College of Liberal Arts and Sciences

# Department of Mathematics

## Analysis and Probability Seminar

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Title: Sobolev spaces that are not algebras
Speaker: Luke Rogers (University of Connecticut)
Time: Friday, February 27, 2015 at 3:30 pm
Place: MSB 109A

Title: Duality of Markov processes with respect to a function
Speaker: Sabine Jansen (Ruhr-Universität Bochum)
Time: Friday, March 6, 2015 at 3:30 pm
Place: MSB 109AAbstract: Duality of Markov processes with respect to a function is a powerful tool in application areas ranging from queuing theory to mathematical population genetics, but despite its usefulness, there is no general theory of this notion. This is in stark contrast to duality with respect to a measure, which has been well-studied in the context of potential theory. This talk presents a systematic study of the notion of duality with respect to a function; the focus is on elucidating the analytic framework, building on notions of dual pairs in functional analysis and convex geometry. In addition, we address questions such as why some dualities are associated with stochastic monotonicity. The talk is based on joint work with Noemi Kurt (Probab. Surveys 11, 59-120 (2014)).

Title: A characterization of rectifiable metric measure spaces
Speaker: David Bate (University of Chicago)
Time: Friday, March 13, 2015 at 3:30 pm
Place: MSB 109AAbstract: We characterize rectifiable metric measure spaces as those spaces that have sufficiently many 1-rectifiable representations of their measure. These representations, known as Alberti representations, have recently been used to describe those spaces that satisfy Cheeger's generalization of Rademacher's theorem and have since become an important tool that allows a geometric approach to solving problems in the purely metric setting. This talk will first cover the relevant background material that relates Alberti representations to Cheeger's generalization of Rademacher's theorem and then give an outline of the proof of the new result. This is joint work with Sean Li.

Title: On the analytic and Cauchy capacities
Speaker: Malik Younsi (Stony Brook University)
Time: Friday, March 27, 2015 at 3:30 pm
Place: MSB 109AAbstract: Analytic capacity of compact plane sets was first introduced by Ahlfors in order to study Painlevé's problem of finding a geometric characterization of the compact sets that are removable for bounded holomorphic functions. Despite recent advances due to Tolsa, the properties of analytic capacity remain rather mysterious. In particular, it is still unknown if analytic capacity is equal to the so-called Cauchy capacity. In this talk, I will present some new sufficient conditions for a compact set $E \subseteq \mathbb{C}$ to satisfy $\gamma(E)=\gamma_c(E)$, where $\gamma$ is the analytic capacity and $\gamma_c$ is the Cauchy capacity. As a consequence, I will describe how to produce examples of compact plane sets such that the above equality holds but the Ahlfors function is not the Cauchy transform of any complex Borel measure supported on the set.

Title: Degenerate martingales arising from various geometric structures, including minimal surfaces
Speaker: Robert Neel (Lehigh University)
Time: Friday, April 10, 2015 at 3:30 pm
Place: MSB 109AAbstract: We first discuss a class of degenerate martingales (which we will call rank-n martingales) that arises naturally as the diffusion associated with minimal submanifolds, mean curvature flow, and some sub-Riemannian structures. This provides a unified approach to "coarse" properties, such as transience, of such structures. We then specialize to minimal surfaces in R^3, in which case the associated rank-2 martingale (which is just Brownian motion on the surface, viewed as a process in R^3) has the additional property that the tangent plane also evolves as a martingale. Taking advantage of this extra structure, we develop an extrinsic analogue of the mirror coupling of two Brownian motions. This allows us to study finer geometric and analytic properties of minimal surfaces, such as intersection results (strong halfspace-type theorems) and Liouville properties.

Title: (cancelled)
Speaker: Theresa Anderson (Brown University)
Time: Friday, April 17, 2015 at 3:30 pm
Place: MSB 109AAbstract: This talk originally scheduled for 4/17/2015 has been cancelled.

Title: On the range of the transient frog model on Z
Speaker: Alexander Roitershtein (Iowa State University)
Time: Friday, April 24, 2015 at 3:30 pm
Place: MSB 109AAbstract: I will discuss a one-dimensional frog model", an infinite system of interacting random walks on Z with an asymmetric underlying random walk. The talk will focus on results concerning the distribution of the models long-run range and its asymptotic behavior as the drift of the underlying random walk vanishes. This a joint work with my colleague at ISU Arka P. Ghosh and our PhD student Steven Noren.

Title: Probability and Related Topics -- in memory of Evarist Gine (special session in New England Statistics Symposium)
Speaker: Multiple speakers (NESS sesssion in memory of Evarist Gine)
Time: Saturday, April 25, 2015 at 3:30 pm
Place: AUST 434Abstract: Rick Vitale (UConn) Welcome Dick Dudley (MIT) Evarist as a student, teacher and friend'' Victor de la Pena (Columbia) Dependence measures: a perspective'' Iddo Ben-Ari (UConn) Evarist's favorite undergraduate proof and where it got me'' Lu Lu (Colby) On the sup-norm behavior of the Bernstein density estimator'' Molly Hahn (Tufts), and others as they would like: Evarist: Reminiscences Link to conference: http://merlot.stat.uconn.edu/ness15 Conference organizer: Rick Vitale

Title: A conformally invariant metric on the Gaussian free field level loops
Speaker: Samuel Watson (MIT)
Time: Friday, May 1, 2015 at 3:30 pm
Place: MSB 109AAbstract: We will discuss an exploration process, introduced by Wendelin Werner and Hao Wu, in which conformal loop ensemble (CLE(kappa)) loops grow uniformly from the boundary of a domain. We relate this process in the case kappa = 4 to the set of level loops of the zero-boundary Gaussian free field, and we use this point of view to show that the exploration process is a deterministic function of the CLE loops. We describe how this gives rise to a conformally invariant metric on CLE(4), which we conjecture can be given a natural geometric interpretation. Based on joint work with Scott Sheffield and Hao Wu.

Title: Intersection of SLE paths
Speaker: Hao Wu (MIT)
Time: Friday, May 8, 2015 at 3:30 pm
Place: MSB 109AAbstract: SLE curves are introduced by Oded Schramm as the candidates of the scaling limit of discrete models. In this talk, we first describe basic properties of SLE curves and their relation with discrete models. Then we summarize the Hausdorff dimension results related to SLE curves, in particular the new results about the dimension of cut points and double points. Third we introduce Imaginary Geometry, and from there give the idea of the proof of the dimension results.

Organizer: Vasileios Chousionis and Oleksii Mostovyi