The standard time for the analysis seminar will be Thursday at 3:30-4:30PM in the French Hall Seminar Room (French Hall 4206). In weeks this conflicts with a department meeting or colloquium, it will alternate to Tuesday at 4:00-5:00PM.
If you are interested in giving a talk contact Gareth Speight (Gareth.Speight<at>uc.edu).
To find out more about our department click here.
The analysis seminar in Fall 2023 is tentatively scheduled for Thursdays at 2:30-3:25pm
Almaz Butaev
Thursday February 16th at 3:30-4:30pm in French Hall Seminar Room (4206)
Higher Order Lipschitz Functions on Riemannian Manifolds of Bounded Geometry
I will talk about higher order Lipschitz spaces on Rn and few if its characterizations. Based on one of these characterizations we will define higher order Lipschitz spaces on Riemannian manifolds of bounded geometry. Finally, we will formulate the Whitney-type extension theorem in this setting.
Qing Liu (Okinawa Institute of Science and Technology Graduate University)
Thursday February 23rd at 3:30-4:30pm in West Charlton 270
Principal eigenvalue problem for infinity Laplacian in metric spaces
This talk is concerned with the Dirichlet eigenvalue problem associated to the infinity Laplacian in metric spaces. We provide a PDE approach to find the principal eigenvalue and eigenfunctions for a bounded domain in a proper geodesic space with no measure structure. We give an appropriate notion of solutions to the infinity eigenvalue problem and show the existence of solutions by adapting Perron’s method. Our method is different from the standard limit process, introduced by Juutinen, Lindqvist and Manfredi in 1999, via the variational eigenvalue formulation for p-Laplacian in the Euclidean space. Several further results and concrete examples will be given in the case of finite metric graphs. This talk is based on joint work with Ayato Mitsuishi at Fukuoka University.
Andrew Lorent
Thursday March 2nd at 3:30-4:30pm in French Hall Seminar Room (4206)
The generalized Aviles Giga functional
We sketch the connections between conservation laws and the Aviles Giga functional. We then explain how the generalized Aviles Giga functional's relationship to the standard AG functional is analogous to relationship between general scalar conservation laws and Burger's equation. Finally we survey recent results, including some done in collaboration with Xavier sLamy and Guanying Peng.
Bingyu Zhang
Thursday March 9th at 3:30-4:30pm in West Charlton 270
Joint Effects of Dispersion and Dissipation in Nonlinear Evolution Equations
Abstract available here.
Rami Ayoush (University of Warsaw)
Tuesday March 21st at 3:30-4:30pm in Seminar Room
On finite configurations in the spectra of singular measures
Xiaoqin Guo
Thursday March 23rd at 3:30-4:30pm in 60W Charlton 270
Optimal homogenization rates for non-divergence form difference operators in a random environment
Diffusions in materials can be described by differential equations with rapidly oscillating coefficients in microscopic scale . Homogenization studies the convergence of equations of rapidly oscillating coefficients to a large scale deterministic "effective" equation. When the coefficient is periodic, it is known that generically the optimal convergence rate is ε. In this talk we will focus on the homogenization of second order non-divergence form difference operators with random coefficients. We will discuss the optimal convergence rate using concentration inequalities in probability and techniques in PDE. Joint work with Hung V. Tran (UW-Madison).
Ivan Caamano Aldemunde (Universidad Complutense Madrid)
Friday March 31st at 3:00-4:00pm in French Hall Seminar Room (4206)
Sobolev-Reshetnyak classes of mappings with values in Banach spaces
For mappings with values in a Banach space a natural generalization of the classical Sobolev spaces arise, namely the Sobolev Reshetnyak spaces introduced by Reshetnyak in 1997. The goal of the talk will be to introduce these classes of mappings, relate them to the classical notion of Sobolev mappings and study further results that characterize Sobolev Reshetnyak space.
Hyogo Shibahara
Thursday April 6th at 3:30-4:30pm in West Charlton 270
Toward C^m Whitney Extension Theorem for Horizontal Curves in Free Step 2 Carnot Groups
Pinamonti, Speight, and Zimmerman established a C^m Whitney Extension Theorem for horizontal curves in the Heisenberg group. While C^1 Whitney Extension Theorem is well understood due to the work of Juillet and Sigalotti as they found an appropriate coordinate-free Taylor polynomial for C^1 horizontal curves, the C^m case is poorly understood. In fact, Pinamonti and Speight found an example that shows that a simple generalization of the condition given in the Heisenberg group does not work in G_3, a free step 2 Carnot group with 3 generators. In this talk, we establish a C^m Whitney Extension Theorem for horizontal curves in G_3. New observations and techniques include a quantitative linear dependence of jets on the horizontal layer and a discretization procedure to correct multiple vertical areas. If time allows, we give a necessary condition for horizontal curves in G_r where the effect of unrelated vertical areas comes into play.
Patricia Alonso Ruiz (Texas A&M)
Friday April 14th at 2:30-3:30pm in French Hall Seminar Room (4206)
Minimal eigenvalue spacing in the Sierpinski gasket
Since its first investigations by the physicists Rammal and Tolouse in the 80s, the spectrum of the Sierpinski gasket has revealed structures with many interesting features not seen in other more classical settings. One example are the large exponential gaps (or spacings), whose existence and properties have extensively been studied in the literature.
Small gaps however had eluded previous investigations and will be the main subject of this talk, that presents yet another remarkable fact: The possibility to describe precisely the least separation between any two distinct consequent eigenvalues in the Dirichlet or in the Neumann spectrum of the Laplacian on the Sierpinski gasket.
The analysis seminar will be held on Thursday at 4-5PM
If you are interested in giving a talk contact Gareth Speight (Gareth.Speight<at>uc.edu).
To find out more about our department click here.
David Freeman
Thursday September 1 at 4-5PM in French Hall Seminar Room (4206)
Lipschitz Functions on Quasiconformal Trees
Given a metric space X, the Lipschitz free space on X is a Banach space in which X isometrically embeds and whose geometry reflects the structure of Lipschitz functions defined on X. In our talk, we briefly survey key facts about Lipschitz free space and prove that the Lipschitz free space of any quasiconformal tree is isomorphic to L^1. Time permitting, we will also discuss connections between this work and the concept of Lipschitz dimension. This is joint work with Chris Gartland and Texas A&M.
Andrew Lorent
Thursday September 15 at 4-5PM in French Hall Seminar Room (4206)
Quantitative rigidity of differential inclusions in two dimensions
We briefly survey Friesecke-James-Muller Quantitative rigidity result for SO(n) and its various generalizations. Then we will state a recent result with Xavier Lamy and Guanying Peng that is a generalization of FJM result to Elliptic connected 1-submanifolds and explain why this is the optimal generalization of FJM the 2 dimensions. We will briefly explain the main new ideas required for the proof.
Michal Wojciechowski
Tuesday September 20 at 4-5PM in French Hall Seminar Room (4206)
On the bi-analytic version of the Mitiagin-DeLeeuw-Mirkhil non-inequality
Using the method of Rudin-Shapiro polynomials we prove the bi-analytic version of the Mitiagin - DeLeeuw - Mirkhil non-inequality for complex partial differential operators with constant coefficients on bi-disc
Michal Wojciechowski (Colloquium)
Thursday September 22 at 4-5PM in French Hall 4221
Microlocal approach to the Hausdorff dimension of measures
Chengcheng Yang
Tuesday October 4 at 1:30-2:30PM in French Hall 4221
Geometric-Analytic Properties of Semi-Algebraic Sets
We will survey the geometric-analytic properties of a real algebraic variety and its generalization into a semi-algebraic set. Then we will talk about the current research in progress. The talk has two parts: global and local geometric structures; and relations between constants that arise in analysis inequalities.
Liding Yao
Thursday October 6 at 3-4PM in W Charlton 277
An In-depth Look at Rychkov’s Universal Extension Operators for Lipschitz Domains
Given a bounded Lipschitz domain $\Omega\subset\mathbb R^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $\Omega$ which is bounded in Besov and Triebel-Lizorkin spaces. In this paper we introduce some new estimates for the extension operator $\mathcal E$ and give some applications. We prove the equivalent norms $\|f\|_{\mathscr A_{pq}^s(\Omega)}\approx\sum_{|\alpha|\le m}\|\partial^\alpha f\|_{\mathscr A_{pq}^{s-m}(\Omega)}$ for general Besov and Triebel-Lizorkin spaces. We also derive some quantitative smoothing estimates of the extended function and all its derivatives in $\overline{\Omega}^c$ up to boundary. This is a joint work with Ziming Shi.
Josh Kline
Thursday October 13 at 4-5PM in French Hall Seminar Room (4206)
Least gradient problems in metric measure spaces
It is known in the Euclidean and metric setting that for a nice domain $\Omega$, the trace of any function of bounded variation in $\Omega$ belongs to $L^1(\partial\Omega)$, and likewise, $L^1$-functions on the boundary have BV-extensions into the domain. For functions of least gradient, that is, BV-energy minimizers, this trace and extension problem can be viewed as the following Dirichlet-type problem: which class of $L^1$-boundary data admit least gradient solutions in the domain? While continuous functions belong to this class, Spradlin and Tamasan (2014) showed that not all $L^1$-functions admit solutions. In this talk, we consider the least gradient problem in the metric setting and discuss some recent existence results for various discontinuous boundary data. By modifying the Spradlin and Tamasan example, we also show how the trace class of least gradient functions does not in general form a vector space.
Liangbing Luo
Thursday October 20 at 4-5PM in French Hall 4221
Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups
We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic operator. Furthermore, these results can be applied in an infinite-dimensional setting to prove a logarithmic Sobolev inequality on an infinite-dimensional Heisenberg group modelled on an abstract Wiener space.
Robert Buckingham
Tuesday October 25 at 1:20-2:20PM in French Hall 4221
Universality of High-Order Rogue Waves
Rogue waves are short-lived, high-intensity pulses that have been observed in the ocean and fiber optics. We will discuss a series of recent results indicating that high-order rogue-wave behavior is universally described for a variety of different equations and initial conditions by a family of functions connected to the Painleve-III hierarchy and first encountered by Suleimanov in 2017. This is joint work with Deniz Bilman, Bob Jenkins, and Peter Miller.
Deniz Bilman
Tuesday November 15 at 4-5PM in French Hall 4221
Wave patterns generated by large-amplitude rogue waves and their universal character
It is known from our recent work that both fundamental rogue wave solutions (with Peter Miller and Liming Ling) and multi-pole soliton solutions (with Robert Buckingham) of the nonlinear Schrödinger equation exhibit the same universal asymptotic behavior in the limit of large order in a shrinking region near the peak amplitude point, despite the quite different boundary conditions these solutions satisfy at infinity. We review these results and show that this profile arises universally from arbitrary background fields. We then show how rogue waves and solitons of arbitrary orders can be placed within a common analytical framework in which the "order" becomes a continuous parameter, allowing one to tune continuously between types of solutions satisfying different boundary conditions. In this framework, solitons and rogue waves of increasing integer orders alternate as the continuous order parameter increases. We show that in a bounded region of the space-time of size proportional to the order, these solutions all appear to be the same when the order is large. However, in the unbounded complementary region one sees qualitatively different asymptotic behavior along different sequences. This is joint work with Peter Miller (U. Michigan).
Sylvester Eriksson-Bique
Tuesday December 6 at 2:30-3:30PM in French Hall Seminar Room
P-Differentiable Structure on Metric Measure Spaces
I will discuss a modulus, and curve wise, approach -similar to that of Shanmugalingam - for defining a differential for a Sobolev function on a complete metric measure space with a doubling measure. The definition of this structure turns out to be something fairly “simple” to describe, although its existence is far from obvious. I will do my best to give a general introduction to how to define this structure - with fairly minimal background - and then mention how it leads to some new questions - and connections.