Contact information

Office at St George: BA6189

Office at UTM: DH3052

Email: ylio [at] math.toronto.edu

Teaching

Fall of 2021: MAT1300HF Differential Topology.

Course webpage: Quercus.

If you want to audit the course, please, email me so I can add you to the Quercus course page.

Fall of 2020: MAT337H1F Introduction to Real analysis.

Spring of 2020: MAT1309HS Geometric Inequalities.

Fall of 2019: MAT337H1F Introduction to Real analysis.

Upcoming talks

1. Rice, Geometry-Analysis Seminar, in Zoom, 5pm EST, February 2, 2022.

2. Rice, Colloquium, in Zoom, 5pm EST, February 3, 2022.

3. Cornell, Olivier club in Zoom, 4pm EST March 17, 2022.

Seminars

I am a co-organizer of the Geometry and Topology Seminar and Fields Geometric Analysis Colloquium.

Students and postdocs

Current: Bruno Staffa (PhD student), Boris Lishak (postdoc), Xinze Li (undergraduate), Xiao Yu (undergraduate), Luis Carlos Soldevilla Estrada (undergraduate), Jialin Xin (visiting PhD student).

Past: Daniel Stern (postdoc), Zhichao Wang (postdoc).

Research

My interests lie at the intersection of Geometric Analysis, Quantitative Topology and Geometric Measure Theory. I'm also interested in problems in 3-manifold topology. My research is supported by NSERC Discovery Grant, NSERC Accelerator Award and Sloan Fellowship.

I'm new to mathematics. Can you explain your work to me?

Many of my results are applications of min-max methods to problems in geometry and topology.

Finding min-max is an optimization problem. The simplest optimization problems that we learn from high school are finding minima and maxima. It’s clear why these are important: a manager may want to maximize profit and an engineer may want to minimize stress concentration in a structure. Finding min-max is a somewhat more advanced optimization problem. It is also sometimes called a "mountain pass problem" for the following reason. Say, a group of tourists wants to get to the other side of a mountain range in the easiest possible way. They are not interested in climbing the peak of a mountain in the range, rather they want to find a mountain pass. The problem of finding a mountain pass can also be described as finding a path with the smallest possible maximum elevation among all paths connecting two points on the two sides of the range. In other words, we are trying to minimize a maximum, hence the name “min-max”.

Min-max problems arise in many areas of science, such as machine learning in Computer Science, differential equations, geometry, and topology in Mathematics. An important problem of such kind in differential geometry is the construction of minimal surfaces. Minimal surfaces are generalizations of soap films or soap bubbles; using min-max techniques one can show that they exist not only in our usual space, but also in a very large class of spaces that can be defined mathematically. Describing minimal surfaces like that makes them sound exotic. Why would anyone study these objects? It turns out that they can be used to prove many important results in mathematics and General Relativity. The reason is because these “soap bubbles” can be used to cut a complicated space into simpler pieces and also because they capture important information about the space they lie in.

Papers

Most of my papers can be found on the arxiv.

“Parametric inequalities and Weyl law for the volume spectrum” (with L. Guth), math arXiv:2202.11805

“Generic density of geodesic nets” (with B. Staffa), math arXiv:2107.12340

“Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions” (with O. Chodosh and C. Li), to appear in Geom. Topol., math arXiv:2105.07306

“Waist inequality for 3-manifolds with positive scalar curvature” (with D. Maximo), to appear in Perspectives in Scalar Curvature (Vol. 2), math arXiv:2012.12478

“Singular behavior and generic regularity of min-max minimal hypersurfaces” (with O. Chodosh and L. Spolaor), math arXiv:2007.11560

“On the existence of minimal Heegaard surfaces” (with D. Ketover and A. Song), math arXiv:1911.07161.

“Filling metric spaces” (with B. Lishak, A. Nabutovsky and R. Rotman), to appear in Duke Math. J., math arXiv:1905.06522

“On the existence of closed C^{1,1} curves of constant curvature” (with D. Ketover), math arXiv:1810.09308

“Area of convex disks” (with G.R. Chambers, C. Croke and H. Wen), Proc. Amer. Math. Soc., math arXiv:1701.06594

“Existence of minimal hypersurfaces in complete manifolds of finite volume” (with G.R. Chambers), Invent. math. (2019), math arXiv:1609.04058

“Weyl law for the volume spectrum” (with F. C. Marques and A. Neves), Annals of Mathematics, Vol. 187 (2018), 933-961, math arXiv:1607.08721

“Determinantal variety and bilipschitz equivalence” (with K.U. Katz, M.G. Katz and D. Kerner), J. Topol. Anal. Vol. 10, No. 01, pp. 27-34 (2018), math arXiv:1602.01227

“Optimal sweepouts of a Riemannian 2-sphere” (with G. R. Chambers), J. Eur. Math. Soc. 21 (2019), 1361-1377.

“Splitting a contraction of a simple curve traversed m times” (with G. R. Chambers), J. Topol. Anal. Vol. 9, No. 3 (2017) 409–418.

“Lengths of three simple periodic geodesics on a Riemannian 2-sphere” (with A. Nabutovsky and R. Rotman), Math. Ann. (2017) 367:831–855.

“Width, Ricci curvature and minimal hypersurfaces” (with P. Glynn-Adey), J. Diff. Geom. Vol. 105 (2017), 33-54.

“Sweeping out 3-manifold of positive Ricci curvature by short 1-cycles via estimates of min-max surfaces” (with X. Zhou), Int. Math. Res. Not. (IMRN) (2016).

“Families of short cycles on Riemannian surfaces”, Duke Math. J. 165 (2016), no. 7, 1363-1379.

“Contracting the boundary of a Riemannian 2-disc” (with A. Nabutovsky and R. Rotman), Geom. Funct. Anal. (GAFA), Vol. 25 (2015), 1543-1574.

“Slicing a 2-sphere”, J. Topol. Anal. Vol. 06 (2014), No. 04, 573-590.

“Converting homotopies to isotopies and dividing homotopies in half in an effective way” (with G. R. Chambers), Geom. Funct. Anal. (GAFA) Vol. 24 (2014), 1080-1100.

“Surfaces of small diameter with large width”, J. Topol. Anal. Vol. 06 (2014), No. 03, 383-396, 2013.

“Spheres of small diameter with long sweep-outs”, Proc. Amer. Math. Soc. 141 (2013), 309-312.