasiconformal analysis, and bilipschitz embedding. The first part of the talk will be a survey. This will be followed by a discussion of some new examples and open questions.

**Event Details: **Thursday, October 8, 2015; 4:00 – 5:00 pm; Math Sciences Building (Gant Science Complex), Room 109A

**About the Speaker: **Bruce Kleiner is a Professor at the Courant Institute for Mathematical Sciences and is the Chair of the Department of Mathematics at New York University. He received a BA and PhD in mathematics from the University of California, Berkeley. Bruce’s diverse research in areas such as geometric analysis, geometric group theory, geometric topology, and metric geometry involves a mixture of ideas from geometry, analysis, topology, and algebra. He was an invited speaker at the ICM in 2006 and received the National Academy of Sciences Award for Scientific Reviewing in 2013 for his exposition with John Lott of Perlman’s proof of Thurston’s geometrization conjecture. He currently has over 50 research publications and has supervised 6 PhD students.

## October 22, 2015: Pertti Mattila (Helsinki)

### Hausdorff Dimension and Projections

**Abstract: **Hausdorff dimension is a parameter which gives us information about metric size of sets. Marstrand proved in 1954 that for sets of Hausdorff dimension *s* almost all orthogonal projections on lines also have Hausdorff dimension *s*, provided *s* is at most 1, and they have positive length, provided *s* is bigger than 1. There have been many generalizations and variations of this result. In the talk I shall discuss some of the recent ones.

**Event Details:**Thursday, October 22, 2015; 4:00 – 5:00pm; Math Sciences Building (Gant Science Complex), Room 109A

**About the Speaker:**Pertti Mattila is a Professor of Mathematics in the Department of Mathematics and Statistics at the University of Helsinki. He is one of the most influential figures in the area of Geometrtic Measure Theory and its applications in Harmonic and Complex Analysis. He has supervised 16 PhD students and his highly cited book “

*Geometry of Sets and Measures in Euclidean Spaces”*is the standard textbook in the area. His latest book “

*Fourier Analysis and Hausdorff Dimension”*was published in the Summer of 2015. He has twice been a member of the Institute of Advanced Study in Princeton and he was an invited speaker at the ICM in 1998.

## October 29, 2015: Nages Shanmugalingam (Cincinnati)

### Geometry behind the Poincare inequalities

**Abstract: **Classical work on Euclidean potential theory and PDE extensively used the control of the variance of a function on a ball in terms of the average energy of the function on that ball. Such control, called Poincare inequality, is available in Euclidean spaces as well as Euclidean Lipschitz domains, but fail for more general Euclidean domains. Much of the recent developments of analysis in metric measure spaces is done for metric measure spaces whose measure is doubling and supports a Poincare inequality. The work of Heinonen and Koskela indicates that there is geometric information of the metric measure space encoded in the Poincare inequality supported by that space. This talk will give a survey of results expanding on this connection between geometry and support of Poincare inequalities.

**Event Details:** Thursday, October 29, 2015; 4:00 – 5:00 pm; Math Sciences Building (Gant Sciences Complex), Room 109A

**About the Speaker:** Nageswari Shanmugalingam is a Professor of Mathematics at University of Cincinnati. She earned her BS in Physics from the University of Rochester and her PhD in Mathematics from the University of Michigan. Her research expertise includes geometric function theory, potential theory, and analysis on metric measure spaces, as exemplified by her highly cited paper (currently over 200 on MathSciNet!) “Newtonian Spaces: An extension of Sobolev spaces to metric measure spaces.” She organized the International Workshop on Harmonic and Quasiconformal Mappings–a Satellite Conference at the 2010 ICM–and gave a mini-course in the IPAM Long Program on Interactions between Analysis and Geometry in 2012.

## November 12, 2015: Luca Capogna (WPI)

### Regularity for Subelliptic PDE through Uniform Estimates in Multiscale Geometries

**Abstract: **Sub-Riemannian geometry and subelliptic PDE provide a useful framework for modeling a broad range of systems, ranging from motion of robot arms, to some mechanisms of perception in the visual cortex and beyond. In most cases the mathematical challenges associated to these objects are due to the lack of ellipticity of the operators involved and on the degeneracy of the ambient metric structure. A natural approach to tackle such difficulties is by approximating the sub-Riemannian metric through a family of collapsing Riemannian approximates. This scheme also immediately provides elliptic approximates for the subelliptic operators in play. In this talk, I will describe some recent results concerning stability of doubling properties, Poincare inequalities, Gaussian estimates on heat kernels, Schauder estimates, and so on, in this approximation scheme. I will also address some concrete applications (that provided the initial motivation for this study), including the study of motion by mean curvature and Liouville type theorems for quasiconformal mappings in the sub-Riemannian setting.

**Event Details:** Thursday, November 12, 2015; 4:00 – 5:00 pm; Math Sciences Building (Gant Sciences Complex), Room 109A

**About the Speaker: **Luca Capogna is the Head of the Department of Mathematics at the Worcester Polytechnic Institute. His research output includes important contributions in Partial Differential Equations, Calculus of Variations, Sub-Riemannian geometry, Minimal Surfaces and Quasiconformal Mappings. He was an Associate Director of the Institute for Mathematics and its Applications (IMA) in Minnesota from 2011 until 2013. In 2002 he received an Early Carrer Award by the NSF. His PhD thesis advisor was Nicola Garofalo at Purdue and his postdoc advisors were Luis Caffarelli and Fang Hua Lin at the Courant Institute. Luca is the author of two books and of around 50 research publications.

## November 19, 2015: Xavier Tolsa (Barcelona)

### Rectifiability, the Riesz Transform, and Harmonic Measure

**Abstract: **In this talk, I will review several characterizations of rectifiable sets and I will explain a connection with the L^2 boundedness of the Riesz transforms in the codimension 1 case. In the second part of the talk, I will shown an application of these results to the study of harmonic measure, which can be considered as a converse to the well known F. and M. Riesz theorem.

**Event Details:** Thursday, November 19, 2015; 4:00 – 5:00 pm; Math Sciences Building (Gant Sciences Complex), Room 109A

**About the Speaker: **Xavier Tolsa is a Research Professor at ICREA, the Catalan Institute for Research and Advanced Studies, and a Professor at the Autonomous University of Barcelona. He is a leading expert in Singular Integrals and their applications in Geometric Measure Theory, Potential Theory and Quasiconformal mappings. He has been awarded prestigious prizes, such as the Salem prize and the European Mathematical prize, for his contributions in Harmonic Analysis. He was an invited speaker at the ICM in 2006. Recently he also received an advanced research grant of 1.1 million Euros by the European Research Council.