College of Liberal Arts and Sciences

# Department of Mathematics

## Analysis Learning Seminar

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Title: Hodge theory on manifolds with boundary I
Speaker: Lihan Wang (University of Connecticut)
Time: Friday, February 2, 2018 at 3:30 pm
Place: MONT 314Abstract: In differential geometry, Hodge theory is well known and fundamental. In this sequence of talks, we will focus on the case of manifolds with boundary. We start with Riemannian case , then go to Symplctic case. We will try to use simple language to make it accessible for everyone.

Title: Hodge theory on manifolds with boundary II
Speaker: Lihan Wang (University of Connecticut)
Time: Friday, February 9, 2018 at 3:30 pm
Place: MONT 314Abstract: In differential geometry, Hodge theory is well known and fundamental. In this sequence of talks, we will focus on the case of manifolds with boundary. We start with Riemannian case , then go to Symplctic case. We will try to use simple language to make it accessible for everyone.

Title: Hodge theory on manifolds with boundary III
Speaker: Lihan Wang (University of Connecticut)
Time: Friday, February 16, 2018 at 3:30 pm
Place: MONT 314Abstract: In differential geometry, Hodge theory is well known and fundamental. In this sequence of talks, we will focus on the case of manifolds with boundary. We start with Riemannian case , then go to Symplctic case. We will try to use simple language to make it accessible for everyone.

Title: Harmonic Measure
Speaker: Matthew Badger (University of Connecticut)
Time: Friday, February 23, 2018 at 1:30 pm
Place: MONT 313Abstract: I will give a gentle introduction to harmonic measure, a fascinating object associated to any Euclidean domain which arises in complex analysis, partial differential equations, and probability theory. The first half of the talk will focus on definitions and basic examples. The second half of the talk will give a flavor of contemporary research questions about harmonic measure. No previous exposure to this topic is required.

Title: Applications of Multiplicative LLN and CLT to Random Matrices
Speaker: Raji Majumdar and Anthony Sisti (University of Connecticut)
Time: Friday, March 2, 2018 at 3:30 pm
Place: MONT 314Abstract: The Lyapunov exponent is the exponential growth rate of the operator norm of the partial products of a sequence of independent and identically distributed random matrices. It usually cannot be computed explicitly from the distribution of the matrices. Furstenberg and Kesten (1960) and Le Page (1982) found analogues to the Law of Large Numbers and Central Limit Theorem, respectively, for the norm of the partial products of a sequence of such random matrices. We use these analogues to efficiently compute the Lyapunov exponent for several random matrix models and numerically estimate the corresponding variances. For random matrices of order 2, with independent components distributed as $\xi\textrm{Bernoulli}(\frac{1}{2})$, where $\xi\in\mathbb{R}$, we obtain analytic estimates for the Lyapunov exponent in terms of a limit involving Fibonacci-like sequences.

Title: An Introduction to Rectifiable Sets and Measures
Speaker: Lisa Naples (University of Connecticut)
Time: Friday, March 23, 2018 at 3:30 pm
Place: MONT 314Abstract: Geometric Measure Theory uses measure theoretic techniques to classify structural properties of sets. A set in Euclidean space can be classified by its Hausdorff dimension, but even sets of a fixed dimension can have drastically different structures. In this talk I will introduce the notion of rectifiability as a structural property of sets, and describe a classification theorem for rectifiable sets of integer dimension. I will end by discussing some more recent extensions to rectifiability of measures.

Title: Phase transitions in the 1-2 model
Speaker: Zhongyang Li (University of Connecticut)
Time: Friday, March 30, 2018 at 3:30 pm
Place: MONT 314Abstract: A configuration in the 1-2 model is a subgraph of the hexagonal lattice, in which each vertex is incident to 1 or 2 edges. By assigning weights to configurations at each vertex, we can define a family of probability measures on the space of these configurations, such that the probability of a configuration is proportional to the product of weights of configurations at vertices. We study the phase transition of the model by investigating the probability measures with varying weights. We explicitly identify the critical weights, in the sense that the edge-edge correlation decays to 0 exponentially in the subcritical case, and converges to a non-zero constant in the supercritical case, under the limit measure obtained from torus approximation. These results are obtained by a novel measure-preserving correspondence between configurations in the 1-2 model and perfect matchings on a decorated graph, which appears to be a more efficient way to solve the model, compared to the holographic algorithm used by computer scientists to study the model. The major difficulty here is the absence of stochastic monotonicity.

Title: An Introduction to Self Similar Structures
Speaker: Christopher Hayes (University of Connecticut)
Time: Friday, April 6, 2018 at 3:30 pm
Place: MONT 314Abstract: One category of fractal is the self similar set. Famous examples include the Sierpinski Carpet and Koch Curve and analysis on these types of fractals is currently an active field of research. In this talk we will go over a variety of examples, discuss some construction methods, definitions, commonly used properties such as the open set condition and having a finite post-critical set, and end with comments on how analysis can be done on these sets, in particular the construction of a Dirichlet form on a post-critically finite set.

Title: BiLipschitz decomposability of Lipschitz functions
Speaker: Sean Li (University of Connecticut)
Time: Friday, April 27, 2018 at 3:30 pm
Place: MONT 314Abstract: Sard's theorem says that the critical values of a differentiable functions between Euclidean spaces have measure zero. We will study a similar phenomenon for Lipschitz functions and review what is known in more general metric spaces.