Gluing constructions in differential geometry

Speaker: 

Nicolaos Kapouleas

Institution: 

Brown University

Time: 

Thursday, October 22, 2015 - 4:00pm to 5:00pm

Host: 

Location: 

Rowland Hall 306

Abstract:

I will discuss various geometric gluing constructions. First I will discuss constructions for Constant Mean Curvature hypersurfaces in Euclidean spaces including my earlier work for two-surfaces in three-space which settled the Hopf conjecture for surfaces of genus two and higher, and recent generalizations in collaboration with Christine Breiner in all dimensions. I will then briefly mention gluing constructions in collaboration with Mark Haskins for special Lagrangian cones in Cn. A large part of my talk will concentrate on doubling and desingularization constructions for minimal surfaces and on applications on closed minimal surfaces in the round spheres, free boundary minimal surfaces in the unit ball, and self-shrinkers for the Mean Curvature flow. Finally I will discuss my collaboration with Simon Brendle on constructions for Einstein metrics on four-manifolds and related geometric objects.

Integral equation modeling for nonlocal diffusion and mechanics

Speaker: 

Max Gunzburger

Institution: 

Florida State University

Time: 

Thursday, December 3, 2015 - 4:00pm to 5:00pm

Host: 

Location: 

Rowland Hall 306

We use the canonical examples of fractional Laplacian and peridynamics equations to discuss their use as models for nonlocal diffusion and mechanics, respectively, via integral equations with singular kernels. We then proceed to discuss theories for the analysis and numerical analysis of the models considered, relying on a nonlocal vector calculus to define weak formulations in function space settings. In particular, we discuss the recently developed asymptotically compatible families of discretization schemes. Brief forays into examples and extensions are made, including obstacle problems and wave problems.

The Mathematical Connections of Juggling

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Thursday, November 12, 2015 - 3:00pm to 4:00pm

Location: 

Natural Science II 1201

In this co-sponsored UCI Illuminations and Juggle Buddies event, we will talk about the math theories associated with the art of juggling, a form of prop manipulation. This theory involves the use of Siteswap notation.

Siteswap is a juggling notation used to describe possible juggling patterns. For example, the most basic three-ball trick called a cascade can be written as "3" in this notation. Another juggling trick called a shower, where balles are thrown in a circular motion, is denoted by "51".

In this talk we start with the definition of Siteswap, and explain the beautiful mathematical theory beind it.

No background knowledge is required.

This event is free an open to the public. Free pizza will be served.

From quantum coin tossing to classical mechanics

Speaker: 

Michael Bjorklund

Institution: 

Chalmers University, Sweden

Time: 

Thursday, April 9, 2015 - 4:00pm to 5:00pm

Host: 

Location: 

Rowland Hall 306

After briefly reviewing some basic aspects of quantum
probability theory, especially questions surrounding a
celebrated theorem of Gleason, we turn to analogues of
quantum probabilities studied in symplectic geometry
known as quasi-states, which are functions on Lie algebras
which are linear on abelian sub-algebras.

A prototypical (and well-studied) example is the so called
Maslov index. We will discuss the existence of non-linear quasi-states on
various families of finite-dimensional Lie algebras. No fluency
in the language of Lie algebras will be assumed.

Joint work with Tobias Hartnick (Technion).

Mean Field Games and the Search for Large Population Dynamic Equilibria

Speaker: 

R. Carmona

Institution: 

Princeton University

Time: 

Thursday, April 2, 2015 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

After discussing a few examples of herding and flocking, we review the mean field game paradigm as introduced by Lasry and Lions. Using a probabilistic reformulation of the problem, we demonstrate how the solutions of these models can be identified with solutions of forward - backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. We give existence and uniqueness results for a large class of these FBSDEs and if time permits, we discuss the similarities and differences with the solutions of the optimal control of McKean-Vlasov stochastic differential equations

Phyllotaxis: Some progress, but a story far from over

Speaker: 

Alan Newell

Institution: 

University of Arizona

Time: 

Thursday, March 5, 2015 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Phyllotaxis, the arrangement of phylla (leaves, bracts, seeds) near the shoot apical meristems of plants has intrigued and mystified natural scientists for over two thousand years. It is surprising that only within the last two decades have quantitative explanations emerged that describe the wonderful architectures which are observed. I will give an overview of two types of explanation, teleological and mechanistic, one based on rules which posit that each new phyllo be placed according to some optimal packing principle and the other which uses plain old biophysics and biochemistry to build mechanistic models which lead to pattern forming pde's. One of the stunning new results is that, while the latter is richer, both approaches lead to completely consistent results. This may well have broader ramifications in that it suggests that nature may use instability driven patterns to achieve optimal outcomes.

The talk should be accessible to students and colleagues in other disciplines.

On Water Waves with Angled Crests

Speaker: 

Sijue Wu

Institution: 

University of Michigan

Time: 

Friday, February 6, 2015 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

We consider the two-dimensional water wave problem in the case where the free interface of the fluid meets a vertical wall at a possibly non-trivial angle; our problem also covers interfaces with angled crests. We assume that the fluid is inviscid, incompressible, and irrotational, with no surface tension and with air density zero. We construct a low-regularity energy and prove a closed energy estimate for this problem, and we show that the two-dimensional water wave problem is solvable locally in time in this framework. Our work differs from earlier work in that, in our case, only a degenerate Taylor stability criterion holds, with $-\frac{\partial P}{\partial \bold{n}} \ge 0$, instead of the strong Taylor stability criterion $-\frac{\partial P}{\partial \bold{n}} \ge c > 0$. This work is partially joint with Rafe Kinsey.

Rigidity of local holomorphic isometries from a Kahler manifold to the product of of complex projective spaces

Speaker: 

Xiaojun Huang

Institution: 

Rutgers University

Time: 

Thursday, February 26, 2015 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

We discuss the global property of a local holomorphic isometry into the product of projective spaces. We prove global extension and rigidity properties for such a map when the source is a Hermitian symmetric space  of compact type. Our work is along the lines of the previous work of Calabi, Clozel-Ullmo and Mok.
This is a joint work with Yuan Yuan from Syracuse University

Modern Optimization Meets Physics: Recent Progress on the Phase Retrieval Problem

Speaker: 

Emmanuel Candes

Institution: 

Stanford University

Time: 

Thursday, November 6, 2014 - 4:00pm to 5:00pm

Location: 

Natural Sciences II Room 1201

In many imaging problems such as X-ray crystallography, detectors can only record the intensity or magnitude of a diffracted wave as opposed to measuring its phase.  Phase retrieval concerns the recovery of an image from such phaseless information.  Although this problem is in general combinatorially hard, it is of great importance because it arises in many applications ranging from astronomical imaging to speech analysis. This talk discusses novel acquisition strategies and novel convex and non-convex algorithms which are provably exact, thereby allowing perfect phase recovery from a minimal number of noiseless and intensity-only measurements. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. This may be of special contemporary interest because phase retrieval is at the center of spectacular current research efforts collectively known under the name of coherent diffraction imaging aimed, among other things, at determining the 3D structure of large protein complexes.  

 

Pages

Subscribe to RSS - Colloquium