Graduate Student Math Conference


Brown University

Saturday, March 1st 2014

11:00 AM - 6:00 PM

Schedule & Abstracts


11:00 - 11:30

Vivian Healey

Brown Probability

Universality, Scaling Limits, and Schramm-Loewner Evolution

This talk explores universality, one of the big ideas of probability, through examples. We'll travel from the central limit theorem to Brownian motion and end up at Schramm-Loewner Evolution (SLE), a recent development in probability theory that models random paths. We'll finish by highlighting some of the discrete models that give rise to SLE and some of SLE's more interesting properties.

11:40 - 12:10

Nathan Pflueger

Harvard Algebraic Geometry

Semigroups and Riemann Surfaces

A numerical semigroup is a subset of the positive integers which is closed under addition and has finite complement. Every point on a Riemann surface defines a numerical semigroup, given by the set of pole orders of meromorphic functions that are regular except at the chosen point. I will discuss some of the interplay between the combinatorics of semigroups and the geometry of curves, emphasizing examples in very low genus.

12:20 - 12:50

Zihan Liu

MIT Geometric Analysis

Singularities of the mean curvature flow

The mean curvature flow is a nonlinear geometric evolution equation of submanifolds of a Riemannian manifolds. Informally, it may be described as the time evolution of a surface trying to minimize its area. Intuitively, a submanifold evolving by the mean curvature flow might be expected to approach a minimal surface, but unfortunately (but more interestingly, depending on the point of view), the mean curvature flow can develop singularities in finite time. We'll discuss these singularities, and the connection with certain special solutions of mean curvature flow, known as self-similar shrinkers.

1:00 - 2:15

Lunch

There are many good and afforadble options located on Thayer Street.

2:30 - 3:00

Amalia Culiuc

Brown Harmonic Analysis

Whitney's extension problem and function interpolation

In 1934, Hassler Whitney published a series of papers dealing with the following question: given a function f defined on an arbitrary subset of a Euclidean space, is it possible to extend it to a function with prescribed properties on the entire space? Whitney answered the question in the affirmative for a space of dimension one and the property of m times continuous differentiability, but despite significant progress, the question in its full generality remained open until recent years, when it was answered by Charles Fefferman. I will present Fefferman's generalized extension theorem and provide a survey of the developments that arose as a result. Time permitting, I will discuss some open problems in the field together with their conjectured solutions.

3:10 - 3:40

Rob Castellano

Columbia Symplectic Geometry

Symplectic Geometry or: How I Learned to Stop Worrying and Love J-holomorphic Curves

In 1985, Gromov proved his celebrated nonsqueezing theorem, which has proven to be fundamental to the field of symplectic geometry. I will give an introduction to the field of symplectic geometry and an overview of the proof the nonsqueezing theorem. The proof of the nonsqueezing theorem is noteworthy for its introduction of J-holomorphic curves, which are the basis for much of symplectic geometry. I will discuss other important uses of J-holomorphic curves.

3:50 - 4:20

Eric Riedl

Harvard Algebraic Geometry

Spaces of Rational Curves on Fano Hypersurfaces

Moduli spaces are a major object of study in algebraic geometry. In this talk, I'll describe what moduli spaces are, provide some examples of moduli spaces, and then say a bit about one of my favorite moduli spaces, the Kontsevich space. Kontsevich spaces were introduced in the early 90's for their uses in string theory and mathematical physics. They are crucial objects of study in Gromov-Witten theory and Mirror Symmetry, and there are many important open problems relating to them. In this talk, I will focus less on their applications to physics, and more on the way that they are useful in answering many natural geometric questions about curves in varieties. In particular, I will try to explain how they can be used to understand the geometry of rational curves in hypersurfaces.

4:30 - 5:00

Break & Refreshments


5:00 - 5:30

Ruthi Hortsch

MIT Number Theory

What does an "average" elliptic curve look like?

Elliptic curves are a specific class of geometric objects that have a natural group structure. This structure turns out to have very important implications for number theory and cryptography, and so much modern research is aimed at a deeper understanding of this structure. However, many of the natural questions have proved to be fairly elusive to answer precisely, which has lead modern mathematicians to consider instead the statistics of what we expect an elliptic curve to look like. We'll discuss what kind of things we might care to measure, why number theorists care about such things, and how to even talk about how to measure an average elliptic curve.

5:40 - 6:10

Robert Haraway

Boston College Hyperbolic Geometry

Smallest Hyperbolic 3-manifolds

A survey of approaches to finding smallest hyperbolic 3-manifolds of various kinds, after a motivation and brief of 3-manifold theory.

6:30 - TBA

Dinner

Dinner will be provided for all registered guests. The location will be announced at the conference.


Location



Edward W. Kassar House
Brown University

All talks will be located in Foxboro Auditorium

The entrance is on the Thayer Street side of Kassar House

Peter Pan Bus
9:00am

This bus departs for 'Providence Downtown' from South Station, Boston.

Amtrak Train
9:40am

This train departs from Back Bay Station, Boston or South Station, Boston.

There are MBTA commuter rail trains departing Providence at 8:52 PM & 10:00 PM


About


The AMS Graduate Student Chapter at Brown is a hosting a graduate student math conference, thanks to the gracious support from the American Math Society and the Brown Math Department.

A conference for graduate students, by graduate students.

All are welcome.

There will be a dinner following the conference.

Please register if you are interested in attending the dinner.

List of Participants Contact us Like us on



Organized by Ken Ascher, Wade Hindes, and David Lowry-Duda at the Brown Math Department

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