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DESCRIPTION:Yevgeny Liokumovich (Toronto) September 18 and 21\, 2020\n\nAnt
 oine Song (UC Berkeley) September 23 and 25\, 2020\n\n \n\nPlease note tha
 t the CRM requires registration in order to obtain a Zoom link\, however t
 he form is simple and it will not take more than a minute:\n\nhttps://docs
 .google.com/forms/d/e/1FAIpQLSeItVLz6SbuDsCQ0MAzjNUrGSWVd3H1NC-PUwiU8HW6pV
 44Bg/viewform\n\n \n\nYevgeny Liokumovich (Toronto) September 18 and 21\, 
 2020\n\nFirst lecture - September 18\, 2020\, 3 p.m.\n\nThis lecture is ai
 med at a general mathematical audience.\n\nMeasuring size and complexity o
 f Riemannian manifolds\n\nGiven a Riemannian manifold how hard is it to sl
 ice it into pieces of smaller dimension? More precisely\, we can ask how l
 arge in terms of their volume\, diameter and topological complexity the sl
 ices have to be? These questions give rise to various notions of width of 
 a manifold\, which turn out to be closely related to questions about minim
 al surfaces. I will describe some recent results relating widths to other 
 notions of size of a Riemannian manifold. The talk will be partly based on
  separate joint works with A. Nabutovsky\, R. Rotman and B. Lishak\, with 
 G.R. Chambers\, and with D. Ketover and A. Song.\n\nSecond lecture - Septe
 mber 21 2020\, 3 p.m.\n\nMinimal surfaces and quantitative topology\n\nTo 
 each cohomology class of the space of codimension 1 cycles\, Min-Max theor
 y associates a minimal hypersurface with some integer multiplicity. Volume
 s of these minimal hypersurfaces are called 'widths' or 'volume spectrum' 
 of the manifold. Gromov conjectured that like eigenvalues of the Laplacian
 \, the volume spectrum has asymptotic behaviour described by a Weyl law. I
  will discuss the proof of this conjecture for hypersurfaces and some rece
 nt progress for the Weyl law in higher codimension. The talk will be based
  on separate joint works with F.C. Marques and A. Neves and with L. Guth.
 \n\n \n\nAntoine Song\, University of California\, Berkeley\n\nFirst lectu
 re - September 23\, 2020\, 3 p.m.\n\nComplexities of minimal hypersurfaces
 \n\nLet M be a closed Riemannian 3-manifold. By min-max theory\, one can c
 onstruct a sequence of minimal surfaces embedded in M which are the geomet
 ric analogue of eigenfunctions of the Laplacian. Motivated by well known q
 uestions about eigenfunctions\, one can ask about the complexity of the mi
 nimal surfaces in that sequence. There are several natural measures of com
 plexity: genus\, area\, or Morse index. I will talk about the interaction 
 between these quantities\, which has been thoroughly studied since the cla
 ssical work of R. Schoen and S.-T. Yau\, and I will present new quantitati
 ve estimates that are relevant for the question previously mentioned.\n\nS
 econd lecture - September 25\, 2020\, 3 p.m.\n\nThis lecture is aimed at a
  general mathematical audience.\n\nAbundance of minimal hypersurfaces\n\nM
 inimal hypersurfaces are higher dimensional analogues of geodesics. In the
  early 80's\, S.-T. Yau conjectured that in any closed Riemannian 3-manifo
 ld\, there is infinitely many minimal surfaces. I will introduce the probl
 em and give an account of the recent series of work by many people\, which
  led to the understanding that minimal hypersurfaces abound in closed Riem
 annian manifolds. In particular\, Yau's conjecture is true and for generic
  metrics\, much stronger properties\, like equidistribution\, hold. I will
  give some ideas from the proofs\, which borrow tools from analysis\, geom
 etric measure theory and topology. This talk is partially based on joint w
 ork with F.C. Marques and A. Neves.\n\nBiographical note: Antoine Song rec
 eived his Ph.D. in 2019 from Princeton University under the supervision of
  F.C. Marques. He has made several spectacular advances in the theory of m
 inimal surfaces. In particular\, in his Ph.D. thesis\, he presented a comp
 lete solution of Yau’s conjecture on the existence of infinitely many mini
 mal hypersurfaces in closed manifolds. Currently\, Antoine Song is a Clay 
 Research Fellow working at the University of California\, Berkeley.\n\n \n
 \n \n\n \n\n \n\nBiographical note: Yevgeny Liokumovich received his Ph.D.
  in 2015 at the University of Toronto under the supervision of A. Nabutovs
 ky and R. Rotman. After a postdoc at MIT and the Institute for Advanced St
 udy\, he returned to Toronto in 2019 as an Assistant Professor. Yevgeny Li
 okumovich has obtained several major results in geometric analysis\, inclu
 ding a solution of Gromov’s conjecture on the Weyl law for the volume spec
 trum in a recent joint work with F.C. Marques and A. Neves.\n
DTSTART:20200917T134500Z
DTEND:20200917T134500Z
SUMMARY:CRM Nirenberg Lectures in Geometric Analysis
URL:/mathstat/channels/event/crm-nirenberg-lectures-ge
 ometric-analysis-324642
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