Difference between revisions of "NTS Spring 2014/Abstracts"
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− | == | + | == February 23 == |
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''John Voight''' (Dartmouth) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: Numerical calculation of three-point branched covers of the projective line | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | Abstract: We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups. This is joint work with Michael Klug, Michael Musty, and Sam Schiavone. | ||
+ | |} | ||
+ | </center> | ||
− | + | <br> | |
− | + | <!-- | |
+ | == September 12 == | ||
− | ---- | + | <center> |
− | + | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall''' (Northwestern) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: Endoscopy and cohomology growth on U(3) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | 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. | ||
+ | |} | ||
+ | </center> | ||
− | + | <br> | |
+ | |||
+ | == September 19 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Valerio Toledano Laredo''' (Northeastern) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: From Yangians to quantum loop algebras via abelian difference equations | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | 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. | ||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | == September 26 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Haluk Şengün''' (Warwick/ICERM) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: Torsion homology of Bianchi groups and arithmetic | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | 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. | ||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | == October 3 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrew Bridy''' (Madison) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: The Artin–Mazur zeta function of a Lattes map in positive characteristic | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | 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 '''P'''<sup>1</sup> to '''P'''<sup>1</sup> is known to always be a rational function. In positive characteristic, the situation is much less clear. Lattes maps are rational maps on '''P'''<sup>1</sup> 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. | ||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | == October 10 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Bogdan Petrenko''' (Eastern Illinois University) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: Generating an algebra from the probabilistic standpoint | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | 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). | ||
+ | |||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | == October 17 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Anthony Várilly-Alvarado''' (Rice) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | 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. | ||
+ | |||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | == October 24 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Paul Garrett''' (Minnesota) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: Spectra of pseudo-Laplacians on spaces of automorphic forms | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | Abstract: Faddeev–Pavlov and Lax–Phillips observed that certain | ||
+ | restrictions of the Laplacian to parts of automorphic continuous | ||
+ | spectrum have discrete spectrum. Colin de Verdiere used this to prove | ||
+ | meromorphic continuation of Eisenstein series, and proposed ways to | ||
+ | exploit this idea to construct self-adjoint operators with spectra | ||
+ | related to zeros of ''L''-functions. We show that simple forms of this | ||
+ | construction produce at most very sparse spectra, due to | ||
+ | incompatibility with pair correlations for zeros. Ways around some of | ||
+ | the obstacles are sketched. (Joint with E. Bombieri.) | ||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | == October 31 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jerry Wang''' (Princeton) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: Pencils of quadrics and the arithmetic of hyperelliptic curves | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | Abstract: In recent joint works with Manjul Bhargava and Benedict Gross, we showed that a positive proportion of hyperelliptic curves over '''Q''' of genus ''g'' have no points over any odd degree extension of '''Q'''. This is done by computing certain 2-Selmer averages and applying a result of Dokchitser–Dokchitser on the parity of the rank of the 2-Selmer groups in biquadratic twists. In this talk, we will see how arithmetic invariant theory and the geometric theory of pencils of quadrics are used to obtain the 2-Selmer averages. | ||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | == November 7 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''who?''' (where?) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: tba | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | Abstract: tba | ||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | == November 12 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Evan Dummit''' (Madison) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: Counting extensions of number fields of given degree, bounded (rho)-discriminant, and specified Galois closure | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | Abstract: A very basic question in algebraic number theory is: how many number fields are there? A natural way to order the fields of a fixed degree n is by discriminant, and classical results of Minkowski then assure us that there are only finitely many fields with a given discriminant. We are also often interested in counting number fields, or relative extensions, with other properties, such as having a particular Galois closure. A folk conjecture sometimes attributed to Linnik states that the number of extensions of degree n and absolute discriminant less than X is on the order of X. A great deal of recent and ongoing work has been focused towards achieving upper bounds on counts of this nature (quite successfully, in degree 5 and lower), but there is comparatively little known in higher degrees, for relative extensions, or for sufficiently complicated Galois closures: the primary results are those of Schmidt and Ellenberg-Venkatesh. I will discuss these results and my thesis work, in which I generalize several of their results and introduce another counting metric, the "rho-discriminant". | ||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | == November 21 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Michael Lipnowski''' (Duke) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: tba | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | Abstract: tba | ||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | == November 26 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Kane''' (Stanford) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: Diffuse decompositions of polynomials | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | Abstract: We study some problems relating to polynomials evaluated | ||
+ | either at random Gaussian or random Bernoulli inputs. We present some | ||
+ | new work on a structure theorem for degree-''d'' polynomials with Gaussian | ||
+ | inputs. In particular, if ''p'' is a given degree-''d'' polynomial, then ''p'' | ||
+ | can be written in terms of some bounded number of other polynomials | ||
+ | ''q''<sub>1</sub>, ..., ''q''<sub>''m''</sub> so that the joint probability density function of | ||
+ | ''q''<sub>1</sub>(''G''), ..., ''q''<sub>''m''</sub>(''G'') is close to being bounded. This says essentially | ||
+ | that any abnormalities in the distribution of ''p''(''G'') can be explained by | ||
+ | the way in which ''p'' decomposes into the ''q''<sub>''i''</sub>. We then present some | ||
+ | applications of this result. | ||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | == December 5 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jennifer Park''' (MIT) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: tba | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | Abstract: tba | ||
+ | |||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | == December 12 == | ||
+ | |||
+ | <center> | ||
+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
+ | |- | ||
+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Vivek Shende''' (Berkeley) | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" align="center" | Title: Equidistribution on the space of rank two vector bundles over the projective line | ||
+ | |- | ||
+ | | bgcolor="#BCD2EE" | | ||
+ | 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 [http://arxiv.org/abs/1307.8237 arxiv posting] of the same name as the talk). This talk presents joint work with Jacob Tsimerman. | ||
+ | |} | ||
+ | </center> | ||
+ | |||
+ | <br> | ||
+ | |||
+ | --> |
Revision as of 08:55, 7 January 2014
February 23
John Voight (Dartmouth) |
Title: Numerical calculation of three-point branched covers of the projective line |
Abstract: We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups. This is joint work with Michael Klug, Michael Musty, and Sam Schiavone. |