The default time for the analysis seminar this semester is Monday at 2:30-3:30pm
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.
Gareth Speight (University of Cincinnati)
Monday February 3 at 2:30pm in French Hall Seminar Room
Maximal Directional Derivatives and Universal Differentiability Sets
Rademacher's theorem states that every Lipschitz function is differentiable almost everywhere. A universal differentiability set (UDS) is a set which contains a point of differentiability for every real-valued Lipschitz function. As a consequence of Rademacher's theorem, every set of positive measure is a UDS. However, many spaces contain a measure zero UDS. This talk describes recent work on UDS in Laakso space, a space that admits a differentiable structure despite not having a linear or group structure.
Gareth Speight (University of Cincinnati)
Monday February 10 at 2:30pm in French Hall Seminar Room
Maximal Directional Derivatives and Universal Differentiability Sets (Continued)
Benham Esmayli (University of Cincinnati)
Monday February 17 at 2:30pm in French Hall Seminar Room
On motivations for measures on the Euclidean space other than the Lebesgue.
I will give examples of contexts in analysis (PDE) and geometry (bi-Lipschitz and quasisymmetric mappings) that encourage us to work with measures on the Euclidean space other than the Lebesgue measure. What are the bare minimum properties we should require of a generic measure so that it behaves nicely with the (metric) properties of the Euclidean space? Under these assumptions, are such measures absolutely continuous with respect to the Lebesgue measure (i.e. is the Lebesgue measure canonical in this sense)?
I will also discuss some subtleties when it comes to defining Sobolev functions with respect to measures. I plan on a mostly expository talk, but I will explain how an open problem regarding “admissible measures” may have already been solved in 2016 but is yet unnoticed in literature!
Ilmari Kangasniemi (University of Cincinnati)
Monday February 24 at 2:30pm in French Hall Seminar Room
Comparing Dirichlet and Newtonian Sobolev spaces in metric measure spaces
In this talk I discuss a joint work with Shanmugalingam and Gibara, where we investigate metric measure spaces (X, d, μ) for which every Dirichlet Sobolev function is a constant term away from a Newtonian Sobolev function. Here, Dirichlet Sobolev functions are real-valued functions with an Lᵖ-integrable upper gradient, and Newtonian Sobolev functions are additionally assumed to be Lᵖ-integrable themselves. We find numerous distinct conditions which prevent most well-behaved spaces X from having this property, including global doubling, p-parabolicity or possessing a p-parabolic end, and various uniformization conditions related to Gromov hyperbolic spaces. We also find one positive example where this condition does hold, which is the standard hyperbolic space ℍⁿ with 1 ≤ p ≤ n-1.
Chun Ho Lau (University of Cincinnati)
Monday March 3 at 2:00pm in French Hall Seminar Room (note the time - 30 minutes earlier than usual)
A very short introduction to Fourier restriction problem and its differentiability
In this talk, I will introduce a famous problem in harmonic analysis, the Fourier restriction problem (aka the Fourier restriction conjecture), and a related problem on the differentiability of Fourier restriction. In the first half of the talk, I will introduce the problem through the properties of the Fourier transform and through a Strichartz estimate of a PDE. I will also list some of the partial results on this problem. In the second half of the talk, I will introduce the Derivative Restriction Problem, the restriction of the (fractional) derivative with respect to the last variable on a surface, and give the result to it. The second part is based on a joint work with Michael Goldberg.
Bingyu Zhang (University of Cincinnati)
Monday March 24 at 2:30pm in French Hall Seminar Room
Global Well-posedness of the Initial-Boundary Value Problem of a Class Generalized KdV Equation on a Finite Interval
Abstract available here
Ivonne Rivas Trivino (Universidad del Valle, Cali, Colombia)
Monday April 7 at 2:30pm in French Hall Seminar Room
Exact Controllability through Flatness Method in space of analytic Functions
Abstract available here.
Mokshay Madiman (University of Delaware)
Friday, April 11 at 1-2pm in French Hall Seminar Room
Log-concavity and Tracy-Widom universality
The Tracy-Widom (TW) distributions, indexed by a positive parameter b, arise as limiting distributions in many different probabilistic models including random matrices and the longest increasing subsequences of random permutations, thus exemplifying a universality phenomenon in probability. However, they are rather mysterious, with no explicit form for their densities or characteristic functions, and little has been proved analytically about their properties. Although simulations suggest that the TW distributions are log-concave, the only (partial) result in this direction is a theorem of Percy Deift that the TW distribution with parameter 2 is log-concave on the positive real line. We settle this as well as several related questions: Not only are all TW distributions shown to be log-concave, we do this by establishing log-concavity of certain pre-limit distributions, including the largest eigenvalues of Gaussian beta-ensembles and the Poissonized Plancherel measure on Young diagrams that arises in the representation theory of the symmetric group. In particular, one consequence of our results is that a Poissonized version of a 2008 conjecture of W.Y.C.Chen— asserting that for any fixed natural number N, the number of permutations in the symmetric group S(N) that have a longest increasing subsequence of length K, is a log-concave sequence in K— is true. The talk is based on joint work with Jnaneshwar Baslingker and Manjunath Krishnapur.
Yunfeng Zhang (University of Cincinnati)
Monday April 21 at 2:30pm in French Hall Seminar Room
The modified KdV equation beyond Sobolev spaces
It has become standard practice to solve a PDE with initial data in a maximal Sobolev space H^s. For the modified KdV equation, significant progress includes the work of Kenig–Ponce–Vega, which established analytic local well-posedness in H^{1/4}, the maximal Sobolev space for analytic well-posedness. Further advancements were made by Colliander–Keel–Staffilani–Takaoka–Tao, Guo, and Kishimoto, who demonstrated analytic global well-posedess in H^{1/4}. More recently, Harrop-Griffiths–Killip–Visan extended these results by proving non-analytic global well-posedness in H^s for the maximal range s>-1/2, covering the entire scaling subcritical range for mKdV. Notably, this reveals a gap between analytic well-posedness and the scaling critical regularity, which is bridged by non-analytic well-posedness.
Interestingly, by employing the Fourier–Lebesgue spaces instead of Sobolev spaces, Grünrock, followed by Grünrock–Vega, achieved analytic local well-posedness across the full scaling subcritical range (for Fourier–Lebesgue spaces). This result—where no gap appears—is in stark contrast to the Sobolev case described earlier, highlighting the advantages of exploring alternative function spaces.
In this talk, I will introduce yet another function space: the modulation space, originally developed by Feichtinger in the 1980s. A natural question arises: does a gap exist between analytic well-posedness and the scaling critical regularity for mKdV in these spaces? The answer is affirmative, and I will demonstrate non-analytic well-posedness for mKdV within a gap range of modulation spaces. As in the work of Harrop-Griffiths–Killip–Visan, our approach leverages the completely integrable structure of mKdV, with equicontinuity of solutions being the key property to establish. This is joint work with Haque, Killip, and Visan.
Sabrina Traver (Syracuse University)
Friday April 25 at 2:30pm in French Hall Seminar Room
Global invertibility of Sobolev mappings with prescribed homeomorphic boundary values
Let $X, Y \subset \mathbb{R}^n$ be Lipschitz domains, and suppose there is a homeomorphism $\varphi \colon \overline{X} \to \overline{Y}$. We consider the class of Sobolev mappings $f \in W^{1,n} (X, \mathbb{R}^n)$ with a strictly positive Jacobian determinant almost everywhere, whose Sobolev trace coincides with $\varphi$ on $\partial X$. We prove that every mapping in this class extends continuously to $\overline{X}$ and is a monotone (continuous) surjection from $\overline{X}$ onto $\overline{Y}$ in the sense of C.~B.~Morrey. This talk will give an overview of this result and some of the predominant tools in the proof. Time permitting, we will apply our result to an energy minimization problem.
The standard time for the analysis seminar will be Friday at 1-2pm in 60W Charlton 240. However it may change if there is reason to do so.
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.
Xining Li (Sun Yat-Sen University)
Thursday August 29 at 2:40-3:40pm in French Hall 4221
Uniformization of Gromov hyperbolic spaces with respect to measures
In this talk we consider a version of uniformization of Gromov hyperbolic metric measure spaces with respect to measure.
Yunfeng Zhang (University of Cincinnati)
Friday September 6 at 1-2PM in 60W Charlton 240
Semiclassical fun with SU(3)
This talk is intended to give an overview of several "semiclassical analytic" questions on compact Lie groups, SU(3) being a typical example, such as Strichartz estimates for the Schrödinger flow, bounds of Laplacian eigenfunctions, etc.
Michal Wojciechowski (IMPAN - Polish Academy of Sciences, Warsaw)
Tuesday September 17 at 3:35-4:30PM in 60W Charlton 273
Unexpected approximation of Gaussian density and Hermite functions by B-splines
We prove, and provide a quantitative error estimate, that a B-spline based on essentially arbitrarily distributed knots approximates the Gaussian density. This construction provides an unexpected inequality for any polynomial with distinct real roots. On the other hand, one can see this as a kind of generalization of CLT for the uniform distribution, which is exactly what arises for equally spaced knots. Moreover, we prove that the approximation could be made in the Sobolev norms which leads to analogous approximations of Hermite functions. The talk is based on a joint work with M. Rzeszut.
Josh Kline (University of Cincinnati)
Friday September 20 at 1-2PM in 60W Charlton 240
Self-improvement of fractional Hardy inequalities on metric measure spaces
A domain satisfies a p-Hardy inequality if for any compactly supported smooth function, the L^p-norm of its gradient controls its L^p-norm, weighted by the distance to the boundary. Such inequalities are important in potential theory, having close connections with uniform p-fatness and boundary regularity of p-harmonic functions, and are known to self-improve. That is, validity of a p-Hardy inequality implies validity of a q-Hardy inequality for q sufficiently close to p. In this talk, we consider a fractional Hardy inequality in the metric setting, replacing the L^p norm of the gradient with the nonlocal Besov energy, and show that these inequalities also self-improve, both in exponent p and the fractional parameter. This talk is based on joint work with Sylvester Eriksson-Bique.
Scott Zimmerman (Ohio State University) - Colloquium
Thursday October 10 at 4-5PM in 4221 French Hall
Bi-Lipschitz arcs in metric spaces
A bi-Lipschitz arc in a metric space $X$ is the image of an interval in the real line under a bi-Lipschitz map, and, from a metric point of view, such a set is indistinguishable from the interval. In this talk, we’ll consider the following question: given a set $K \subset \mathbb{R}$ and a bi-Lipschitz map $f:K \to X$, is $f(K)$ always contained in a bi-Lipschitz arc? This question was answered positively in the case $X = \mathbb{R}^n$ by David and Semmes for $n \geq 3$ and later by MacManus when $n = 2$. I will present a large class of metric spaces in which we can also answer this question positively and discuss the ideas behind the constructions. This is joint work with Jacob Honeycutt and Vyron Vellis.
Jani Onninen (Syracuse University) - Colloquium
Thursday October 17 at 4-5PM in 4221 French Hall
Existence of 2D frictionless minimal deformations
In Geometric Function Theory we seek, as a generalization of the Riemann Mapping Problem, homeomorphisms that minimize certain energy integrals. No boundary values of such homeomorphisms are prescribed. This is interpreted as saying that the deformations are allowed to slip along the boundary, known as frictionless problems. This leads us to determine the infimum of a given energy functional among homeomorphisms from X onto Y . However, even in the model of the Dirichlet energy, it is often unrealistic to expect the infimum to be achieved within the class of homeomorphisms. Expanding the class of admissible mappings may alter the energy-minimizing solutions. To avoid the Lavrentiev gap, we turn to the study of monotone Sobolev mappings, which is the focus of my talk.
Pavel Zatitskii (University of Cincinnati)
Friday October 25 at 1-2PM in 60W Charlton 240
Poisson analog of the Stein-Weiss theorem on Hilbert transform
The classical theorem by E. Stein and G. Weiss gives an explicit formula for the distribution of a Hilbert transform of an indicator function of a set of finite measure. We will discuss this beautiful result and its Poisson analog, recently discovered jointly with L. Slavin.
Benham Esmayli (University of Cincinnati)
Friday November 8 at 1-2PM in 60W Charlton 240
Surjective Lipschitz maps with low-rank derivative everywhere.
If the derivative of a C^1 map from R^m to R^n, m >n, has full rank at a point x, then its image contains an open ball around f(x). This follows from the implicit function theorem.
The converse is not true. There exists a surjective C^1 map from R^3 to R^2 with rank less than or equal to 1 at every point. Something (geometric) must be degenerate about these maps, but what is it?
In work with P. Hajlasz, we prove that a Lipschitz map from a cube in a Euclidean space into any metric space factors through a metric tree if and only if the rank of its derivative is less than or equal to 1 at almost every point. In other words, the way these maps cover the range is by compressing and folding a (huge) metric tree onto the target space.
I will explain the construction of this tree and, time permitting, I will discuss the fate of the implicit function theorem for low rank maps.
Sylvester Eriksson-Bique (University of Jyvaskyla)
Friday November 15 at 1-2PM in 60W Charlton 240
Fractals of “Laakso type”
Laakso gave examples of fractals with non-integer Hausdorff dimension and satisfying a Poincaré inequality. I will describe in this talk a closely related construction, which leads to answers to a host of other problems (Kleiners conjecture in quasiconformal geometry, biLipshitz embeddings, Sobolev spaces and potential theory of fractals, Nagata dimension and analytic dimension, increase of Analytic dimension…). Besides the construction, I’ll explain the key features that make it simple to construct interesting examples that answer these different problems.
Juan Manfredi (University of Pittsburgh) - Taft Lecture
Thursday November 21 at 4-5pm in Rec Center 3250
Asymptotic Mean Value Expansions for Solutions to Nonlinear Equations
Harmonic functions in Euclidean space are characterized by the mean value property and by expectations of stopped Brownian motion processes. This equivalence has a long history with fundamental contributions by Doob, Hunt, Ito, Kakutani, Kolmogorov, Levy, and many others. In this lecture, I will propose ways to extend this characterization to solutions of non-linear elliptic and parabolic equations. The non-linearity of the equation requires that the rigid mean value property be replaced by asymptotic mean value expansions and Brownian motion by stochastic games, but the main equivalence remains when formulated with the help of the theory of viscosity solutions. Moreover, this local equivalence also holds on more general ambient spaces like Riemannian manifolds and the Heisenberg group.
I will present the details of this equivalence for the p-Laplace equation in Euclidean space and in the Heisenberg group.