Colloquia 2012-2013: Difference between revisions
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== Abstracts == | == Abstracts == | ||
===Oct 14: Alex Kontorovich (Yale)=== | ===Fri, Oct 14: Alex Kontorovich (Yale)=== | ||
''On Zaremba's Conjecture'' | ''On Zaremba's Conjecture'' | ||
It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of "low-lying" divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a "congruence" analog of the renewal method in the thermodynamical formalism. This is joint work with Jean Bourgain. | It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of "low-lying" divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a "congruence" analog of the renewal method in the thermodynamical formalism. This is joint work with Jean Bourgain. | ||
===Wed, Oct 19: Bernd Sturmfels (Berkeley)=== | |||
''Convex Algebraic Geometry'' | |||
Convex algebraic geometry is an emerging field at the interface of convex optimization and algebraic geometry. A primary focus lies on the | |||
mathematical underpinnings of semidefinite programming. This lecture offers a self-contained introduction. Starting with elementary questions about | |||
multifocal ellipses in the plane, we move on to discuss the geometry of spectrahedra, orbitopes, and convex hulls of a real algebraic varieties. | |||
===Thu, Oct 20: Bernd Sturmfels (Berkeley)=== | |||
''Quartic Curves and Their Bitangents'' | |||
A smooth quartic curve in the complex projective plane has 36inequivalent representations as a symmetric determinant of linear forms | |||
and 63 representations as a sum of three squares. We compute these objects from the 28 bitangents. This expresses Vinnikov quartics | |||
as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and the variety of Cayley octads. Interwoven | |||
is a delightful exposition of the 19th century theory of plane quartics. | |||
===Fri, Oct 21: Bernd Sturmfels (Berkeley)=== | |||
''Multiview Geometry'' | |||
The study of two-dimensional images of three-dimensional scenes is a foundational subject for computer vision, known as multiview geometry. | |||
We present recent work with Chris Aholt and Rekha Thomas on the polynomials defining images taken by n cameras. Our varieties are threefolds that vary | |||
in a family of dimension 11n-15 when the cameras are moving. We use toric geometry and multigraded Hilbert schemes to characterize degenerations | |||
of camera positions. | |||
Revision as of 11:37, 13 August 2011
Mathematics Colloquium
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.
Fall 2011
| date | speaker | title | host(s) | |
|---|---|---|---|---|
| Sep 9 | Manfred Einsiedler (ETH-Zurich) | TBA | Fish | |
| Sep 16 | Richard Rimanyi (UNC-Chapel Hill) | TBA | Maxim | |
| Sep 23 | Andrei Caldararu (UW-Madison) | The Hodge theorem as a derived self-intersection | (local) | |
| Oct 7 | Hala Ghousseini (University of Wisconsin-Madison) | TBA | Lempp | |
| Oct 14 | Alex Kontorovich (Yale) | On Zaremba's Conjecture | Shamgar | |
| oct 19, Wed | Bernd Sturmfels (UC Berkeley) | Convex Algebraic Geometry | distinguished lecturer | Shamgar |
| oct 20, Thu | Bernd Sturmfels (UC Berkeley) | Quartic Curves and Their Bitangents | distinguished lecturer | Shamgar |
| oct 21 | Bernd Sturmfels (UC Berkeley) | Multiview Geometry | distinguished lecturer | Shamgar |
| oct 28 | Peter Constantin (University of Chicago) | TBA | distinguished lecturer | |
| oct 31, Mon | Peter Constantin (University of Chicago) | TBA | distinguished lecturer | |
| Nov 4 | Sijue Wu (U Michigan) | TBA | Qin Li | |
| Nov 11 | Henri Berestycki (EHESS and University of Chicago) | TBA | Wasow lecture | |
| Nov 18 | Benjamin Recht (UW-Madison, CS Department) | TBA | Jordan | |
| Dec 2 | Robert Dudley (University of California, Berkeley) | From Gliding Ants to Andean Hummingbirds: The Evolution of Animal Flight Performance | Jean-Luc | |
| dec 9 | Xinwen Zhu (Harvard University) | TBA | Tonghai |
Spring 2012
| date | speaker | title | host(s) |
|---|---|---|---|
| Feb 24 | Malabika Pramanik (University of British Columbia) | TBA | Benguria |
Abstracts
Fri, Oct 14: Alex Kontorovich (Yale)
On Zaremba's Conjecture
It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of "low-lying" divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a "congruence" analog of the renewal method in the thermodynamical formalism. This is joint work with Jean Bourgain.
Wed, Oct 19: Bernd Sturmfels (Berkeley)
Convex Algebraic Geometry
Convex algebraic geometry is an emerging field at the interface of convex optimization and algebraic geometry. A primary focus lies on the mathematical underpinnings of semidefinite programming. This lecture offers a self-contained introduction. Starting with elementary questions about multifocal ellipses in the plane, we move on to discuss the geometry of spectrahedra, orbitopes, and convex hulls of a real algebraic varieties.
Thu, Oct 20: Bernd Sturmfels (Berkeley)
Quartic Curves and Their Bitangents
A smooth quartic curve in the complex projective plane has 36inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. We compute these objects from the 28 bitangents. This expresses Vinnikov quartics as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and the variety of Cayley octads. Interwoven
is a delightful exposition of the 19th century theory of plane quartics.
Fri, Oct 21: Bernd Sturmfels (Berkeley)
Multiview Geometry
The study of two-dimensional images of three-dimensional scenes is a foundational subject for computer vision, known as multiview geometry. We present recent work with Chris Aholt and Rekha Thomas on the polynomials defining images taken by n cameras. Our varieties are threefolds that vary in a family of dimension 11n-15 when the cameras are moving. We use toric geometry and multigraded Hilbert schemes to characterize degenerations of camera positions.