Graduate Student Math Conference


Brown University

Saturday, February 21st 2015

11:00 AM - 6:00 PM

Schedule / Titles & Abstracts


11:00 - 11:30

Peter McGrath

Brown Geometric PDE

Geometric Maximum principles

The maximum principle is one of the most fundamental techniques in Partial Differential Equations. In the last sixty years, geometric arguments using the maximum principle have been enormously successful in uniqueness and rigidity theorems. I will discuss several of these geometric maximum principles, including the method of moving planes and the Alexandrov reflection principle.

11:40 - 12:10

Federica Fanoni

CUNY/Fribourg Hyperbolic Geometry

Kissing numbers of hyperbolic surfaces

An important object in the study of hyperbolic surfaces (and of more general manifolds) is the systole, i.e. a shortest non-trivial curve. One of the basic questions one can ask about systoles is: how many systoles can we have on a given surface? The number of systoles is the so-called kissing number.

12:20 - 12:50

Anna Medvedovsky

Brandeis Number Theory

Recursion operators, nilpotence, and applications to mod-p Hecke algebras

A recursion operator on a polynomial algebra over a field is a linear operator T: F[y] -> F[y] so that the images of powers of y satisfy a linear recurrence over F[y]. Suppose every polynomial f in F[y] is killed by some power of T, and write N(f) for the least such power. We will analyze the growth of this nilpotence index N(y^n) in the case that F has characteristic p, and describe applications to determining the structure of mod-p Hecke algebras.

1:00 - 2:00 + ε

Lunch

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

2:20 - 2:50

Sarah Bray

Tufts Topological Dynamics

The dynamics of Hilbert geometries

Any properly convex open domain in real projective space admits a projectively invariant metric, attributed to Hilbert, which realizes such a domain as a geometry in it's own right. The first example is the ellipse, isometric to hyperbolic space with this Hilbert geometry. In this talk, we will see compact quotients of such domains which naturally generalize Riemannian hyperbolic manifolds. These manifolds are very distinct from the nice, classical hyperbolic case, geometrically and dynamically. My objective is to share some of what we know and still ponder as we leave the hyperbolic world for it's real projective counterpart, with an infusion of concrete examples along the way.

3:00 - 3:30

Francesco Cavazzani

Harvard Algebraic Geometry

Geometric Invariant Theory

In 1965, in an age where algebra was giving completely new insights on algebraic geometry, Mumford created Geometric Invariant Theory; in practice, using the good old invariant theory (that had its greatest success at the end of the 19^th century), he described a way to produce geometric quotients for group actions. Since then, this extremely powerful tool has been used massively, producing explicit constructions of moduli spaces in algebraic geometry (but also in Symplectic and Differential Geometry). In this talk, we will describe the basics of this constructions, with a few examples to understand its geometry.

3:40 - 4:10

Umut Varolgunes

MIT Symplectic Geometry

Nearby Lagrangian conjecture and Lagrangian Floer homology

It is an old conjecture of Arnold that any closed exact Lagrangian K in the cotangent bundle of a closed manifold L is Hamiltonian isotopic to the zero section L\subset T^*L. Even though we are still far away from proving (or disproving) the result in this original form, there has been impressive results at the topological level, culminating in the statement that the natural map from K to L is a homotopy equivalence under a mild assumption. This, and indeed many other results in symplectic geometry, rely on an invariant called Lagrangian Floer homology, which involves counting discs that solve a certain PDE. After introducing all this in a more detailed manner (which will constitute most of the talk), I plan to sketch an argument which shows that when L is a sphere of any dimension, H^*(K)=H^*(L), if H^1(K)=0. If time permits, I will briefly mention what extra ingredients go into the proofs of the more general statements.

4:20 - 4:50

Break & Refreshments


4:50 - 5:20

Peter Smillie

Harvard Geometric Analysis

Constant mean curvature hypersurfaces in general relativity

Constant mean curvature (CMC) hypersurfaces are an important tool in general relativity, with applications to gravitational radiation, the structure of singularities, gravitational dynamics, and numerical computation. For example, certain spatially compact universes admit a unique foliation by CMC surfaces with mean curvature increasing from leaf to leaf. The mean curvature of the leaf through a point then gives a canonical global ‘time’ function on the universe, using which you can think of the n+1 dimensional spacetime more classically as an n-manifold evolving dynamically. I will explain how to construct CMC hypersurfaces in some specific cases.

5:30 - 6:00

Yumeng Ou

Brown Harmonic Analysis

What is dyadic shift operator and why do we care?

In modern harmonic analysis, dyadic models not only act as illuminating toy models where ideas can be tested, but also provide a way to directly solve problems in the non-dyadic world. We will illustrate the latter point by introducing the simplest dyadic model, Haar analogues, and the so-called dyadic shift operators, which is one of the most recent hot tools in singular integral theory. It has been proved by T. Hytönen that any Calderón-Zygmund operator can be represented as an average of dyadic shift operators. As an application, we will discuss how these dyadic objects could be used in the study of iterated commutators.

6:00 + ε - TBA

Dinner

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




Organized by Ken Ascher, Vivian Healey, and David Lowry-Duda, at the Brown Math Department

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