## Seminars and Colloquia by Series

### Approximate Schauder Frames for Banach Sequence Spaces

Series
Dissertation Defense
Time
Friday, April 16, 2021 - 16:00 for 1.5 hours (actually 80 minutes)
Location
ONLINE
Speaker
Yam-Sung ChengGeorgia Institute of Technology

The main topics of this thesis concern two types of approximate Schauder frames for the Banach sequence space $\ell_1$. The first main topic pertains to finite-unit norm tight frames (FUNTFs) for the finite-dimensional real sequence space $\ell_1^n$. We prove that for any $N \geq n$, FUNTFs of length $N$ exist for real $\ell_1^n$. To show the existence of FUNTFs, specific examples are constructed for various lengths. These constructions involve repetitions of frame elements. However, a different method of frame constructions allows us to prove the existence of FUNTFs for real $\ell_1^n$ of lengths $2n-1$ and $2n-2$ that do not have repeated elements.

The second main topic of this thesis pertains to normalized unconditional Schauder frames for the sequence space $\ell_1$. A Schauder frame provides a reconstruction formula for elements in the space, but need not be associated with a frame inequality. Our main theorem on this topic establishes a set of conditions under which an $\ell_1$-type of frame inequality is applicable towards unconditional Schauder frames. A primary motivation for choosing this set of hypotheses involves appropriate modifications of the Rademacher system, a version of which we prove to be an unconditional Schauder frame that does not satisfy an $\ell_1$-type of frame inequality.

The final topic of our thesis introduces the concept of $\ell_1$-boundedness in the Hilbert sequence space $\ell_2$. We prove various properties of $\ell_1$-bounded sets of $\ell_2$ and state some potentially mathematically significant open problems in this subject area.

### Mathematical and Data-driven Pattern Representation with Applications in Image Processing, Computer Graphics, and Infinite Dimensional Dynamical Data Mining

Series
Dissertation Defense
Time
Friday, April 9, 2021 - 10:00 for 1.5 hours (actually 80 minutes)
Location
Online
Speaker
Yuchen (Roy) HeSchool of Math, Georgia Institute of Technology

Patterns represent the spatial or temporal regularities intrinsic to various phenomena in nature, society, art, and science. From rigid ones with well-defined generative rules to flexible ones implied by unstructured data, patterns can be assigned to a spectrum. On one extreme, patterns are completely described by algebraic systems where each individual pattern is obtained by repeatedly applying simple operations on primitive elements. On the other extreme, patterns are perceived as visual or frequency regularities without any prior knowledge of the underlying mechanisms. In this presentation, we aim at demonstrating some mathematical techniques for representing patterns traversing the aforementioned spectrum, which leads to qualitative analysis of the patterns’ properties and quantitative prediction of the modeled behaviors from various perspectives. We investigate lattice patterns from material science, shape patterns from computer graphics, submanifold patterns encountered in point cloud processing, color perception patterns applied in underwater image processing, dynamic patterns from spatial-temporal data, and low-rank patterns exploited in medical image reconstruction. For different patterns and based on their dependence on structured or unstructured data, we introduce suitable mathematical representations using techniques ranging from group theory to deep neural networks.

Join Zoom Meeting

https://zoom.us/j/97642529845?pwd=aS9aTGloQnBGVVNQMHd6d0I4eGFNQT09

Meeting ID: 976 4252 9845

Passcode: 42PzXb

### On the stationary/uniformly rotating solutions of active scalar euquations

Series
Dissertation Defense
Time
Tuesday, April 6, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Jaemin ParkGeorgia tech

We study qualitative and quantitative properties of stationary/uniformly-rotating solutions of the 2D incompressible Euler equation.

For qualitative properties, we aim to establish sufficient conditions for such solutions to be radially symmetric. The proof  is based on variational argument, using the fact that a uniformly-rotating solution can be formally thought of as  a critical point of an energy functional. It turns out that if positive vorticity is rotating with angular velocity, not in (0,1/2), then the corresponding energy functional has a unique critical point, while radial ones are always critical points. We apply similar ideas to more general active scalar equations (gSQG) and vortex sheet equation. We also prove that for rotating vortex sheets, there exist  non-radial rotating vortex sheets, bifurcating from radial ones. This work is based on the joint work with Javier Gomez-Serrano, Jia Shi and Yao Yao.

It is well-known that there are non-radial rotating patches with angular velocity in (0,1/2). Using the variational argument, we derive some quantitative estimates for their angular velocities and the difference from the radial ones.

### Numerical Estimation of Several Topological Quantities of the First Passage Percolation Model

Series
Dissertation Defense
Time
Monday, April 5, 2021 - 13:00 for 2 hours
Location
ONLINE
Speaker
Yuanzhe MaGeorgia Institute of Technology

In this thesis, our main goal is to use numerical simulations to study some quantities related to the growing set B(t). Motivated by prior works, we mainly study quantities including the boundary size, the hole size, and the location of each hole for B(t). We discuss the theoretical background of this work, the algorithm we used to conduct simulations, and include an extensive discussion of our simulation results. Our results support some of the prior conjectures and further introduce several interesting open problems.

This defense will be conducted on bluejeans, at https://bluejeans.com/611615950.

### Branched Covers and Braided Embeddings

Series
Dissertation Defense
Time
Friday, March 26, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Sudipta KolayGeorgia Tech

We study braided embeddings, which is a natural generalization of closed braids in three dimensions. Braided embeddings give us an explicit way to construct lots of higher dimensional embeddings; and may turn out to be as instrumental in understanding higher dimensional embeddings as closed braids have been in understanding three and four dimensional topology. We will discuss two natural questions related to braided embeddings, the isotopy and lifting problem.

### Interaction energies, lattices, and designs

Series
Dissertation Defense
Time
Wednesday, May 13, 2020 - 13:30 for 1 hour (actually 50 minutes)
Location
Bluejeans: https://bluejeans.com/9024318866/
Speaker
Josiah ParkGeorgia Tech

This thesis has four chapters. The first three concern the location of mass on spheres or projective space, to minimize energies. For the Columb potential on the unit sphere, this is a classical problem, related to arranging electrons to minimize their energy. Restricting our potentials to be polynomials in the squared distance between points, we show in the Chapter 1 that there exist discrete minimal energy distributions. In addition we pose a conjecture on discreteness of minimizers for another class of energies while showing these minimizers must have empty interior.

In Chapter 2, we discover that highly symmetric distributions of points minimize energies over probability measures for potentials which are completely monotonic up to some degree, guided by the work of H. Cohn and A. Kumar. We make conjectures about optima for a class of energies calculated by summing absolute values of inner products raised to a positive power. Through reformulation, these observations give rise to new mixed-volume inequalities and conjectures. Our numerical experiments also lead to discovery of a new highly symmetric complex projective design which we detail the construction for. In this chapter we also provide details on a computer assisted argument which shows optimality of the $600$-cell for such energies (via interval arithmetic).

In Chapter 3 we also investigate energies having minimizers with a small number of distinct inner products. We focus here on discrete energies, confirming that for small $p$ the repeated orthonormal basis minimizes the $\ell_p$-norm of the inner products out of all unit norm configurations. These results have analogs for simplices which we also prove.

Finally, in Chapter 4 we show that real tight frames that generate lattices must be rational, and that the same holds for other vector systems with structured matrices of outer products. We describe a construction of lattices from distance transitive graphs which gives rise to strongly eutactic lattices. We discuss properties of this construction and also detail potential applications of lattices generated by incoherent systems of vectors.

### Rayleigh-Taylor instability with heat transfer

Series
Dissertation Defense
Time
Saturday, May 9, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Qianli HuGeorgia Tech

In this thesis, the Rayleigh-Taylor instability effects in the setting of the Navier-Stokes equations, given some three-dimensional and incompressible fluids, are investigated. The existence and the uniqueness of the temperature variable in the the weak form is established under suitable initial and boundary conditions, and by the contraction mapping principle we investigate further the conditions for the solution to belong to some continuous class; then a positive minimum temperature result can be proved, and with the aid of the RT instability effect in the density and the velocity, the instability for the temperature is established.

### Numerical Estimates for Arm Exponents and the Acceptance Profile of Invasion Percolation

Series
Dissertation Defense
Time
Thursday, April 23, 2020 - 14:00 for 2 hours
Location
Online via BlueJeans: https://bluejeans.com/127628065?src=calendarLink
Speaker
Jiaheng LiSchool of Mathematics

The main work of this thesis is to numerically estimate some conjectured arm exponents when there exist certain number of open paths and closed dual paths that extend to the boundary of a box of sidelength N centering at the origin in bond invasion percolation on a plane square lattice by Monte-Carlo simulations. The result turns out to be supportive for the conjectured value. The numerical estimate for the acceptance profile of invasion percolation at the critical point is also obtained. An efficient algorithm to simulate invasion percolation and to find disjoint paths on most regular 2-dimensional lattices are also discussed.

### Finding and cerifying roots of systems of equations

Series
Dissertation Defense
Time
Tuesday, April 21, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://gatech.bluejeans.com/481175204
Speaker
Kisun LeeGeorgia Tech

Numerical algebraic geometry studies methods to approach problems in algebraic geometry numerically. Especially, finding roots of systems of equations using theory in algebraic geometry involves symbolic algorithm which requires expensive computations, numerical techniques often provides faster methods to tackle these problems. We establish numerical techniques to approximate roots of systems of equations and ways to certify its correctness.

As techniques for approximating roots of systems of equations, homotopy continuation method will be introduced. Combining homotopy method with monodromy group action, we introduce techniques for solving parametrized polynomial systems. Since numerical approaches rely on heuristic method, we study how to certify numerical roots of systems of equations. Based on Newton’s method, we study Krawczyk method and Smale’s alpha theory. These two method will be mainly used for certifying regular roots of systems. Furthermore, as an approach for multiple roots, we establish the local separation bound of a multiple root. For multiple roots whose deflation process terminates by only one iteration, we give their local separation bound and study how to certify an approximation of such multiple roots.

### Spectrum Reconstruction Technique and Improved Naive Bayes Models for Text Classification Problems

Series
Dissertation Defense
Time
Thursday, April 16, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Bluejeans Meeting 866242745
Speaker
Zhibo DaiGeorgia Tech

My thesis studies two topics. In the first part, we study the spectrum reconstruction technique. As is known to all, eigenvalues play an important role in many research fields and are foundation to many practical techniques such like PCA (Principal Component Analysis). We believe that related algorithms should perform better with more accurate spectrum estimation. There was an approximation formula proposed by Prof. Matzinger. However, they didn't give any proof. In our research, we show why the formula works. And when both number of features and dimension of space go to infinity, we find the order of error for the approximation formula, which is related to a constant C-the ratio of dimension of space and number of features.

In the second part, we focus on some applications of Naive Bayes models in text classification problems. Especially we focus on two special situations: 1) there is insufficient data for model training; 2) partial labeling problem. We choose Naive Bayes as our base model and do some improvement on the model to achieve better performance in those two situations. To improve model performance and to utilize as many information as possible, we introduce a correlation factor, which somehow relaxes the conditional independence assumption of Naive Bayes. The new estimates are biased estimation compared to the traditional Naive Bayes estimate, but have much smaller variance, which give us a better prediction result.