Penn Graduate Student Seminar

Nonstandard analysis studies problems of calculus using modern logic rather than epsilons and deltas. We'll go through a gentle introduction to nonstandard reasoning, and include a somewhat less gentle discussion of the construction that makes it rigorous.

Somewhat recently, polynomials have been used to provide brief and elegant solutions to problems that had previously been considered very hard. I'll talk about some of these problems/solutions.

In the 1970s, E. Stein and other mathematicians studying fundamental questions related to pointwise convergence of Fourier series discovered surprising new links between this very old problem and the geometry of submanifolds of Euclidean space. These discoveries paved the way for many of the questions at the forefront of modern harmonic analysis. A common element in many of these areas is the role of a strange sort of curvature condition which arises naturally from Fourier-theoretic roots but is poorly understood outside the extreme cases of curves and hypersurfaces. In this talk, I will discuss recent work which combines elements of Geometric Invariant Theory, Convex Geometry, Signal Processing, and other areas to shed light on this problem in intermediate dimensions

A perceptron network is one of the simplest artificial neural networks, in which the activation function of each neuron is a step function. We impose the additional restriction that each neuron can only have a fixed number of inputs. Given an N x N binary image (all pixels either black or white), can such a network detect whether or not the black pixels are connected? The answer is no (kind of).

Arithmetic Geometry applies techniques of Algebraic Geometry to tackle problems in Number Theory. Even though the techniques are usually modern and elaborate, this strategy itself is not new. In this talk, we will explore how this interaction between geometry and arithmetic evolved, starting from Pythagoras and his contemporaries, and aiming towards modern arithmetic geometry. Elliptic curves (and their rational points) will be our reference point throughout this journey. Disclaimer: This will NOT be a talk on arithmetic geometry.

What is the probability that a royal name will eventually go extinct? The study of branching processes gives an explicit answer in terms of the distribution of the number children of each person. We'll see this result as well as many other properties of branching processes and percolation on random trees. The goal of this talk is not only to cover this topic, but also to illustrate some common techniques in probability.

This is a continuation of my Pizza Seminar Talk from April 7, 2017, and also the first of a three part talk in Pizza Seminar happening this November. The questions I will discuss comes from quadratic forms theory. For instance, can one give a good but biased overview of the history of quadratic forms, or coherently explain the recent work of Manjul Bhargava on composition laws and the Bhargava cube, focusing on core ideas and examples? The prerequisites for this talk are just honors level algebra (but the proofs - which I will not really go into - involve very deep mathematics).

The Brauer group Br(k) of a field k is an important object of study in number theory. A convenient condition to force the Brauer group to be trivial is for k to be quasi-algebraically closed. Quasi-algebraically closed fields are part of a more general notion of C_i fields, introduced in Lang's thesis in 1951. The goal of this talk is to prove that finite fields and function fields over algebraically closed fields are quasi-algebraically closed hence have trivial Brauer groups.

The u-invariant arises naturally from the study quadratic forms since it gives a uniform bound on the number of variables n such that any quadratic forms of n variables are universal. Particularly, u-invariant is the minimal such n satisfying such property. In this talk, I’ll give a survey of some known results in the u-invariant. The main goal is to compute u-invariants of some standard fields and discuss the relationship of u-invariants between field extensions.

We will talk about one of the simplest, oldest and hardest problems in geometry, and its interactions with the rest of math.

The Atiyah Singer Index theorem is one of the great achievements of 20th century mathematics. By studying generalizations and analogues, I have taken a journey through a lot of different parts of mathematics. I will explain what the index theorem is and describe some of my destinations.