Seminars and Colloquia by Series

Stable phase retrieval for infinite dimensional subspaces of L_2(R)

Series
Analysis Seminar
Time
Wednesday, March 4, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel FreemanSt. Louis University

 The problem of phase retrieval for a set of functions $H$ can be thought of as being able to identify a function $f\in H$ or $-f\in H$ from the absolute value $|f|$.  Phase retrieval for a set of functions is called stable if when $|f|$ and $|g|$ are close then $f$ is proportionally close to $g$ or $-g$.  That is, we say that a set $H\subseteq L_2({\mathbb R})$ does stable phase retrieval if there exists a constant $C>0$ so that
$$\min\big(\big\|f-g\big\|_{L_2({\mathbb R})},\big\|f+g\big\|_{L_2({\mathbb R})}\big)\leq C \big\| |f|-|g| \big\|_{L_2({\mathbb R})} \qquad\textrm{ for all }f,g\in H.
$$
 It is known that phase retrieval for finite dimensional spaces is always stable.  On the other hand, phase retrieval for infinite dimensional spaces using a frame or a continuous frame is always unstable.  We prove that there exist infinite dimensional subspaces of $L_2({\mathbb R})$ which do stable phase retrieval.  This is joint work with Robert Calderbank, Ingrid Daubechies, and Nikki Freeman.

Optimization over the Diffeomorphism Group Using Riemannian BFGS with Application

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, March 4, 2020 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dr. Darshan Bryner Naval Surface Warfare Center, Panama City Division

Please Note: This is a part of IEEE Signal Processing Society Lecture Series, organized by Dr. Alessio Medda (alessiomedda@ieee.org). PLEASE RSVP to https://events.vtools.ieee.org/m/222947

The set of diffeomorphisms of the unit interval, or “warping functions,” plays an important role in many in functional data analysis applications. Most prominently, the problem of registering, or aligning, pairs of functions depends on solving for an element of the diffeomorphism group that, when applied to one function, optimally aligns it to the other.
The registration problem is posed as the unconstrained minimization of a cost function over the infinite dimensional diffeomorphism function space. We make use of its well-known Riemannian geometry to implement an efficient, limited memory Riemannian BFGS optimization scheme. We compare performance and results to the benchmark algorithm, Dynamic Programming, on several functional datasets. Additionally, we apply our methodology to the problem of non-parametric density estimation and compare to the benchmark performance of MATLAB’s built-in kernel density estimator ‘ksdensity’.

Invariant objects and Arnold diffusion. From theory to computation.

Series
Research Horizons Seminar
Time
Wednesday, March 4, 2020 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rafael de la LlaveGeorgia Tech

We consider the problem whether small perturbations of integrable mechanical systems can have very large effects.

Since the work of Arnold in 1964, it is known that there are situations where the perturbations can accumulate (Arnold diffusion). 

This can be understood by noting that the small perturbations generate some invariant objects in phase space that act as routes which allow accumulation of effects. 

We will present some rigorous results about geometric objects lead to Arnold diffusion as well as some computational tools that allow to find them in concrete applications.

Thanks to the work of many people, an area which used to be very speculative, is becoming an applicable tool.

Modeling Phytoplankton Blooms with a Reaction-Diffusion Predator-Prey Model

Series
Mathematical Biology Seminar
Time
Wednesday, March 4, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Seth CowallMercer University

Phytoplankton are the base of the marine food web. They are also responsible for much of the oxygen we breathe, and they remove carbon dioxide from the atmosphere. The mechanisms that govern the timing of seasonal phytoplankton blooms is one of the most debated topics in oceanography. Here, we present a macroscale plankton ecology model consisting of coupled, nonlinear reaction-diffusion equations with spatially and temporally changing coefficients to offer insight into the causes of phytoplankton blooms. This model simulates biological interactions between nutrients, phytoplankton and zooplankton. It also incorporates seasonally varying solar radiation, diffusion and depth of the ocean’s upper mixed layer because of their impact on phytoplankton growth. The model is analyzed using seasonal oceanic data with the goals of understanding the model’s dependence on its parameters and of understanding seasonal changes in plankton biomass. A study of varying parameter values and the resulting effects on the solutions, the stability of the steady-states, and the timing of phytoplankton blooms is carried out. The model’s simulated blooms result from a temporary attraction to one of the model’s steady-states.

Inverse Problems, Imaging and Tensor Decomposition

Series
Job Candidate Talk
Time
Tuesday, March 3, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joe KileelProgram in Applied and Computational Mathematics, Princeton University

Perspectives from numerical optimization and computational algebra are  
brought to bear on a scientific application and a data science  
application.  In the first part of the talk, I will discuss  
cryo-electron microscopy (cryo-EM), an imaging technique to determine  
the 3-D shape of macromolecules from many noisy 2-D projections,  
recognized by the 2017 Chemistry Nobel Prize.  Mathematically, cryo-EM  
presents a particularly rich inverse problem, with unknown  
orientations, extreme noise, big data and conformational  
heterogeneity. In particular, this motivates a general framework for  
statistical estimation under compact group actions, connecting  
information theory and group invariant theory.  In the second part of  
the talk, I will discuss tensor rank decomposition, a higher-order  
variant of PCA broadly applicable in data science.  A fast algorithm  
is introduced and analyzed, combining ideas of Sylvester and the power  
method.

Computation of invariants and Hankel index on a variety of almost minimal degree

Series
Student Algebraic Geometry Seminar
Time
Monday, March 2, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jaewoo JungGeorgia Tech

We have seen that Hankel index of varieties can be controlled by some invariants such as $$N_{2,p}$$ or p-base point free property. Also, we know that the Hankel index of (a linear join of) variety of minimal degree is infinity (and all invariant above are same as infinity). As next case, I will share some computations of invariants on a variety that projecting rational normal curve away from a point (which is a variety of almost minimal degree).

Joint UGA-GT Topology Seminar at UGA: The dihedral genus of a knot

Series
Geometry Topology Seminar
Time
Monday, March 2, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
Patricia CahnSmith College

We consider dihedral branched covers of $S^4$, branched along an embedded surface with one non-locally flat point, modelled on the cone on a knot $K\subset S^3$. Kjuchukova proved that the signature of this cover is an invariant $\Xi_p(K)$ of the $p$-colorable knot $K$. We prove that the values of $\Xi_p(K)$ fall in a bounded range for homotopy-ribbon knots. We also construct a family of (non-slice) knots for which the values of $\Xi_p$ are unbounded. More generally, we introduce the notion of the dihedral 4-genus of a knot, and derive a lower bound on the dihedral 4-genus of $K$ in terms of $\Xi_p(K)$. This work is joint with A. Kjuchukova.

Toric Vector Bundles and the tropical geometry of piecewise-linear functions

Series
Algebra Seminar
Time
Monday, March 2, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chris ManonUniversity of Kentucky

Like toric varieties, toric vector bundles are a rich class of algebraic varieties that can be described with combinatorial data.  Klyachko gave a classification of toric vector bundles in terms of certain systems of filtrations in a vector space.  I'll talk about some recent work with Kiumars Kaveh showing that Klyachko's data has an interesting interpretation in terms of tropical geometry.  In particular, we show that toric vector bundles can be classified by points on tropicalized linear spaces over a semifield of piecewise-linear functions.   I'll discuss how to use this recipe and a closely related tropicalization map to produce toric vector bundles and more general flat toric families.  

Joint UGA-GT Topology Seminar at UGA: Stein domains in complex two space with prescribed boundary

Series
Geometry Topology Seminar
Time
Monday, March 2, 2020 - 14:30 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
Bolent TosunUniversity of Alabama

A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, the existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. In this talk we will consider the existence of such structures in the ambient settings, that is, manifolds/domains with various degree of convexity as open/compact subsets of a complex manifold, e.g. complex 2-space C^2. In particular, I will discuss the following question: Which homology spheres embed in C^2 as the boundary of a Stein domain? This question was first considered and explored in detail by Gompf. At that time, he made a fascinating conjecture that no non-trivial Brieskorn homology sphere, with either orientation, embeds in C^2 as a Stein boundary. In this talk, I will survey what we know about this conjecture, and report on some closely related recent work in progress that ties to an interesting symplectic rigidity phenomena in low dimensions.

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