College of Liberal Arts and Sciences

# Department of Mathematics

## Analysis Learning Seminar

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Title: Geodesics in the Heisenberg group
Speaker: Scott Zimmerman (University of Connecticut)
Time: Friday, September 1, 2017 at 3:30 pm
Place: MONT 314Abstract: The Heisenberg group was first described by Herman Weyl, and it is named after Werner Heisenberg (as he was the first to study the Lie algebra associated with the group). Since it's origin, the Heisenberg group has been an object of study in many fields in analysis, geometry, and physics. In this talk, I will introduce the group and provide some of its interesting analytic and geometric properties. As a point of illustration, I will discuss the structure of geodesic curves in the group. In the case n=1, this is simple. However, deducing the structure of these curves in higher dimensions requires proof, and I will present such a proof from a paper by Piotr Hajlasz and me.

Title: The Whitney Extension Theorem for curves in the Heisenberg group
Speaker: Scott Zimmerman (University of Connecticut)
Time: Friday, September 8, 2017 at 3:30 pm
Place: MONT 314Abstract: The Heisenberg group was first described by Herman Weyl, and it is named after Werner Heisenberg (as he was the first to study the Lie algebra associated with the group). Since it's origin, the Heisenberg group has been an object of study in many fields in analysis, geometry, and physics. In this talk, I will introduce the group and provide some of its interesting analytic and geometric properties. As a point of illustration, I will discuss the structure of geodesic curves in the group. In the case n=1, this is simple. However, deducing the structure of these curves in higher dimensions requires proof, and I will present such a proof from a paper by Piotr Hajlasz and me.

Title: Sobolev extension of Lipschitz mappings into metric spaces
Speaker: Scott Zimmerman (University of Connecticut)
Time: Friday, September 15, 2017 at 3:30 pm
Place: MONT 314Abstract: The Heisenberg group was first described by Herman Weyl, and it is named after Werner Heisenberg (as he was the first to study the Lie algebra associated with the group). Since it's origin, the Heisenberg group has been an object of study in many fields in analysis, geometry, and physics. In this talk, I will introduce the group and provide some of its interesting analytic and geometric properties. As a point of illustration, I will discuss the structure of geodesic curves in the group. In the case n=1, this is simple. However, deducing the structure of these curves in higher dimensions requires proof, and I will present such a proof from a paper by Piotr Hajlasz and me.

Title: Quasi-invariance of Wiener measure on (sub)-Riemannian manifolds
Speaker: Qi Feng (University of Connecticut)
Time: Friday, September 29, 2017 at 3:30 pm
Place: MONT 314Abstract: In this talk, I will present the idea of quasi-invariance of Wiener measure on $\mathbb{R}^n$, Riemannian manifolds and sub-Riemannian manifolds. First, I will show what is quasi-invariance of Wiener measure (Cameron-Martin theorem) in $\mathbb{R}^n$. Second, I will construct Brownian motion on a Riemannian manifold $M$ by using the orthonormal frame bundles on $M$ and show the quasi-invariance of Wiener measure on the path space of a Riemannian manifold. In the end, I will present the quasi-invariance of horizontal Wiener measure on compact foliated manifolds. In particular, I will give an example for the Heisenberg group.

Title: Riesz transform on Riemannian manifolds and beyond I
Speaker: Li Chen (University of Connecticut)
Time: Friday, October 6, 2017 at 3:30 pm
Place: MONT 314Abstract: In this series of talks, we give a survey of Riesz transforms on Riemannian manifolds and other geometric settings. The first talk is introductory. We start from the background and applications of Riesz transforms. Then we introduce classical results for the boundedness of Riesz transforms on Euclidean spaces from both analytic and probabilistic methods. In the second talk, we focus on introducing previous results on Riemannian manifolds, particularly under assumptions of volume doubling property and Gaussian heat kernel estimate. The last talk is about my own work on this topic. We discuss the boundedness of Riesz transform under sub-Gaussian heat kernel estimates.

Title: Riesz transform on Riemannian manifolds and beyond II
Speaker: Li Chen (University of Connecticut)
Time: Friday, October 13, 2017 at 3:30 pm
Place: MONT 314Abstract: TBA

Title: Riesz transform on Riemannian manifolds and beyond III
Speaker: Li Chen (University of Connecticut)
Time: Tuesday, October 24, 2017 at 4:00 pm
Place: MONT 214

Title: A Brief Introduction to Morse Theory
Speaker: Gianmarco Molino (University of Connecticut)
Time: Friday, October 27, 2017 at 3:30 pm
Place: MONT 314Abstract: Morse Theory commences by considering the critical points of smooth functions on manifolds; when all of these points are isolated and nondegenerate (that is the matrix of second derivatives is nonsingular), we are able to recover significant topological information about the manifold. This leads to the celebrated Morse Inequalities, which are the starting point for many ideas in differential geometry.

Title: An Introduction to Trudinger-Moser Inequalities
Speaker: Jungang Li (University of Connecticut)
Time: Friday, November 3, 2017 at 3:30 pm
Place: MONT 314Abstract: Trudinger-Moser inequality is treated as the borderline case (p = n) of the Sobolev inequality. In this talk, I will briefly introduce the history of Trudinger-Moser ineuqalities together with the applications to differential geometry and PDE. I will also introduce some recent results in which Trudinger-Moser type inequalities are established in different settings.

Title: Nonsmooth Analysis: Postdoc Workshop
Speaker: Conference Speakers
Time: Thursday, November 9, 2017 at 8:45 am
Place: MONT 214Abstract: 8:45-9:45 Max Engelstein (MIT) 10:00-11:00 Murat Akman (UConn) 12:40-1:40 Panel Discussion on Grants & Finding Tenure-Track Positions 2:30-3:30 Patricia Alonso-Ruiz (UConn)

Title: Nonsmooth Analysis: Postdoc Workshop
Speaker: Conference Speakers
Time: Friday, November 10, 2017 at 8:30 am
Place: MONT 214Abstract: 8:30-9:30 Irina Holmes (Michigan State) 9:45-10:45 Vyron Vellis (UConn) 11:00-12:00 Li Chen (UConn) 2:45-3:45 Manki Cho (Rochester Institute of Technology) 4:00-5:00 Simon Bortz (Minnesota)

Title: Nonsmooth Analysis: Postdoc Workshop
Speaker: Conference Speakers
Time: Saturday, November 11, 2017 at 8:30 am
Place: MONT 225Abstract: 8:30-9:30 Bobby Wilson (MIT) 9:45-10:45 Biwas Chandan (Cincinnati) 11:00-12:00 Scott Zimermann (UConn) 2:00-3:00 Jose Conde (Brown) 3:15-4:15 Dali Nimer (Chicago)

Title: TBA
Speaker: Lihan Wang (University of Connecticut)
Time: Friday, December 1, 2017 at 3:30 pm
Place: MONT 314Abstract: TBA

Title: TBA
Speaker: Lihan Wang (University of Connecticut)
Time: Friday, December 8, 2017 at 3:30 pm
Place: MONT 314Abstract: TBA