Beyond the Willmore Conjecture

Speaker: 

Robert Kusner

Institution: 

University of Massachusetts at Amherst

Time: 

Thursday, June 7, 2012 - 4:00pm

Location: 

RH 306

The squared-mean-curvature integral was introduced two centuries
ago by Sophie Germain to model the bending energy and vibration patterns of
thin elastic plates. By the 1920s the Hamburg geometry school realized
this energy is invariant under the Möbius group of conformal
transformations, and thus that minimal surfaces in R^3 or S^3 are
equilibria. In 1965 Willmore observed that the round sphere minimizes the
bending energy among all closed surfaces, and he conjectured that a certain
torus of revolution - the stereographic projection of the Clifford minimal
torus in S^3 - minimizer for surfaces of genus 1. This conjecture was
proven this spring by Fernando Coda and Andre Neves; they use the
Almgren-Pitts minimax construction, the Hersch-Li-Yau notion of conformal
area, and Urbano's characterization of the Clifford torus by its Morse
index. We will discuss what is known or conjectured for other topological
types of bending-energy minimizing or equilibrium surfaces, and how
bending-energy gradient flow might be applied to problems in
low-dimensional topology.

Modeling, Calibration, and Simulation of Spot Price Paths

Speaker: 

Hua Lv

Institution: 

UCI, Mathematics Department

Time: 

Tuesday, June 5, 2012 - 2:00pm

Location: 

RH 440R

Advisor: Professor Knut Solna

Abstract: We introduce some popular models in the energy markets. Then we
propose to incorporate a stochastic volatility feature to an existing
multi-factor deterministic volatility model in order to take into account
the observed implied volatility skews for each of the commodities in the
simulation of monthly forward prices. As examples we consider natural gas,
crude oil and heating oil price and option data. Our objective is to
explore the role of stochastic volatility modeling for calibration and
simulation of price paths and scenario analysis. The linkage between
price, option data and modeling is captured by the so called "Vs" in our
approach. These are the effective group market parameters that capture the
main impact of an uncertain and fluctuating volatility, in particular how
these affect prices. To explore the significance of incorporating this
link we carry out an initial calibration test to explore the role of the
"Vs" in the commodity price distribution. We find that indeed the
distribution of the commodity prices are significantly affected by
incorporating the leading correction that accounts for the effect of
uncertain volatility parameters which manifests itself in the data via
strong "skew" effect in the option pricing data. An added benefit of this
modeling framework is that it enables us to use observations around and
not only at the money in a consistent way, thus, providing robustness and
stability in calibration also at the order one level.

A gentle introduction to quantum walks

Speaker: 

David Damanik

Institution: 

Rice University

Time: 

Friday, June 1, 2012 - 2:00pm to 3:00pm

Host: 

Anton Gorodetski

Location: 

RH 340N

 

In this talk we explain the concept of a quantum walk and survey some of
the results obtained for them recently by various authors. We will also
address the special case of coins given by the Fibonacci sequence, both
in a spatial and in a temporal context.

Realizing Cubic Hypersurfaces

Speaker: 

Robert Campbell

Institution: 

UCI, Mathematics Department

Time: 

Tuesday, June 5, 2012 - 10:00am

Location: 

RH 340N

Advisor: Vladimir Baranovsky
Abstract:
We begin by exploring algebraic codes created using cubic hypersurfaces. This leads to the questions of classification, realization, and construction of cubic hypersurfaces. Given the classification by Manin, Frame, Swinnerton-Dyer, etc., we will look at methods of realization and construction of these cubics. Specifically, we will focus on two approaches. The first approach involves looking at the blow-downs of cubics. The second approach involves automorphisms of well-defined cubics.

Soliton Equations and Geometric Curve Flows

Speaker: 

Zhiwei Wu

Institution: 

UCI, Mathematics Department

Time: 

Tuesday, May 22, 2012 - 2:00pm

Location: 

RH 440R

Advisor: Professor Chuu-Lian Terng

In this thesis defense, I will explain

(1) properties of a classical soliton equation----the KdV equation,

(2) the symmetry and Hamiltonian properties of the Matrix modified Constrained KP hierarchy,

(3) an integrable curve flow on the affine n-space.

 

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