Seminars and Colloquia by Series

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
https://bluejeans.com/603353375/4347?src=calendarLink
Speaker
Qianli HuGeorgia Tech

Please Note: Online at https://bluejeans.com/603353375/4347?src=calendarLink

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.

The Maxwell-Pauli Equations

Series
Dissertation Defense
Time
Tuesday, March 10, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Forrest KiefferGeorgia Institute of Technology

Please Note: Thesis Defense

Energetic stability of matter in quantum mechanics, which refers to the ques-
tion of whether the ground state energy of a many-body quantum mechanical
system is finite, has long been a deep question of mathematical physics. For a
system of many non-relativistic electrons interacting with many nuclei in the
absence of electromagnetic fields this question traces back to the seminal work
of Tosio Kato in 1951 and Freeman Dyson and Andrew Lenard in 1967/1968.
In particular, Dyson and Lenard showed the ground state energy of the many-
body Schrödinger Hamiltonian is bounded below by a constant times the total
particle number, regardless of the size of the nuclear charges. This says such a
system is energetically stable (of the second kind). This situation changes dra-
matically when electromagnetic fields and spin interactions are present in the
problem. Even for a single electron with spin interacting with a single nucleus
of charge $Z > 0$ in an external magnetic field, Jurg Fröhlich, Elliot Lieb, and
Michael Loss in 1986 showed that there is no ground state energy if $Z$ exceeds
a critical charge $Z_c$ and the ground state energy exists if $Z < Z_c$ . In other
words, if the nuclear charge is too large, the one-electron atom is energetically
unstable.


Another notion of stability in quantum mechanics is that of dynamic stabil-
ity, which refers to the question of global well-posedness for a system of partial
differential equations that models the dynamics of many electrons coupled to
their self-generated electromagnetic field and interacting with many nuclei. The
central motivating question of our PhD thesis is whether energetic stability has
any influence over dynamic stability. Concerning this question, we study the
quantum mechanical many-body problem of $N \geq 1$ non-relativistic electrons with
spin interacting with their self-generated classical electromagnetic field and $K \geq 0$
static nuclei. We model the dynamics of the electrons and their self-generated
electromagnetic field using the so-called many-body Maxwell-Pauli equations.
The main result presented is the construction time global, finite-energy, weak
solutions to the many-body Maxwell-Pauli equations under the assumption that
the fine structure constant $\alpha$ and the nuclear charges are sufficiently small to
ensure energetic stability of this system. This result represents an initial step
towards understanding the relationship between energetic stability and dynamic
stability. If time permits, we will discuss several open problems that remain.


Committee members: Prof. Michael Loss (Advisor, School of Mathematics,
Georgia Tech), Prof. Brian Kennedy (School of Physics, Georgia Tech), Prof.
Evans Harrell (School of Mathematics, Georgia Tech), Prof. Federico Bonetto
(School of Mathematics, Georgia Tech), Prof. Chongchun Zeng (School of Math-
ematics, Georgia Tech).

Small torsion generating sets for mapping class groups

Series
Dissertation Defense
Time
Monday, March 9, 2020 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Justin LanierGeorgia Tech

A surface of genus $g$ has many symmetries. These form the surface’s mapping class group $Mod(S_g)$, which is finitely generated. The most commonly used generating sets for $Mod(S_g)$ are comprised of infinite order elements called Dehn twists; however, a number of authors have shown that torsion generating sets are also possible. For example, Brendle and Farb showed that $Mod(S_g)$ is generated by six involutions for $g \geq 3$. We will discuss our extension of these results to elements of arbitrary order: for $k > 5$ and $g$ sufficiently large, $Mod(S_g)$ is generated by three elements of order $k$. Generalizing this idea, in joint work with Margalit we showed that for $g \geq 3$ every nontrivial periodic element that is not a hyperelliptic involution normally generates $Mod(S_g)$. This result raises a question: does there exist an $N$, independent of $g$, so that if $f$ is a periodic normal generator of $Mod(S_g)$, then $Mod(S_g)$ is generated by $N$ conjugates of $f$? We show that in general there does not exist such an $N$, but that there does exist such a universal bound for the class of non-involution normal generators.

The proxy point method for rank-structured matrices

Series
Dissertation Defense
Time
Friday, October 25, 2019 - 13:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 311
Speaker
Xin XingSchool of Mathematics, Georgia Tech

Rank-structured matrix representations, e.g., H2 and HSS, are commonly used to reduce computation and storage cost for dense matrices defined by interactions between many bodies. The main bottleneck for their applications is the expensive computation required to represent a matrix in a rank-structured matrix format which involves compressing specific matrix blocks into low-rank form.
We focus on the study and application of a class of hybrid analytic-algebraic compression methods, called the proxy point method. We address several critical problems concerning this underutilized method which limit its applicability. A general form of the method is proposed, paving the way for its wider application in the construction of different rank-structured matrices with kernel functions that are more general than those usually used. Further, we extend the applicability of the proxy point method to compress matrices defined by electron repulsion integrals, which accelerates one of the main computational steps in quantum chemistry. 

Committee members: Prof. Edmond Chow (Advisor, School of CSE, Georgia Tech), Prof. David Sherrill (School of Chemistry and Biochemistry, Georgia Tech), Prof. Jianlin Xia (Department of Mathematics, Purdue University), Prof. Yuanzhe Xi (Department of Mathematics, Emory University), and Prof. Haomin Zhou (School of Mathematics, Georgia Tech).

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