NTS Fall 2013/Abstracts: Difference between revisions
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Abstract: Del Pezzo surfaces X of degree 4 are smooth (complete) intersections of two quadrics in four-dimensional projective space. They are some of the simplest surfaces for which there can be cohomological obstructions to the existence of rational points, mediated by the Brauer group Br X of the surface. I will explain how to construct, for every non-trivial, non-constant element A of Br X, a rational genus-one fibration X --> P^1 such that A is | Abstract: Del Pezzo surfaces X of degree 4 are smooth (complete) intersections of two quadrics in four-dimensional projective space. They are some of the simplest surfaces for which there can be cohomological obstructions to the existence of rational points, mediated by the Brauer group Br X of the surface. I will explain how to construct, for every non-trivial, non-constant element A of Br X, a rational genus-one fibration X --> P^1 such that A is "vertical" for this map. This implies, for example, that if there is a cohomological obstruction to the existence of a point on X, then there is a genus-one fibration X --> P^1 where none of the fibers are locally soluble, giving a concrete, geometric way of "seeing" a Brauer-Manin obstruction. The construction also gives a fast, practical algorithm for computing the Brauer group of X. Conjecturally, this gives a mechanical way of testing for the existence of rational points on these surfaces. This is joint work with Bianca Viray. | ||
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Revision as of 14:24, 4 October 2013
September 5
| Guillermo Mantilla-Soler (EPFL) |
| Title: The spinor genus of the integral trace and local arithmetic equivalence |
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Abstract: In this talk I'll explain my recent results on the spinor genus of the integral trace form of a number field. I'll show how from them one can decide in terms of finitely many ramification invariants, and under some restrictions, whether or not a pair of number fields have isometric integral trace forms. Inspired by the work of R. Perlis on number fields with the same zeta function I'll define the notion of local arithmetic equivalence, and I'll show that under certain hypothesis this equivalence determines the local root numbers of the number field, and the isometry class of integral trace form. |
September 12
| Simon Marshall (Northwestern) |
| Title: Endoscopy and cohomology growth on U(3) |
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Abstract: I will use the endoscopic classification of automorphic forms on U(3) to determine the asymptotic cohomology growth of families of complex-hyperbolic 2-manifolds. |
September 19
| Valerio Toledano Laredo (Northeastern) |
| Title: From Yangians to quantum loop algebras via abelian difference equations |
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Abstract: For a semisimple Lie algebra g, the quantum loop algebra and the Yangian are deformations of the loop algebra g[z, z − 1] and the current algebra g[u], respectively. These infinite-dimensional quantum groups share many common features, though a precise explanation of these similarities has been missing so far. In this talk, I will explain how to construct a functor between the finite-dimensional representation categories of these two Hopf algebras which accounts for all known similarities between them. The functor is transcendental in nature, and is obtained from the discrete monodromy of an abelian difference equation canonically associated to the Yangian. This talk is based on a joint work with Sachin Gautam. |
September 26
| Haluk Şengün (Warwick/ICERM) |
| Title: Torsion homology of Bianchi groups and arithmetic |
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Abstract: Bianchi groups are groups of the form SL(2, R) where R is the ring of integers of an imaginary quadratic field. They form an important class of arithmetic Kleinian groups and moreover they hold a key role for the development of the Langlands program for GL(2) beyond totally real fields. In this talk, I will discuss several interesting questions related to the torsion in the homology of Bianchi groups. I will especially focus on the recent results on the asymptotic behavior of the size of torsion, and the reciprocity and functoriality (in the sense of the Langlands program) aspects of the torsion. Joint work with N. Bergeron and A. Venkatesh on the cycle complexity of arithmetic manifolds will be discussed at the end. The discussion will be illustrated with many numerical examples. |
October 3
| Andrew Bridy (Madison) |
| Title: The Artin–Mazur zeta function of a Lattes map in positive characteristic |
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Abstract: The Artin–Mazur zeta function of a dynamical system is a generating function that captures information about its periodic points. In characteristic zero, the zeta function of a rational map from P1 to P1 is known to always be a rational function. In positive characteristic, the situation is much less clear. Lattes maps are rational maps on P1 that are finite quotients of endomorphisms of elliptic curves, and they have many interesting dynamical properties related to the geometry and arithmetic of elliptic curves. I show that the zeta function of a separable Lattes map in positive characteristic is a transcendental function. |
October 10
| Bogdan Petrenko (Eastern Illinois University) |
| Title: Generating an algebra from the probabilistic standpoint |
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Abstract: Let A be a ring whose additive group is free Abelian of finite rank. The topic of this talk is the following question: what is the probability that several random elements of A generate it as a ring? After making this question precise, I will show that it has an interesting answer which can be interpreted as a local-global principle. Some applications will be discussed. This talk will be based on my joint work with Rostyslav Kravchenko (University of Chicago) and Marcin Mazur (Binghamton University). |
October 17
| Anthony Várilly-Alvarado (Rice) |
| Title: Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups |
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Abstract: Del Pezzo surfaces X of degree 4 are smooth (complete) intersections of two quadrics in four-dimensional projective space. They are some of the simplest surfaces for which there can be cohomological obstructions to the existence of rational points, mediated by the Brauer group Br X of the surface. I will explain how to construct, for every non-trivial, non-constant element A of Br X, a rational genus-one fibration X --> P^1 such that A is "vertical" for this map. This implies, for example, that if there is a cohomological obstruction to the existence of a point on X, then there is a genus-one fibration X --> P^1 where none of the fibers are locally soluble, giving a concrete, geometric way of "seeing" a Brauer-Manin obstruction. The construction also gives a fast, practical algorithm for computing the Brauer group of X. Conjecturally, this gives a mechanical way of testing for the existence of rational points on these surfaces. This is joint work with Bianca Viray. |
October 24
| Paul Garrett (Minnesota) |
| Title: tba |
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October 31
| Jerry Wang (Princeton) |
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November 7
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November 14
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November 21
| Michael Lipnowski (Duke) |
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December 5
| Jennifer Park (MIT) |
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December 12
| Vivek Shende (Berkeley) |
| Title: Equidistribution on the space of rank two vector bundles over the projective line |
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Abstract: I will discuss how the algebraic geometry of hyperelliptic curves gives an approach to a function field analogue of the 'mixing conjecture' of Michel and Venkatesh. (For a rather longer abstract, see the arxiv posting of the same name as the talk). This talk presents joint work with Jacob Tsimerman. |
Organizer contact information
Sean Rostami
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