The seminar takes place every Wednesday from 13:30 to 14:30 at AA-4336.
Organizers:
Changjie Chen, email: changjie.chen@umontreal.ca
Jérémy Perazzelli, email: jeremy.perazzelli@umontreal.ca
September 18: Denis Vinokurov
Title: Harmonic maps from S^2 to S^n, Part I
Abstract: Harmonic maps appear in Differential Geometry as a natural generalization of harmonic functions. During the talk, we will see that harmonic maps into sheres have a strong connection with the Laplace eigenvalue optimization problem. In the case of harmonic mas from S^2, we will be able to establish an almost bijective correspondence of such maps with certain holomorphic curves. This will give us a useful description of their moduli space and the opportunity to study it by the methods of Complex Geometry.
September 25: Denis Vinokurov
Title: Harmonic maps from S^2 to S^n, Part II
Abstract: During the first talk we discussed the importance of harmonic maps into spheres for Laplace-Beltrami eigenvalue optimization problem. In the forthcoming talk, we are going to analyze the structure of harmonic maps from S^2 and will be able to relate them with certain holomorphic curves in projective spaces.
References:
Calabi, E. (1967). Minimal immersions of surfaces in Euclidean spheres. Journal of Differential Geometry, 1(1-2), 111-125.
Barbosa, J. L. M. (1975). On minimal immersions of S2 into S2m. Trans. Am.
October 2: Alain Didier Noutchegueme
Title:
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October 9: Alain Didier Noutchegueme
Title:
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October 16: Irene Silvestre Rosello
Title: Courant's nodal theorem and false generalizations
Abstract: Let Ω be a domain in R^d and let f be a continuous function on Ω. If Z(f) denotes the zero set of f, we will call a nodal domain each of the connected components of Ω\Z(f).
When studying the eigenfunctions of the Laplace operator, we can ask ourselves if the number of nodal domains is related somehow with the associated eigenvalue. This is partially answered by Courant's theorem, which tells us that and eigenfunction associated to the n-th eigenvalue cannot have more than n nodal domains.
In the first talk we will review the proof of Courant's theorem and an example showing that it cannot be generalized to linear combinations of eigenfunctions.
In the second talk we will see a way to generalize Courant's theorem via Topological Persistence.
References:
- Buhovsky, L. ; Payette, J. ; Polterovich, I. ; Polterovich, L. ; Shelukhin, E. et Stojisavljević, V. (2022).Coarse nodal count and topological persistence. Journal of the European Mathematical Society, https://doi.org/10.4171/jems/1521.
- Levitin, M., Mangoubi, D., et Polterovich, I. (2023). Topics in Spectral Geometry. Graduate Studies in Mathematics.
- Buhovsky, Lev & Logunov, Alexander & Sodin, Mikhail. (2019). Eigenfunctions with infinitely many isolated critical points.International Mathematics Research Notices. doi:10.1093/imrn/rnz181.
October 23: Irene Silvestre Rosello
Title: Courant's nodal theorem and false generalizations
Abstract: Let Ω be a domain in R^d and let f be a continuous function on Ω. If Z(f) denotes the zero set of f, we will call a nodal domain each of the connected components of Ω\Z(f).
When studying the eigenfunctions of the Laplace operator, we can ask ourselves if the number of nodal domains is related somehow with the associated eigenvalue. This is partially answered by Courant's theorem, which tells us that and eigenfunction associated to the n-th eigenvalue cannot have more than n nodal domains.
In the first talk we will review the proof of Courant's theorem and an example showing that it cannot be generalized to linear combinations of eigenfunctions.
In the second talk we will see a way to generalize Courant's theorem via Topological Persistence.
References:
- Buhovsky, L. ; Payette, J. ; Polterovich, I. ; Polterovich, L. ; Shelukhin, E. et Stojisavljević, V. (2022).Coarse nodal count and topological persistence. Journal of the European Mathematical Society, https://doi.org/10.4171/jems/1521.
- Levitin, M., Mangoubi, D., et Polterovich, I. (2023). Topics in Spectral Geometry. Graduate Studies in Mathematics.
- Buhovsky, Lev & Logunov, Alexander & Sodin, Mikhail. (2019). Eigenfunctions with infinitely many isolated critical points.International Mathematics Research Notices. doi:10.1093/imrn/rnz181.
October 30: Catherine Pfaff
Title: Pants & coordinates on Teichmuller space & projective measured foliations
Abstract: We describe coordinates new & old on the Teichmuller space (space of hyperbolic metrics) & space of projective measured foliations on a fixed surface. The new coordinates will realize the Thurston compactification as the radial compactification of Teichmuller space. New results presented are joint work in progress with Daryl Cooper.
November 6: Jérémy Perazzelli
Title: Mostow's rigidity theorem
Abstract: While a closed hyperbolic surface of genus g has a deformation space of dimension 6g-6, Mostow's rigidity theorem asserts that closed hyperbolic manifolds of dimension n>2 cannot be deformed. In other words, a closed manifold of dimension n>2 admits at most one hyperbolic metric. In these talks, we will discuss Gromov's proof of Mostow's rigidity as well as some corollaries and generalizations.
Reference: 1- Benedetti, R.; Petronio, C., Lectures on Hyperbolic Manifolds
2- Thurston, W., The Geometry and Topology of Three-Manifolds
November 13: Jérémy Perazzelli
Title: Mostow's rigidity theorem
Abstract: While a closed hyperbolic surface of genus g has a deformation space of dimension 6g-6, Mostow's rigidity theorem asserts that closed hyperbolic manifolds of dimension n>2 cannot be deformed. In other words, a closed manifold of dimension n>2 admits at most one hyperbolic metric. In these talks, we will discuss Gromov's proof of Mostow's rigidity as well as some corollaries and generalizations.
Reference: 1- Benedetti, R.; Petronio, C., Lectures on Hyperbolic Manifolds
2- Thurston, W., The Geometry and Topology of Three-Manifolds
November 20: Maxime Fortier Bourque
Title: Hyperbolic trigonometry
Abstract: I will describe the hyperboloid model of the hyperbolic plane and then use it to prove trigonometric formulas for hyperbolic triangles and right-angled hexagons. In particular, we will see why the lengths of three non-consecutive sides of a right-angled hexagon can be prescribed arbitrarily.
References: 1. John H. Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics, Volume 1
2. Peter Buser, Geometry and spectra of compact Riemann surfaces
November 27: Maxime Fortier Bourque
Title: Hyperbolic trigonometry
Abstract: I will describe the hyperboloid model of the hyperbolic plane and then use it to prove trigonometric formulas for hyperbolic triangles and right-angled hexagons. In particular, we will see why the lengths of three non-consecutive sides of a right-angled hexagon can be prescribed arbitrarily.
References: 1. John H. Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics, Volume 1
2. Peter Buser, Geometry and spectra of compact Riemann surfaces
December 4: Changjie Chen
Title: From building hyperbolic surfaces to the Teichmüller space
Abstract: Following Maxime’s talks on hyperbolic trigonometry, I will talk about how to use hyperbolic right-angled hexagons to construct hyperbolic surfaces. On an observation of the change of topology, this can bring in the idea of the Teichmüller space, for which the construction provides a coordinate system, namely the Fenchel-Nielson coordinates.