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String theory relates certain seemingly different manifolds through a relationship called mirror symmetry. Discovered about 25 years ago, this story is still very mysterious from a mathematical point of view. Despite the name, mirror symmetry is not entirely symmetric -- several distinct spaces can be mirrors to a given one. When this happens it is expected that certain invariants of these "double mirrors" match up. For a long time the only known examples of double mirrors arose through a simple construction called a flop, which led to the conjecture that this would be a general phenomenon. In joint work with Lev Borisov we constructed the first counterexample to this, which I shall present. Explicitly, I shall construct two Calabi-Yau threefolds which are not related by flops, but are derived equivalent, and therefore are expected to arise through a double mirror construction. The talk will be accessible to a wide audience, in particular to graduate students. There will even be several pictures! | String theory relates certain seemingly different manifolds through a relationship called mirror symmetry. Discovered about 25 years ago, this story is still very mysterious from a mathematical point of view. Despite the name, mirror symmetry is not entirely symmetric -- several distinct spaces can be mirrors to a given one. When this happens it is expected that certain invariants of these "double mirrors" match up. For a long time the only known examples of double mirrors arose through a simple construction called a flop, which led to the conjecture that this would be a general phenomenon. In joint work with Lev Borisov we constructed the first counterexample to this, which I shall present. Explicitly, I shall construct two Calabi-Yau threefolds which are not related by flops, but are derived equivalent, and therefore are expected to arise through a double mirror construction. The talk will be accessible to a wide audience, in particular to graduate students. There will even be several pictures! | ||
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===Fri, Sept 23: Scott Armstrong (UW-Madison)=== | ===Fri, Sept 23: Scott Armstrong (UW-Madison)=== | ||
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and some interesting recent developments. We will even play a | and some interesting recent developments. We will even play a | ||
probabilistic game called "tug-of-war". | probabilistic game called "tug-of-war". | ||
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===Fri, Oct 14: Alex Kontorovich (Yale)=== | ===Fri, Oct 14: Alex Kontorovich (Yale)=== |
Revision as of 13:53, 23 September 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) | Periodic orbits on homogeneous spaces | Fish | |
Sep 16 | Richard Rimanyi (UNC-Chapel Hill) | Global singularity theory | Maxim | |
Sep 23 | Andrei Caldararu (UW-Madison) | The Pfaffian-Grassmannian derived equivalence | (local) | |
Sep 30 | Scott Armstrong (UW-Madison) | Optimal Lipschitz extensions, the infinity Laplacian, and tug-of-war games | (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 | Roman Holowinsky (OSU) | TBA | Street | |
Nov 4 | Sijue Wu (U Michigan) | TBA | Qin Li | |
Nov 7, Mo, 3pm, SMI 133 | Sastry Pantula (NSCU and DMS/NSF) | TBA | Joint Math/Stat Colloquium | |
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) |
---|---|---|---|
Jan 26, Thu | Peter Constantin (University of Chicago) | TBA | distinguished lecturer |
Jan 27 | Peter Constantin (University of Chicago) | TBA | distinguished lecturer |
Feb 24 | Malabika Pramanik (University of British Columbia) | TBA | Benguria |
March 2 | Guang Gong (University of Waterloo) | TBA | Shamgar |
March 23 | Martin Lorenz (Temple University) | TBA | Don Passman |
March 30 | Paolo Aluffi (Florida State University) | TBA | Maxim |
April 6 | Spring recess | ||
April 13 | Ricardo Cortez (Tulane) | TBA | Mitchell |
April 20 | Robert Guralnick (University of South California) | TBA | Shamgar |
April 27 | Tentatively Scheduled | Street | |
May 4 | Mark Andrea de Cataldo (Stony Brook) | TBA | Maxim |
Abstracts
Fri, Sept 9: Manfred Einsiedler (ETH-Zurich)
Periodic orbits on homogeneous spaces
We call an orbit xH of a subgroup H<G on a quotient space Gamma \ G periodic if it has finite H-invariant volume. These orbits have intimate connections to a variety of number theoretic problems, e.g. both integer quadratic forms and number fields give rise periodic orbits and these periodic orbits then relate to local-global problems for the quadratic forms or to special values of L-functions. We will discuss whether a sequence of periodic orbits equidistribute in Gamma \ G assuming the orbits become more complicated (which can be measured by a discriminant). If H is a diagonal subgroup (also called torus or Cartan subgroup), this is not always the case but can be true with a bit more averaging. As a theorem of Mozes and Shah show the case where H is generated by unipotents is well understand and is closely related to the work of M. Ratner. We then ask about the rate of approximation, where the situation is much more complex. The talk is based on several papers which are joint work with E.Lindenstrauss, Ph. Michel, and A. Venkatesh resp. with G. Margulis and A. Venkatesh.
Fri, Sept 16: Richard Rimanyi (UNC)
Global singularity theory
The topology of the spaces A and B may force every map from A to B to have certain singularities. For example, a map from the Klein bottle to 3-space must have double points. A map from the projective plane to the plane must have an odd number of cusp points.
To a singularity one may associate a polynomial (its Thom polynomial) which measures how topology forces this particular singularity. In the lecture we will explore the theory of Thom polynomials and their applications in enumerative geometry. Along the way, we will meet a wide spectrum of mathematical concepts from geometric theorems of the ancient Greeks to the cohomology ring of moduli spaces.
Fri, Sept 23: Andrei Caldararu (UW-Madison)
The Pfaffian-Grassmannian derived equivalence
String theory relates certain seemingly different manifolds through a relationship called mirror symmetry. Discovered about 25 years ago, this story is still very mysterious from a mathematical point of view. Despite the name, mirror symmetry is not entirely symmetric -- several distinct spaces can be mirrors to a given one. When this happens it is expected that certain invariants of these "double mirrors" match up. For a long time the only known examples of double mirrors arose through a simple construction called a flop, which led to the conjecture that this would be a general phenomenon. In joint work with Lev Borisov we constructed the first counterexample to this, which I shall present. Explicitly, I shall construct two Calabi-Yau threefolds which are not related by flops, but are derived equivalent, and therefore are expected to arise through a double mirror construction. The talk will be accessible to a wide audience, in particular to graduate students. There will even be several pictures!
Fri, Sept 23: Scott Armstrong (UW-Madison)
Optimal Lipschitz extensions, the infinity Laplacian, and tug-of-war games
Given a nice bounded domain, and a Lipschitz function defined on its boundary, consider the problem of finding an extension of this function to the closure of the domain which has minimal Lipschitz constant. This is the archetypal problem of the calculus of variations "in the sup-norm". There can be many such minimal Lipschitz extensions, but there is there is a unique minimizer once we properly "localize" this Lipschitz minimizing property. This minimizer is characterized by the infinity Laplace equation: the Euler-Lagrange equation for our optimization problem. This PDE is a very highly degenerate nonlinear elliptic equation which does possess smooth solutions in general. In this talk I will discuss what we know about the infinity Laplace equation, what the important open questions are, and some interesting recent developments. We will even play a probabilistic game called "tug-of-war".
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
This lecture concerns convex bodies with an interesting algebraic structure. A primary focus lies on the geometry of semidefinite optimization. Starting with elementary questions about ellipses in the plane, we move on to discuss the geometry of spectrahedra, orbitopes, and convex hulls of real varieties.
Thu, Oct 20: Bernd Sturmfels (Berkeley)
Quartic Curves and Their Bitangents
We present a computational study of plane curves of degree four, with primary focus on writing their defining polynomials as sums of squares and as symmetric determinants. Number theorists will enjoy the appearance of the Weyl group [math]E_7[/math] as the Galois group of the 28 bitangents. Based on joint work with Daniel Plaumann and Cynthia Vinzant, this lecture spans a bridge from 19th century algebra to 21st century optimization.
Fri, Oct 21: Bernd Sturmfels (Berkeley)
Multiview Geometry
The study of two-dimensional images of three-dimensional scenes is foundational for computer vision. We present work with Chris Aholt and Rekha Thomas on the polynomials characterizing images taken by [math]n[/math] cameras. Our varieties are threefolds that vary in a family of dimension [math]11n-15[/math] when the cameras are moving. We use toric geometry and Hilbert schemes to characterize degenerations of camera positions.