Lillian Pierce : Burgess bounds for short mixed character sums
- Number Theory ( 134 Views )A celebrated result of Burgess proves nontrivial bounds for short multiplicative character sums. In general, bounds for short character sums have utility in a wide range of problems in number theory, and it would be highly desirable to extend Burgesss method to apply to more general character sums. This talk presents new work in this direction, joint with Roger Heath-Brown, that proves nontrivial bounds for short mixed character sums in which the additive character is evaluated at a real-valued polynomial. Our approach, via a version of the Burgess method, includes a novel application of the recent results of Wooley on the Vinogradov mean value theorem.
Robert Rhoades : The story of a strange function
- Number Theory ( 167 Views )In a 1997 Fields Medalist Maxim Kontsevich suggested that the function F(q) = 1 + (1-q) + (1-q)(1-q^2) + (1-q)(1-q^2)(1-q^3)+ , defined only for q a root of unity, is similar to certain functions arising from the computation of Feynman integrals in quantum field theory. In the last sixteen years this function has been connected to interval orders in decision making theory, ascent sequences and matchings in combinatorics, and Vassiliev invariants in knot theory. Don Zagier related the asymptotic properties of this function to the half-derivatives of modular forms and was led to define a notion of quantum modular form. In a trilogy of papers, my collaborators (Andrews, Bryson, Ono, Pitman, Zwegers) and I have connected this function to Ramanujans mock theta functions and the combinatorics of unimodal sequences. I will tell the story of this function and these many relationships.
Ali Altug : Beyond Endoscopy via the Trace Formula
- Number Theory ( 207 Views )In his recent paper,\Beyond Endoscopy", Langlands proposed an approach to (ultimately) attack the general functoriality conjectures by means of the trace formula. For a (reductive algebraic) group G over a global field F and a representation of its L-group, the strategy, among other things, aims at detecting those automorphic representations of G for which the L-function, L(s;\pi ;\rho ), has a pole at s = 1. The method suggested using the the trace formula together with an averaging process to capture these poles. In this talk we will start by recalling the functoriality conjectures and brie y describe the method suggested by Langlands. Then, specializing on the group GL(2) we will discuss some recent work on Beyond Endoscopy. More precisely, we will discuss the elliptic part of the trace formula and the analytic problems caused by the volumes of tori, singularities of orbital integrals and the non-tempered terms. We will then describe how one can use an approximate functional equation in the trace formula to rewrite the elliptic part which resolves these issues. Finally, we will talk about applications of the resulting formula.
Wei Ho : Families of lattice-polarized K3 surfaces
- Number Theory ( 97 Views )There are well-known explicit families of K3 surfaces equipped with a low degree polarization, e.g., quartic surfaces in P^3. What if one specifies multiple line bundles instead of a single one? We will discuss representation-theoretic constructions of such families, i.e., moduli spaces for K3 surfaces whose Neron-Severi groups contain specified lattices. These constructions, inspired by arithmetic considerations, also involve some fun geometry and combinatorics. This is joint work with Manjul Bhargava and Abhinav Kumar.
Frank Thorne : Secondary Terms in Counting Functions for Cubic Fields
- Number Theory ( 122 Views )I will speak about recent progress on the enumeration of number fields, with particular attention to joint work with Taniguchi, which proved the existence of a negative secondary term in the counting function for cubic fields by discriminant. Among other results, we also found surprising biases in arithmetic progressions -- e.g., cubic field discriminants are more likely to be 5 (mod 7) than 3 (mod 7). Our work applies the analytic theory of the Shintani zeta function, which I will describe briefly. I will also discuss other approaches to related questions (and in particular an independent, and different, proof of the secondary term due to Bhargava, Shankar, and Tsimerman), using approaches as diverse as the geometry of numbers, algebraic geometry, and class field theory.
Michael Lipnowski : Torsion in the cohomology of arithmetic groups
- Number Theory ( 133 Views )The remarkable Cheeger-Muller theorem, of differential geometric origin, provides an analytic means of studying torsion in the cohomology of Riemannian manifolds. We describe how this theorem can be applied to prove a numerical form of Langlands' base change functoriality for torsion in cohomology.
P. E. Herman : ON PATTERSONS CONJECTURE: SUMS OF EXPONENTIAL SUMS.
- Number Theory ( 117 Views )It is well known that for an exponential sum with a prime modulus the best bound for the sum comes from Weil's famous estimation. In this talk, we discuss when this bound can be improved on average over integral modulus in a number field. Investigations into exponential sums on average, or sums of exponential sums, have many applications including the Riemann hypothesis and the Ramanujan conjecture for automorphic forms. In particular, we will get an asymptotic for sums of quartic exponential sums over the Gaussian integers. Tools we will use to get this asymptotic include automorphic forms and the trace formula.
Majid Hadian : On a Motivic Method in Diophantine Geometry
- Number Theory ( 124 Views )By studying universal motivic unipotent representations of fundamental group of varieties and comparing their different realizations, we combine Kim's recent method in Diophantine geometry with Deligne-Goncharov's theory of motivic fundamental groups to develop a machinery for approaching Diophantine problems concerning integral points.
Emmanuel J. Candes : Exact Matrix Completion by Convex Optimization Theory and Algorithms
- Number Theory ( 107 Views )The recovery of a data matrix from a sampling of its entries is a problem of considerable practical interest. In partially filled out surveys, for instance, we would like to infer the many missing entries. In the area of recommender systems, users submit ratings on a subset of entries in a database, and the vendor provides recommendations based on the user's preferences. Because users only rate a few items, we would like to infer their preference for unrated items (the famous Netflix problem). Formally, suppose that we observe m entries selected uniformly at random from a matrix. Can we complete the matrix and recover the entries that we have not seen? Surprisingly, one can recover low-rank matrices exactly from what appear to be highly incomplete sets of sampled entries; that is, from a minimally sampled set of entries. Further, perfect recovery is possible by solving a simple convex optimization program, namely, a convenient semi-definite program. We show that our methods are optimal and succeed as soon as recovery is possible by any method whatsoever. Time permitting, we will also present a very efficient algorithm based on iterative singular value thresholding, which can complete matrices with about a billion entries in a matter of minutes on a personal computer.
Mason A. Porter : Communities in Networks
- Number Theory ( 166 Views )Networks (graphs) arise pervasively in biology, physics, technology, the social sciences, and myriad other areas. They typically exhibit a complicated mixture of random and structured features. Over the past several years, my collaborators and I have conducted several studies of cohesive mesoscopic structures known as "communities," which consist of groups of nodes that are closely related. In this talk, I will discuss the idea of network community structure and discuss results my collaborators and I have obtained using networks constructed from data such as Facebook friendships, Congressional committee assignments and voting/legislation cosponsorship, and NCAA football schedules. arXiv.org:0902.3788
Stuart Kauffman : The Open Universe
- Number Theory ( 128 Views )Laplace gave the simplest early statement of reductionism. His Demon, if supplied with the positions and momenta of all the particles in the universe, could, using Newton's laws, calculate the entire future and past of the universe. Add fields, quantum mechanics, and General Relativity and you have, roughly, modern physics. There are four features to Laplace's reductionism: (I) Everything that happens is deterministic, called into question a century later by quantum mechanics and the familiar Copenhagen interpretation and Born rule. (ii) All that is ontologically real are "nothing but" particles in motion. (iii) All that happens in the universe is describable by universal laws. (iv) There exists at least one language able to describe all of reality. Quantum mechanics is evidence against (i). I will argue that biological evolution, the coming into existence in the universe of hearts and humming birds co-evolving with the flowers that feed them and that they pollenate, cannot be deduced or simulated from the basic laws of physics. In Weinberg's phrase, they are not entailed by the laws of physics. I will then claim that at levels above the atom, the universe will never make all possible proteins length 200 amino acids, all possible organisms, or all possible social systems. The universe is indefinitely open upwards in complexity. More, proteins, organisms, and social systems are ontologically real, not just particles in motion. Most radically, I will contest (iii). I will try to show that we cannot pre-state Darwinian pre-adaptations, where a pre-adaptation is a feature of an organism of no use in the current selective environment, but of use in a different environment, hence selected for a novel function. Swim bladders are an example. Let me define the "adjacent possible" of the biosphere. Once there were the lung fish that gave rise to swim bladders, swim bladders were in the adjacent possible of the biosphere. Before there were multi-celled organisms, swim bladders were not in the adjacent possible of the biosphere. What I am claiming is that we cannot pre-state the adjacent possible of the biosphere. How could we pre-state the selective conditions? How could we pre-specify the features of one or several organisms that might become pre-adaptations? How could we know that we had completed the list? The implications are profound, if true. First, we can make no probability statement about pre-adaptations, for we do not know the sample space, so can formulate no probability measure. Most critically, if a natural law is a compact description before hand and afterward of the regularities of a process, then there can be no natural law sufficient to describe the emergence of swim bladders. Thus, the unfolding of the universe is partially lawless! This contradicts our settled convictions since Descartes, Galileo, Newton, Einstein and Schrödinger. It says that (iii) is false. In place of law is a ceaseless creativity, a self consistent self construction of the biosphere, the economy, our cultures, partially beyond law. Were reductionism sufficient, the existence of swim bladders in the universe would be entailed by physical law, hence "explained". But it appears that physics, as stated, is not sufficient in its reductionist version. Then we must explain the existence in the universe of swim bladders and humming birds pollenating flowers that feed them, on some different ground. We need a post-reductionist science. Autocatalytic mutualisms of organisms, the biosphere, and much of the economy, may be part of the explanation we seek. In turn this raises profound questions about how causal systems can coordinate their behaviors, let alone the role of energy, work, power, power efficiency, in the self-consistent construction of a biosphere. There is a lot to think about.