## Past Talks

Mean curvature flow with free boundary
Monday, 22 June 2020.
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.

#### Abstract

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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Some recent results on multilinear pseudo-differential operators with exotic symbols
Monday, 15 June 2020.
Professor @ the Tokyo Woman's Christian University, Japan.
Professor Miyachi received his Ph.D. @ the University of Tokyo, Japan. He was Associate Professor @ Hitotsubashi University in Tokyo from 1983 until 1995. Since then he was appointed to Professor @ the Tokyo Woman's Christian University.

#### Abstract

The theory of linear pseudo-differential operators is now well-known and, in particular, the mapping properties of linear pseudo-differential operators of Hörmander's class $S^m_{\rho, \delta}$ is well understood. Multilinear pseudo-differential operators were introduced by Coifman and Meyer before 1980 but detailed study of their mapping properties were developed after 2000. Some interesting facts peculiar to the multilinear case have been found.

In this lecture, I survey some features of multilinear pseudo-differential operators and introduce some recent results concerning the multilinear version of Hörmander's class.

This talk is based on joint works with Naohito Tomita (Osaka University) and Tomoya Kato (Gunma University).

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On pseudoconformal blow-up solutions to the self-dual Chern-Simons-Schrödinger equation
Monday, 8 June 2020.
Associate Professor @ Korea Advanced Institute of Science and Technology, Daejeon, Korea
Professor Kwon received his Ph.D. at UCLA in 2008. Then he worked at Princeton University as an instructor in 2008-2010. Since 2010, he is appointed as Associate Professor at the Korea Advanced Institute of Science and Technology, Daejeon, Korea.

#### Abstract

In this talk, I 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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New Sharp Inequalities in Analysis and Geometry
Monday, 1 June 2020.
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.

#### Abstract

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).

#### Video recording to the talk

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Motion by mean curvature and coupled KPZ from particle systems
Monday, 25 May 2020.
Professor @Waseda University
Funaki was a Professor @The University of Tokyo from 1995 until 2017 and is now a Professor @Waseda University from 2017. He is a Professor Emeritus of the University of Tokyo.

#### Abstract

It's one of common interests in probability group to derive nonlinear PDEs or stochastic PDEs from microscopic interacting systems via scaling limits in space and time, usually by some averaging effect caused by the local ergodic property of the microscopic systems.

In the talk, we discuss the derivation of two objects: Motion by mean curvature (MMC) and coupled KPZ (Kardar-Parisi-Zhang) equation. The microscopic system we take is that of particles which perform random walks with interaction. To derive MMC, we allow creation and annihilation of particles. The system exhibits a phase separation to sparse and dense regions of particles and, macroscopically, the interface separating these two regions evolves under the MMC. We pass through the Allen-Cahn equation with nonlinear Laplacian at an intermediate level. On the other hand, the coupled KPZ equation is a system of nonlinear singular stochastic PDEs. It is ill-posed in a classical sense and requires renormalizations. We derive it under the fluctuation limit of multi-species weakly-asymmetric particle system of interacting random walks without creation and annihilation. The so-called Boltzmann-Gibbs principle plays a fundamental role for both.

References

The first part is joint work with S. Sethuraman, D. Hilhorst, P. El Kettani and H. Park ( arXiv:2004.05276), while the second is with C. Bernardin and S. Sethuraman ( arXiv:1908.07863).

#### Video recording to the talk

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The vanishing discount problem for systems of Hamilton-Jacobi equations
Monday, 18 May 2020.
Research Fellow @Tsuda University (Japan), Professor Emeritus @Waseda University (Japan), Fellow @American Mathematical Society
Professor Ishii received his Ph.D. from Waseda University in June 1975. He worked at Chuo University from 1976 until 1996, at Metropolitan University from 1996 until 2001, and at Waseda University from 2001 until 2018.

#### Abstract

In the talk, I discuss the recent developments of the vanishing discount problem. The main focus concerns that for monotone systems of Hamilton-Jacobi equations. The principal tool in the analysis for the vanishing discount is the use of Mather measures from Aubry-Mather theory or their generalizations. I explain a way of constructing Mather measures and an approach to the vanishing discount problem.

This talk bases on joint work with Liang Jin of Nanjing University of Science and Technology.

References
• [1]  Hitoshi Ishii and Liang Jin, The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 2: nonlinear coupling, arXiv:1906.07979 [math.AP], to appear in Calculus of Variations and PDE's.
• [2]  Hitoshi Ishii, The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling, arXiv:1903.00244 [math.AP].
• [3]  Andrea Davini and Maxime Zavidovique, Convergence of the solutions of discounted Hamilton-Jacobi systems, Adv. Calc. Var. Online publication (2019), DOI: 10.1515/acv-2018-0037.

#### Video recording to the talk

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Recent progress on a sharp lower bound for Steklov eigenvalue
Monday, 11 May 2020.
Professor
@ Xiamen University, China
Professor Xia received his PhD in July 2012 from the Albert Ludwig University of Freiburg in Germany. He was a doctoral student of Professor Guofang Wang. Professor Xia was a Postdoc Fellow at the Max Planck Institute for Mathematics and Sciences at Leipzig in Germany from 2012-2014, and Postdoc Fellow at McGill University in Montreal, Canada, from 2015-2016.

#### Abstract

Lichnerowicz-Obata's theorem says that for a closed $n$-manifold with Ricci curvature $Ric\ge (n-1)K>0$, the first (nonzero) eigenvalue is greater than or equal to $nK$, with equality holding only on a round $n$-sphere. Similar results for the first Dirichlet eigenvalue and Neumann eigenvalue have been shown by Reilly, Escobar and C.Y.Xia respectively. For the first (nonzero) Steklov eigenvalue, a conjecture has been made by Escobar saying that for a compact manifold with boundary which has nonnegative Ricci curvature and boundary principal curvatures bounded below by some $c>0$, the first (nonzero) Steklov eigenvalue is greater than or equal to $c$, with equality holding only on a Euclidean ball. This conjecture is true in two dimensions due to Payne and Escobar. In this talk, we present a resolution to this conjecture in the case of nonnegative sectional curvature in any dimensions. We will also discuss a sharp comparison result between the first (nonzero) Steklov eigenvalue and the boundary first eigenvalue.

This is a joint work with Changwei Xiong.