The unreasonable effectiveness of Bregman iteration for L1 type optimization

Speaker: 

Professor Stanley Osher

Institution: 

UCLA

Time: 

Thursday, February 18, 2010 - 11:00am

Location: 

RH 306

Bregman iteration has been around since 1967. It turns out to be unreasonably effective for optimization problems involving L1, BV and related penalty terms. This is partly because of a miraculous cancellation of error. We will discuss this and give biomedical imaging applications, related to compressive sensing and Total Variation based restoration.

New Algorithms in Information Science

Speaker: 

Professor Stanley Osher

Institution: 

UCLA

Time: 

Wednesday, February 17, 2010 - 4:00pm

Location: 

RH 306

The past few years have seen an incredible explosion of new (or revival of old) fast and effective algorithms for various imaging and information science applications. These include: nonlocal means, compressive sensing, graph cuts, Bregman iteration, as well as relatively old favorites such as the level set method and PDE based image restoration. I'll give my view of where we are, hopefully giving credit to all the people involved.

The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces

Speaker: 

William Yessen

Institution: 

UC Irvine

Time: 

Friday, November 13, 2009 - 2:00pm

Location: 

RH 440R

Given a diffeomorphism of a two-dimensional manifold with a class of smoothness greater than one. Given a horseshoe of this diffeomorphism, R. Mane (1979) showed, based partially on a program introduced by R. Bowen (1973), that the Hausdorff dimension of this horseshoe depends smoothly on the diffeomorphism. We shall give a general discussion of Mane's aforementioned paper, and the techniques used therein.

Modeling and computation of the optical responses of a nano medium

Speaker: 

Yuanchang Sun

Institution: 

postdoctoral researcher, UCI (Xin)

Time: 

Monday, November 23, 2009 - 4:00pm

Location: 

RH 306

Light-matter interactions on the nanometer scale are at the heart of nano optics. The fact that some quantum effects begin to dominate on nanoscale makes nanostructures possess strikingly different optical properties from their bulk. This talk will consider the excitons and biexcitons confinement effects in an optically excited nanocrystal.
Modeling and computational studies will be presented.

Generalized Borcherds products and two number theoretic applications

Speaker: 

Ken Ono

Institution: 

University of Wisconsin

Time: 

Saturday, November 7, 2009 - 2:30pm

Location: 

RH 101

At the 1994 ICM in Zurich, Borcherds introduced the notion of an automorphic infinite product. In this lecture I will discuss joint work with Jan Bruinier which gives automorphic infinite products arising from harmonic Maass forms. We will discuss two number theoretic applications:

1) Partitions
2) Central derivatives of modular L-functions.

Genus bounds for curves with fixed Frobenius eigenvalues

Speaker: 

Everett Howe

Institution: 

CCR

Time: 

Saturday, November 7, 2009 - 11:30am

Location: 

RH 101

This talk is based on joint work with Noam Elkies and
Christophe Ritzenthaler.

Suppose you are given a finite set S of simple abelian varieties
over a finite field k. Is there a bound on the genera of the
curves over k whose Jacobians are isogenous to products
of powers of elements of S?

Serre, using results of Tsfasman and Vladuts, showed that the
answer is yes. We give explicit bounds on the genus, in terms
of the "Frobenius eigenvalues" (the roots of the characteristic
polynomials of Frobenius) of the elements of S.

We show, for example, that the maximal genus of a curve over
F_2 whose Jacobian splits completely (up to isogeny) into
a product of elliptic curves is 26 --- a bound that is
attained by a certain model of the modular curve X(11).

Serre curves in one-parameter families

Speaker: 

David Grant

Institution: 

University of Colorado

Time: 

Saturday, November 7, 2009 - 10:00am

Location: 

RH 101

Serre famously proved that for elliptic curves $E$ over number fields $k$ without complex multiplication, the galois group $H$ of the field generated over $k$ by all the torsion points $E_{\text{tor}}$ of $E$ is a subgroup of finite index in $G=\displaystyle\lim_{\leftarrow\atop n} \text{GL}_2(\Bbb Z/n\Bbb Z)$. When $k=\Bbb Q$, the smallest the index of $H$ in $G$ can be is 2, and if it is, we say $E$ is a Serre curve over $\Bbb Q$. Now let $E$ be an elliptic curve over $\Bbb Q(t)$. So long as the galois group generated over $\Bbb Q(t)$ by $E_{\text{tor}}$ is all of $G$, ``almost all" specializations $t_0$ of $t$ in $\Bbb Q$ give rise to elliptic curves $E_{t_0}$ which are Serre curves, and if we consider those $t_0$ of height bounded by some $B$, we give bounds for the number of $E_{t_0}$ which are not Serre curves in terms of $B$.

Hybrid continuum-discrete multiscale models of vascular tumor growth

Speaker: 

John Lowengrub

Institution: 

UC Irvine Math

Time: 

Monday, November 9, 2009 - 4:00pm

Location: 

RH 306

We present a hybrid continuum-discrete mathematical multiscale modeling framework for vascular solid tumor growth. A continuum model based on mixture theory is used to describe the evolution of volume fractions of multiple tumor cell clones, extracellular matrix (ECM), host cells, nutrients and water. The continuum model is coupled to an agent-based, lattice-free model that describes the evolution of discrete tumor cell clones. The models couple through mass and momentum exchange. In addition, a model for tumor-induced angiogenesis and vascular growth is incorporated. Simulations are presented that demonstrate the effectiveness of the hybrid approach in describing hypoxia-induced transitions from collective to individual based motion and vice-versa when cells are in oxygen-rich environments near the flowing, dynamic neovasculature network.

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