Robert Haslhofer Department of Mathematics University of Toronto 40 St George St Toronto, Ontario M5S 2E4 Canada offices: BA6208 (downtown campus), IA4132 (Scarborough campus) email: roberth(at)math(dot)toronto(dot)edu

I am a professor at the University of Toronto. Click here for a recent CV.

Recent awards: Frontiers of Science Award, Coxeter-James Prize, Aisenstadt Prize, Sloan Fellowship

I acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). / Je remercie le Conseil de recherches en sciences naturelles et en genie du Canada (CRSNG) de son soutien.

2024 Summer School at CRM Montreal

• Lecture Notes
• slides1, slides2, slides3, slides4, slides5

Research Interests

• Geometric Analysis, Differential Geometry, Partial Differential Equations, Calculus of Variations, Stochastic Analysis, General Relativity

Teaching

• current: MAT1300 Differential Topology (Fall 23), MATC46 Differential Equations II (Winter 24), MATC63 Differential Geometry (Winter 24)
• some prior courses: MAT1061 Partial Differential Equations II (Winter 16, 21), MAT1062 Mean Curvature Flow (Fall 17), MAT1502 Brownian Motion on Manifolds (Fall 18)

Students and Postdocs

• current: Krzysztof Ciosmak (postdoc), Akashdeep Dey (postdoc), Lorenzo Sarnataro (postdoc), Renato Velozo Ruiz (postdoc), Yueheng Bao (PhD student)
  
• past: Mathias Braun (postdoc; now Bernoulli Instructor at EPFL), Artur Kravtsov (MSc student; now PhD student at UofT), Wenkui Du (PhD student; now Moore Instructor at MIT), Christopher Kennedy (PhD student; now postdoc at Queen's University), Jiayin Pan (postdoc; now tenure-track professor at UC Santa Cruz), Gong Chen (postdoc; now tenure-track professor at Georgia Tech), Xinze Li (BSc student; now PhD student at UofT), Zhichao Wang (postdoc; next postdoc at UBC, now tenure-track professor at Fudan University), Beomjun Choi (postdoc; now tenure-track professor at POSTECH), Christian Ketterer (postdoc; then postdoc at KIT, now substitute professor at Uni Freiburg), Mohammad Ivaki (postdoc; now tenure-track professor at TU Vienna), Hussain Kadhem (BSc student; now PhD student at UC Berkeley), Daniel Stern (postdoc; next NSF postdoc at UChicago, now tenure-track professor at Cornell), Xincheng Zhang (BSc student; now PhD student at UofT), Salim Deaibes (BSc student; now personal trainer), Cameron Martin (BSc student; now PhD student at UofT), Panagiotis Gianniotis (postdoc; now tenure-track professor at Uni Athens), Jianhui Gao (BSc student; now PhD student in statistics at UofT), Anmol Bhullar (BSc student; now Operations coordinator), Eva Kopfer (visiting Phd student; now postdoc at Uni Bonn), Ilyas Khan (visiting PhD student; now postdoc at Oxford, now postdoc at Duke), Keegan Dasilva Barbosa (MSc student; next PhD student at UofT, now Simons postdoc at Fields), Mona Sayehban (MSc student; now data analyst), Nicolau Sarquis Aiex (postdoc; next postdocs at UBC, Aukland, and NTNU), Daniel Sigal (MSc student; next medical school Western University, now residency at UofT), Or Hershkovits (Phd student; next postdoc at Stanford, now professor at Hebrew University)
  

Seminars and Events

• current: Fields GA colloquium, UofT seminars (geometry, PDE, etc)
• past: Fields thematic program on Geometric Analysis, McGill workshop, Banff workshop, Fields workshop

Articles and preprints

• Free boundary minimal disks in convex balls (joint with Dan Ketover), preprint [arxiv]
  
• Free boundary flow with surgery, preprint [arxiv]
  
• Flows with surgery revisited, preprint [arxiv]
  
• The blowdown of ancient noncollapsed mean curvature flows (joint with Wenkui Du), preprint [arxiv]
  
• Ricci limit flows and weak solutions (joint with Beomjun Choi), J. Eur. Math. Soc. (to appear) [arxiv]
  
• Classification of bubble-sheet ovals in R^4 (joint with Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du and Natasa Sesum), Geom. Topol. (to appear) [arxiv]
  
• On uniqueness and nonuniqueness of ancient ovals (joint with Wenkui Du), Amer. J. Math. (to appear) [arxiv]
  
• Hearing the shape of ancient noncollapsed flows in R^4 (joint with Wenkui Du), Comm. Pure Appl. Math. 77(1):543--582, 2024 [journal | arxiv]
  
• A nonexistence result for wing-like mean curvature flows in R^4 (joint with Kyeongsu Choi and Or Hershkovits), Geom. Topol. 28(7):3095--3134, 2024 [journal | arxiv]
  
• On kappa-solutions and canonical neighborhoods in 4d Ricci flow, J. Reine Angew. Math. 811:257--265, 2024 [journal | arxiv]
  
• Enhanced profile estimates for ovals and translators (joint with Kyeongsu Choi and Or Hershkovits), Adv. Math. 453#109853:1--22, 2024 [journal | arxiv]
  
• Classification of noncollapsed translators in R^4 (joint with Kyeongsu Choi and Or Hershkovits), Camb. J. Math. 11(3):563--698, 2023 [journal | arxiv]
  
• A nonexistence result for rotating mean curvature flows in R^4 (joint with Wenkui Du), J. Reine Angew. Math. 802:275--285, 2023 [journal | arxiv]
  
• Moving plane method for varifolds and applications (joint with Or Hershkovits and Brian White), Amer. J. Math. 145(4):1051--1076, 2023 [journal | arxiv]
  
• Ancient asymptotically cylindrical flows and applications (joint with Kyeongsu Choi, Or Hershkovits and Brian White), Invent. Math. 229:139--241, 2022 [journal | arxiv]
  
• Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness (joint with Kyeongsu Choi and Or Hershkovits), Acta Math. 228(2):217--301, 2022 [journal | arxiv]
  
• Mean convex mean curvature flow with free boundary (joint with Nick Edelen, Mohammad Ivaki and Jonathan Zhu), Comm. Pure Appl. Math. 75(4):767--817, 2022 [journal | arxiv]
  
• Differential Harnack inequalities on path space (joint with Eva Kopfer and Aaron Naber), Adv. Math. 410#108714:1--47, 2022 [journal | arxiv]
  
• Hitting estimates on Einstein manifolds and applications (joint with Beomjun Choi), J. Reine Angew. Math. 793:261--280, 2022 [journal | arxiv ]
  
• Uniqueness and stability of singular Ricci flows in higher dimensions, Proc. Amer. Math. Soc. 150(12):5433--5437, 2022 [journal | arxiv]
  
• A note on blowup limits in 3d Ricci flow (joint with Beomjun Choi), Math. Res. Lett. 29(5):1377--1386, 2022 [journal | arxiv]
  
• Heat flow on time-dependent manifolds (joint with Beomjun Choi, Jianhui Gao and Daniel Sigal), J. Geom. Anal. 32(1)#11:1--16, 2022 [journal | pdf]
  
• The moduli space of two-convex embedded spheres (joint with Reto Buzano and Or Hershkovits), J. Differential Geom. 118(2):189--221, 2021 [journal | pdf]
  
• A note on the selfsimilarity of limit flows (joint with Beomjun Choi and Or Hershkovits), Proc. Amer. Math. Soc. 149(3):1239--1245, 2021 [journal | pdf]
  
• Diameter and curvature control under mean curvature flow (joint with Panagiotis Gianniotis), Amer. J. Math. 142(6):1877--1896, 2020 [journal | pdf]
  
• Brownian motion on Perelman's almost Ricci-flat manifold (joint with Esther Cabezas-Rivas), J. Reine Angew. Math. 764:217--239, 2020 [journal | pdf]
  
• Minimal 2-spheres in 3-spheres (joint with Dan Ketover), Duke Math. J. 168(10):1929--1975, 2019 [journal | pdf]
  
• The moduli space of two-convex embedded tori (joint with Reto Buzano and Or Hershkovits), Int. Math. Res. Not. 2019(2):392--406, 2019 [journal | pdf]
  
• Low complexity solutions of the Allen-Cahn equation on three-spheres (joint with Mohammad Ivaki), Canad. Math. Bull. 62(2):287--291, 2019 [journal | pdf]
  
• Ricci curvature and Bochner formulas for martingales (joint with Aaron Naber), Comm. Pure Appl. Math. 71(6):1074--1108, 2018 [journal | pdf]
  
• Characterizations of the Ricci flow (joint with Aaron Naber), J. Eur. Math. Soc. 20(5):1269--1302, 2018 [journal | pdf]
  
• Singularities of mean convex level set flow in general ambient manifolds (joint with Or Hershkovits), Adv. Math. 329:1137--1155, 2018 [journal | pdf]
  
• Mean curvature flow with surgery (joint with Bruce Kleiner), Duke Math. J. 166(9):1591--1626, 2017 [journal | pdf]
  
• Mean curvature flow of mean convex hypersurfaces (joint with Bruce Kleiner), Comm. Pure Appl. Math. 70(3):511--546, 2017 [ journal | pdf]
  
• Ancient solutions of the mean curvature flow (joint with Or Hershkovits), Comm. Anal. Geom. 24(3):593--604, 2016 [journal | pdf]
  
• Uniqueness of the bowl soliton, Geom. Topol. 19(4):2393--2406, 2015 [journal | pdf]
  
• A note on the compactness theorem for 4d Ricci shrinkers (joint with Reto Müller), Proc. Amer. Math. Soc. 143(10):4433--4437, 2015 [journal | pdf]
  
• On Brendle's estimate for the inscribed radius under mean curvature flow (joint with Bruce Kleiner), Int. Math. Res. Not. 2015(15):6558--6561, 2015 [journal | pdf]
  
• Quantitative stratification and the regularity of harmonic map flow (joint with Jeff Cheeger and Aaron Naber), Calc. Var. PDE 53(1-2):365--381, 2015 [journal | pdf]
  
• The stability inequality for Ricci-flat cones (joint with Stuart Hall and Michael Siepmann), J. Geom. Anal. 24(1):472--494, 2014 [journal | pdf]
  
• Dynamical stability and instability of Ricci-flat metrics (joint with Reto Müller), Math. Ann. 360(1-2):547--553, 2014 [journal | pdf]
  
• Quantitative stratification and the regularity of mean curvature flow (joint with Jeff Cheeger and Aaron Naber), Geom. Funct. Anal. 23(3):828--847, 2013 [journal | pdf]
  
• Perelman's lambda-functional and the stability of Ricci-flat metrics, Calc. Var. PDE 45(3-4):481--504, 2012 [journal | pdf]
• A mass-decreasing flow in dimension three, Math. Res. Lett. 19(4):927--938, 2012 [journal | pdf]
  
• A renormalized Perelman-functional and a lower bound for the ADM-mass, J. Geom. Phys. 61(11):2162--2167, 2011 [journal | pdf]
  
• A compactness theorem for complete Ricci shrinkers (joint with Reto Müller), Geom. Funct. Anal. 21(5):1091--1116, 2011 [journal | pdf]
  

Conference reports, lecture notes, etc

• Lectures on mean curvature flow of surfaces (from summer schools at UT Austin and CRM Montreal), 2024 [arxiv | pdf]
  
• Free boundary flow with surgery and applications (extended abstract), Oberwolfach Reports, 2023 [mfo | pdf]
  
• Classification of noncollapsed translators in R^4 (extended abstract), Oberwolfach Reports, 2022 [mfo | pdf]
  
• Mean curvature flow through neck-singularities (for the Andre Aisenstadt prize lecture), 2021 [crm | pdf]
  
• Some recent applications of mean curvature flow with surgery (book chapter), 2020 [book | pdf]
  
• Differential Harnack inequalities on path space (extended abstract), Oberwolfach Reports, 2019 [mfo | pdf]
  
• The bounded diameter conjecture for two-convex mean curvature flow (extended abstract), Oberwolfach Reports, 2018 [mfo | pdf]
  
• Applications of mean curvature flow (extended abstract), Oberwolfach Reports, 2017 [mfo | pdf]
  
• Minimal two-spheres in three-spheres (extended abstract), Oberwolfach Reports, 2017 [mfo | pdf]
  
• Ricci curvature and martingales (extended abstract), Oberwolfach Reports, 2016 [mfo | pdf]
  
• The moduli space of 2-convex embedded spheres (extended abstract), Oberwolfach Reports, 2016 [mfo | pdf]
  
• Lectures on curve shortening flow (notes from the last three weeks of my PDE II course), 2016 [pdf]
  
• Weak solutions for the Ricci flow (extended abstract), Oberwolfach Reports, 2015 [mfo | pdf]
  
• Lectures on mean curvature flow (notes from summer schools at KIAS Seoul and SNS Pisa), 2014 [arxiv | pdf]
  
• Mean curvature flow of mean convex hypersurfaces (extended abstract), Oberwolfach Reports, 2013 [mfo | pdf]
  
• Quantitative stratification and the regularity of mean curvature flow (extended abstract), Oberwolfach Reports, 2012 [mfo | pdf]
  
• Singularities in 4d Ricci flow (extended abstract), Oberwolfach Reports, 2012 [mfo | pdf]
  
• A mass-decreasing flow in dimension three (extended abstract), Oberwolfach Reports, 2012 [mfo | pdf]
  
• Stability of Ricci-flat spaces and Singularities in 4d Ricci flow (PhD thesis), ETH Zürich, 2012 [e-collection | pdf]