I will give an introduction to vector-valued modular forms and describe some of my results on the arithmetic of their Fourier coefficients. The collection of vector-valued modular forms form a graded module over the graded ring of modular forms. I will explain how understanding the structure of this module allows one to show that the component functions of vector-valued modular forms satisfy an ordinary differential equation whose coefficients are modular forms. In certain cases, one may use a Hauptmodul to transform such a differential equation into a Fuchsian differential equation on the projective line minus three points. I will explain how these ideas can be used to prove certain cases of the unbounded denominator conjecture for vector-valued modular forms.

Please contact the organizer for the WebEx link.

The two main techniques of integration taught in calculus courses are integration by substitution and integration by parts. This talk will describe and illustrate a third technique of integration, almost never
taught in math courses, called differentiation under the integral sign. It can handle integrals that appear inaccessible to simpler methods. The physicist Richard Feynman had great affection for differentiation under the integral sign, writing once
"I caught on how to use that method, and I used that one damn tool again and again."


This talk will assume familiarity with partial derivatives from multivariable calculus.



Note: Join the meeting at https://uconnvtc.webex.com/meet/mathclub

Title: Information Based Complexity for High Dimensional Statistical Models

Abstract: I will introduce a coherent framework to quantify the complexity of high dimensional models that appropriately accounts for both statistical accuracy and computational cost and better understand the potential trade-off between the two types of efficiencies. As an example, I will use this notion of complexity to examine high-dimensional and sparse nonparametric problems to illustrate how this can lead to the development of novel and optimal sampling and estimation strategies, and in particular reveal the role of experimental design in alleviating computational burden.

Title: Initial boundary value problem for vacuum Einstein equations

Abstract: In general relativity, spacetime metrics are solutions to the Einstein equations, which are wave equations moduli gauge. The Cauchy problem for the vacuum Einstein equations has been well-understood since the work of Choquet-Bruhat. For certain given initial conditions, there is a geometrically unique solution to the vacuum Einstein equations. On contrast, the initial boundary value problem has been much less understood. To solve for a spacetime metric in a domain with time-like boundary, one needs to impose boundary conditions to guarantee geometric uniqueness of the solution. However, due to gauge issues occurring on the boundary, there has not been a satisfying choice of boundary conditions.
In this talk we will first talk about spacetimes, Einstein equations and the Cauchy problem. Then we will discuss obstacles to establishing a well-defined initial boundary value problem and new results on it (joint work with Michael Anderson).

Join us for a talk by Lenore Blum (Carnegie Mellon) in the Logic Colloquium!

"What Can Theoretical Computer Science Contribute to the Discussion of Consciousness?"

We propose a mathematical model, which we call the Conscious Turing Machine (CTM), as a formalization of neuroscientist Bernard Baars’ Theater of Consciousness. The CTM is proposed for the express purpose of understanding consciousness. In settling on this model, we look not for complexity but simplicity, not for a complex model of the brain or cognition but a simple mathematical model sufficient to explain consciousness. Our approach, in the spirit of mathematics and theoretical computer science, proposes formal definitions to fix informal notions and deduce consequences. We are inspired by Alan Turing’s extremely simple formal model of computation that is a fundamental first step in the mathematical understanding of computation. This mathematical formalization includes a precise definition of chunk, a precise description of the competition that Long Term Memory (LTM) processors enter to gain access to Short Term Memory (STM)), and a precise definition of conscious awareness in the model. Feedback enables LTM processors to learn from their mistakes and successes and emerging links enable conscious processing to become unconscious. The reasonableness of the formalization lies in the breadth of concepts that the model explains easily and naturally. The model provides some understanding of the Hard Problem of consciousness, which we explore in the particular case of pain and pleasure. The understanding depends on the dynamics of the CTM, not on chemicals like serotonin, dopamine, and so on. We set ourselves the problem of explaining the feeling of consciousness in ways that apply as well to machines made of silicon and gold as to animals made of flesh and blood. With regard to suggestions for AI, the CTM is well suited to giving succinct explanations for whatever high level decisions it makes. This is because the chunk in STM either articulates an explanation or points to chunks that do.

Please contact Marcus Rossberg for log-in information.

https://logic.uconn.edu/calendar/

Abstract: This will be a series of short (about 10-15 minute) talks that will help graduate students learn about the research conducted by four members of the analysis and probability group (Iddo, Masha, Sasha, and Zhongyang).

The talks will be accessible to newcomers.

MATH 5020 (Alvaro Lozano-Robledo): The Arithmetic of Elliptic Curves.

This course will be an introduction to elliptic curves, which roughly speaking are smooth cubic curves in the projective plane (turns out they have a simple model of the form $$y^2 = x^3 + ax + b$$). The surprising feature of ellip-
tic curves is that their points can be made into an abelian group, and this group is finitely generated when we focus on points with coordinates in the rational numbers lying on an elliptic curve with rational coefficients.

Elliptic curves are central in modern number theory, e.g., they were essential in the proof of Fermat’s Last Theorem. The goal of the course will be to understand and calculate the group of all rational points on an elliptic curve (i.e., calculate its torsion and rank), and a number of more refined
invariants (such as the order of the Shafarevich-Tate group).
The prerequisites for this course are the Abstract Algebra sequence (Math 5210 and 5211) and a basic understanding of algebraic number theory and algebraic geometry, although I will adjust the material to the audience background as much as I can. Our textbook will be ”The Arithmetic of Elliptic Curves,” by Silverman, which is the standard graduate-level textbook for the subject.

Math 5030 (Ovidiu Munteanu): Geometric Analysis on Manifolds:

This course is an introduction to the linear theory of partial differential equations on open manifolds. Assuming some appropriate information on
curvature, we will study properties of solutions to the Laplace and heat equations, which in turn will give us more insight about the geometry and
topology of the underlying manifold. For example, we will use harmonic functions to count the number of ends of open manifolds, and see applications to rigidity results for manifolds with more than one end.

The techniques developed in this theory are essential to many other problems in geometric analysis, such as in the study of geometric flows on manifolds.

The course follows Peter Li’s book Geometric Analysis, Cambridge Studies in Advanced Mathematics.

Prerequisites: Basic knowledge of Riemannian geometry will be assumed. The course is self-contained on the PDE side.

https://uconn-cmr.webex.com/uconn-cmr/j.php?MTID=mf97d7a52926e3c9ed827574468a04310