Jingyi Chen

Professor

Research Classification

Research Interests

Differential Geometry, Partial Differential Equations

Relevant Degree Programs

 
 

Recruitment

Master's students
Doctoral students
Postdoctoral Fellows
2022

Geometric Analysis 

I am open to hosting Visiting International Research Students (non-degree, up to 12 months).

Complete these steps before you reach out to a faculty member!

Check requirements
  • Familiarize yourself with program requirements. You want to learn as much as possible from the information available to you before you reach out to a faculty member. Be sure to visit the graduate degree program listing and program-specific websites.
  • Check whether the program requires you to seek commitment from a supervisor prior to submitting an application. For some programs this is an essential step while others match successful applicants with faculty members within the first year of study. This is either indicated in the program profile under "Admission Information & Requirements" - "Prepare Application" - "Supervision" or on the program website.
Focus your search
  • Identify specific faculty members who are conducting research in your specific area of interest.
  • Establish that your research interests align with the faculty member’s research interests.
    • Read up on the faculty members in the program and the research being conducted in the department.
    • Familiarize yourself with their work, read their recent publications and past theses/dissertations that they supervised. Be certain that their research is indeed what you are hoping to study.
Make a good impression
  • Compose an error-free and grammatically correct email addressed to your specifically targeted faculty member, and remember to use their correct titles.
    • Do not send non-specific, mass emails to everyone in the department hoping for a match.
    • Address the faculty members by name. Your contact should be genuine rather than generic.
  • Include a brief outline of your academic background, why you are interested in working with the faculty member, and what experience you could bring to the department. The supervision enquiry form guides you with targeted questions. Ensure to craft compelling answers to these questions.
  • Highlight your achievements and why you are a top student. Faculty members receive dozens of requests from prospective students and you may have less than 30 seconds to pique someone’s interest.
  • Demonstrate that you are familiar with their research:
    • Convey the specific ways you are a good fit for the program.
    • Convey the specific ways the program/lab/faculty member is a good fit for the research you are interested in/already conducting.
  • Be enthusiastic, but don’t overdo it.
Attend an information session

G+PS regularly provides virtual sessions that focus on admission requirements and procedures and tips how to improve your application.

 

Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - Nov 2019)
Uniqueness of Lagrangian mean curvature flow and minimal immersions with free boundary (2013)

In this thesis we investigate some problems on the uniqueness of mean curvatureflow and the existence of minimal surfaces, by geometric and analyticmethods. A summary of the main results is as follows.(i) The special Lagrangian submanifolds form a very important class ofminimal submanifolds, which can be constructed via the method ofmean curvature flow. In the graphical setting, the potential functionfor the Lagrangian mean curvature ow satisfies a fully nonlinearparabolic equation [formula omitted]where the ⋋j's are the eigenvalues of the Hessian D²u.We prove a uniqueness result for unbounded solutions of (1) withoutany growth condition, via the method of viscosity solutions ([4], [13]):for any continuous u₀ in ℝn, there is a unique continuous viscositysolution to (1) in ℝn x [0;∞).(ii) Let N be a complete, homogeneously regular Riemannian manifoldof dimN ≥ 3 and let M be a compact submanifold of N. Let Ʃ bea compact Riemann surface with boundary. A branched immersionu : (Σ,∂Σ) → (N,M) is a minimal surface with free boundary in Mif u(Σ) has zero mean curvature and u(Σ) is orthogonal to M alongu(∂Σ)⊑ M.We study the free boundary problem for minimal immersions of compactbordered Riemann surfaces and prove that Σ if is not a disk, then there exists a free boundary minimalimmersion of Σ minimizing area in any given conjugacy class ofa map in C⁰(Σ,∂Σ;N,M) that is incompressible; the kernel of i* : π₁(M) → π₁(N) admits a generating set suchthat each member is freely homotopic to the boundary of an areaminimizing disk that solves the free boundary problem. (iii) Under certain nonnegativity assumptions on the curvature of a 3-manifold N and convexity assumptions on the boundary M=∂N, we investigate controlling topology for free boundary minimal surfaces of low index:• We derive bounds on the genus, number of boundary components;• We prove a rigidity result;• We give area estimates in term of the scalar curvature of N.

View record

Master's Student Supervision (2010 - 2018)
The fattening phenomenon for level set solutions of the mean curvature flow (2017)

Level set solutions are an important class of weak solutions to the mean curvature flow which allow the flow to be extended past singularities. Unfortunately, when singularities do develop it is possible for the Hausdorff dimension of the level set solution to increase. This behaviour is referred to as the fattening phenomenon. The purpose of this thesis is to discuss this phenomenon and to provide concrete examples, focusing especially on its relation to the uniqueness of smooth solutions. We first discuss the definition of level set solutions in arbitrary codimension, due to Ambrosio and Soner. We then prove some technical results about distance solutions, a type of set-theoretic subsolution to level set solutions. These include a new method of gluing together distance solutions. Next, we present several known results on the fattening phenomenon in the context of distance solutions. Finally, we provide a new example by proving that fattening occurs when immersed curves in ℝ³ develop self-intersections.

View record

 

Membership Status

Member of G+PS
View explanation of statuses

Program Affiliations

 

If this is your researcher profile you can log in to the Faculty & Staff portal to update your details and provide recruitment preferences.

 
 

Learn about our faculties, research, and more than 300 programs in our 2021 Graduate Viewbook!