Godement-Jacquet use the Schwartz space of n-by-n matrices to construct the standard L-function for GL_n. Ben-Zvi, Sakellaridis and Venkatesh conjecture that the local unramified part of this theory can be categorified to an equivalence between an 'analytic' category of constructible sheaves and a 'spectral' category of dg modules. In this talk I will explain the proof of this equivalence and some of its properties. I will also discuss connections to conjectures of Braverman-Kazhdan on constructions of general automorphic L-functions. This is joint work with Tsao-Hsien Chen (in preparation).
I will discuss the relative trace formula approach to the global Gan-Gross-Prasad conjectures for unitary groups. The focus will be on the spectral side. I will present the various terms that appear in the spectral development of the relative trace formula and discuss what is still missing. This is a joint work with Pierre-Henri Chaudouard.
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
Bruce Berndt : The Circle and Divisor Problems, Bessel Function Series, and Ramanujans Lost Notebook- Number Theory ( 154 Views )
A page in Ramanujan's lost notebook contains two identities for trigonometric sums in terms of doubly infinite series of Bessel functions. One is related to the famous ``circle problem'' and the other to the equally famous ``divisor problem.'' We discuss these classical unsolved problems. Each identity can be interpreted in three distinct ways. We discuss various methods that have been devised to prove the identities under these different interpretations. Weighted divisor sums naturally arise, and new methods for estimating trigonometric sums need to be developed. Trigonometric analogues and extensions of Ramanujan's identities to Riesz and logarithmic sums are discussed. The research to be described is joint work with Sun Kim and Alexandru Zaharescu.
Waves in free-space diffractively spread, while waves
in a spatially non-homogeneous medium undergo a combination of
scattering and localization.
In many applications, e.g. photonic and quantum systems, one is interested in controlled localization of wave energy. Edge states are a type of localization along a line-defect, the interface between different media.
Topologically protected edge states are a class of edge states which are robust to strong local distortions of the edge. They are therefore potential vehicles for robust energy-transfer in the presence of defects and random imperfections. These states arise, for example, in graphene and its photonic analogues.
We first review the mathematics of dispersive waves in periodic media and discuss examples of wave localization by a defect.
We then specialize to the case of honeycomb structures (such as grapheme) and discuss their novel properties. Finally we introduce and discuss a rich family of continuum partial differential equation (Schroedinger) models, admitting edge states which are topologically protected and those which are not.
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.
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.
We show that the distributions occurring in the geometric and spectral side of the twisted Arthur-Selberg trace formula extend to non-compactly supported test functions. The geometric assertion is modulo a hypothesis on root systems proven among other cases, when the group is split. This result extends the work of Finis-Lapid (and Muller, spectral side) in the non-twisted setting. In the end, we will give an application towards residues of Rankin-Selberg L-functions suggested by J. Getz.
In this talk, I will discuss the geometric expansion of a local twisted trace formula for some special varieties. This generalizes the local (twisted) trace formula for reductive groups proved by Arthur and Waldspurger. By applying the trace formula, we prove a multiplicity formula for these spherical varieties. And I will also discuss some applications to the multiplicity of the Galois model and the unitary Shalika model. This is a joint work with Raphael Beuzart-Plessis.