The cohomology of arithmetic groups is built from certain automorphic forms, allowing for explicit computation of Hecke eigenvalues using topological techniques in some cases. For modular forms attached to the general linear group over a number field F of class number one, these cohomological forms can be described in terms an associated Voronoi polyhedron coming from the study of perfect n-ary forms over F. In this talk, we describe this relationship and report on some recent computational investigations of the modularity of elliptic curves over the cubic field of discriminant -23. This is joint work with Donnelly, Gunnells, and Klages-Mundt.

By a non-standard comparison between (relative) trace formulas we mean one where the scalar "transfer factors" are substituted by non-scalar "transfer operators". The problem of global triviality of transfer factors now becomes a problem of proving a Poisson summation formula for such non-scalar operators. I will discuss the adelic analysis behind such a non-standard comparison, that leads to a new proof of Waldspurger's theorem on toric periods for GL(2) and the analytic continuation of the quadratic base change L-function in the spirit of "Beyond Endoscopy".

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 Burgess’s 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.

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 Ramanujan’s mock theta functions and the combinatorics of unimodal sequences. I will tell the story of this function and these many relationships.

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.

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.

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.

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.

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 field. 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.

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.

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.

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.

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