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Abstracts

Changfeng Gui
Dan Parman Endowed Distinguished Professor @University of Texas at San Antonion
Professor Gui received his Ph.D. from University of Minnesota (United States), he held positions @University of British Columbia, Vancouver (Canada) and @University of Connecticut. Professor Gui was a Simons Fellow in 2019 and a Fellow @American Mathematical Society in 2013.
New Sharp Inequalities in Analysis and Geometry

The classical Moser-Trudinger inequality is a borderline case of Soblolev inequalities and has important applications in geometric analysis and PDEs. On the two dimensional sphere, Aubin in 1979 showed that the best constant in the Moser-Trudinger inequality can be reduced by half if the functions are restricted to a subset of the Sobolev space $H^1$ with mass center of the functions at the origin, while Onofri in 1982 discovered an elegant optimal form of Moser-Trudinger inequality. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without constraints.

One such inequality, for example, incorporates the mass center deviation (from the origin) into the optimal inequality of Onofri on the sphere. In another view point, this inequality also generalizes to the sphere the Lebedev-Milin inequality and the second inequality in the Szegö limit theorem on the Toeplitz determinants on the circle, which is useful in the study of isospectral compactness for metrics defined on compact surfaces, among other applications.

The talk is based on a joint work with Amir Moradifam (University of California, Riverside) and a recent joint work with Alice Chang (Princeton).

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Professor @Korea Advanced Institute of Science and Technology, Daejeon, Korea.
On pseudoconformal blow-up solutions to the self-dual Chern-Simons-Schrödinger equation

I will present a blow up construction - jointly with Kihyun Kim - on the self-dual Chern-Simons-Schrödinger equation (CSS), also known as a gauged nonlinear Schrödinger equation (NLS). CSS is $L^{2}$-critical, admits solitons, and has the psuedoconformal symmetry. These features are similar to the $L^{2}$-critical NLS. In this work, we consider pseudoconformal blow-up solutions under $m$-equivariance, $m\geq1$. Our result is threefold. Firstly, we construct a pseudoconformal blow-up solution $u$ with given asymptotic profile $z^{\ast}$: \[ \Big[u(t,r)-\frac{1}{|t|}Q\Big(\frac{r}{|t|}\Big)e^{-i\frac{r^{2}}{4|t|}}\Big] e^{im\theta}\to z^{\ast}\qquad\text{in }H^{1} \] as $t\to0^{-}$, where $Q(r)e^{im\theta}$ is a static solution. Secondly, we show that such blow-up solutions are unique in a suitable class. Lastly, yet most importantly, we exhibit an instability mechanism of $u$. We construct a continuous family of solutions $u^{(\eta)}$, $0\leq\eta\ll1$, such that $u^{(0)}=u$ and for $\eta>0$, $u^{(\eta)}$ is a global scattering solution. Moreover, we exhibit a rotational instability as $\eta\to0^{+}$: $u^{(\eta)}$ takes an abrupt spatial rotation by the angle \[ \Big(\frac{m+1}{m}\Big)\pi \] on the time interval $|t|\lesssim\eta$.

We are inspired by works in the $L^{2}$-critical NLS. In the seminal work of Bourgain and Wang (1997), they constructed such pseudoconformal blow-up solutions. Merle, Raphaël, and Szeftel (2013) showed an instability of Bourgain-Wang solutions. Although CSS shares many features with NLS, there are essential differences and obstacles over NLS. Firstly, the soliton profile to CSS shows a slow polynomial decay $r^{-(m+2)}$. This causes many technical issues for small $m$. Secondly, due to the nonlocal nonlinearities, there are strong long-range interactions even between functions in far different scales. This leads to a nontrivial correction of our blow-up ansatz. Lastly, the instability mechanism of CSS is completely different from that of NLS. Here, the phase rotation is the main source of the instability. On the other hand, the self-dual structure of CSS is our sponsor to overcome these obstacles. We exploited the self-duality in many places such as the linearization, spectral properties, and construction of modified profiles.

In the talks, I will present background of the problem, main theorems, and outline of the proof with emphasis on heuristics of main features, such as the long- range interaction between blow up profile and asymptotic profile $z$, and rotational instability mechanism.

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Martin Li
Assistant Professor @The Chinese University of Hong Kong
Professor Li received his Ph.D. @Standford (United States). His thesis advisor was Professor Richard Schoen. He was a Postdoctoral Fellow @University of British Columbia, Vancouver (Canada) from 2011 until 2013, and @Massachusetts Institute of Technoogy from 2013 until 2014. Since 2014, he is Assistant Professor @The Chinese University of Hong Kong.
Mean curvature flow with free boundary

Mean curvature flow (MCF) is the negative gradient flow for the area functional in Euclidean spaces, or more generally in Riemannian manifolds. Over the past few decades, there have been substantial progress towards our knowledge on the analytic and geometric properties of MCF. For compact surfaces without boundary, we have a fairly good understanding of the convergence and singularity formation under the flow. The corresponding boundary value problems, however, are relatively less studied.

In this talk, we will discuss some recent results on MCF of surfaces with boundary. In the presence of boundary, suitable boundary conditions have to be imposed to ensure the evolution equations are well-posed. Two such boundary conditions are the Dirichlet (fixed or prescribed) and Neumann (free boundary or prescribed contact angle) boundary conditions. We will mention some new phenomena in contrast with the classical MCF without boundary. Using a new perturbation technique, we establish new convexity and pinching estimates for MCF with free boundary lying on an arbitrary convex barrier with bounded geometry. These imply the smooth convergence to shrinking hemispheres along the flow, provided that the surface is initially convex enough. This can be compared to Huisken’s celebrated convergence results for MCF in Riemannian manifolds.

This is joint work with Sven Hirsch. (These works are partially supported by RGC grants from the Hong Kong Government.)

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Connor Mooney
Assistant Professor @ University of California, Irvine.
Professor Mooney received his Ph.D. @ Columbia University in 2015. He was an NSF Postdoctoral Research Fellow @ UT Austin from 2015-16, and a Postdoctoral Researcher @ ETH Zurich from 2016-18. Since 2018, he has been an Assistant Professor @ University of California, Irvine.
The Bernstein problem for elliptic functionals

The Bernstein problem asks whether entire minimal graphs in $\mathbb{R}^{n+1}$ are necessarily hyperplanes. This problem was completely solved by the late 1960s in combined works of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De Giorgi-Giusti.

We will discuss the analogue of this problem for more general elliptic functionals, and some recent progress in the case $n = 6$.

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