Seminars and Colloquia by Series

Topics On the Length of the Longest Common Subsequences With Blocks In Binary Random Words

Series
Dissertation Defense
Time
Thursday, August 8, 2019 - 13:00 for
Location
Skiles 246
Speaker
Yuze Zhang – Georgia Institute of Technology –

The study of LIn, the length of the longest increasing subsequences, and of LCIn, the length of the longest common and increasing subsequences in random words is classical in computer science and bioinformatics, and has been well explored over the last few decades. This dissertation studies a generalization of LCIn for two binary random words, namely, it analyzes the asymptotic behavior of LCbBn, the length of the longest common subsequences containing a fixed number, b, of blocks. We first prove that after proper centerings and scalings, LCbBn, for two sequences of i.i.d. Bernoulli random variables with possibly two different parameters, converges in law towards limits we identify. This dissertation also includes an alternative approach to the one-sequence LbBn problem, and Monte-Carlo simulations on the asymptotics of LCbBn and on the growth order of the limiting functional, as well as several extensions of the LCbBn problem to the Markov context and some connection with percolation theory.

Compactness and singularity related to harmonic maps

Series
PDE Seminar
Time
Friday, July 26, 2019 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jiayu Li – University of Science and Technology of China –

In this talk we will review compactness results and singularity theorems related to harmonic maps. We first talk about maps from Riemann surfaces with tension fields bounded in a local Hardy space, then talk about stationary harmonic maps from higher dimensional manifolds, finally talk about heat flow of harmonic maps.

Quantum torus methods for Kauffman bracket skein modules

Series
Dissertation Defense
Time
Friday, July 26, 2019 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Jonathan Paprocki – Georgia Institute of Technology –

We investigate aspects of Kauffman bracket skein algebras of surfaces and modules of 3-manifolds using quantum torus methods. These methods come in two flavors: embedding the skein algebra into a quantum torus related to quantum Teichmuller space, or filtering the algebra and obtaining an associated graded algebra that is a monomial subalgebra of a quantum torus. We utilize the former method to generalize the Chebyshev homomorphism of Bonahon and Wong between skein algebras of surfaces to a Chebyshev-Frobenius homomorphism between skein modules of marked 3-manifolds, in the course of which we define a surgery theory, and whose image we show is either transparent or (skew)-transparent. The latter method is used to show that skein algebras of surfaces are maximal orders, which implies a refined unicity theorem, shows that SL_2C-character varieties are normal, and suggests a conjecture on how this result may be utilized for topological quantum compiling.

On the Independent Spanning Tree Conjectures and Related Problems

Series
Dissertation Defense
Time
Wednesday, July 10, 2019 - 10:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Alexander Hoyer – Georgia Institute of Technology –

We say that trees with common root are (edge-)independent if, for any vertex in their intersection, the paths to the root induced by each tree are internally (edge-)disjoint. The relationship between graph (edge-)connectivity and the existence of (edge-)independent spanning trees is explored. The (Edge-)Independent Spanning Tree Conjecture states that every k-(edge-)connected graph has k-(edge-)independent spanning trees with arbitrary root.

We prove the case k=4 of the Edge-Independent Spanning Tree Conjecture using a graph decomposition similar to an ear decomposition, and give polynomial-time algorithms to construct the decomposition and the trees. We provide alternate geometric proofs for the cases k=3 of both the Independent Spanning Tree Conjecture and Edge-Independent Spanning Tree Conjecture by embedding the vertices or edges in a 2-simplex, and conjecture higher-dimension generalizations. We provide a partial result towards a generalization of the Independent Spanning Tree Conjecture, in which local connectivity between the root and a vertex set S implies the existence of trees whose independence properties hold only in S. Finally, we prove and generalize a theorem of Györi and Lovász on partitioning a k-connected graph, and give polynomial-time algorithms for the cases k=2,3,4 using the graph decompositions used to prove the corresponding cases of the Independent Spanning Tree Conjecture.

Lattice points, zonotopes, and oriented matroids

Series
Dissertation Defense
Time
Wednesday, July 3, 2019 - 11:00 for
Location
Skiles 006
Speaker
Marcel Celaya – Georgia Tech –

The first half of this dissertation concerns the following problem: Given a lattice in $\mathbf{R}^d$ which refines the integer lattice $\mathbf{Z}^d$, what can be said about the distribution of the lattice points inside of the half-open unit cube $[0,1)^d$? This question is of interest in discrete geometry, especially integral polytopes and Ehrhart theory. We observe a combinatorial description of the linear span of these points, and give a formula for the dimension of this span. The proofs of these results use methods from classical multiplicative number theory.

In the second half of the dissertation, we investigate oriented matroids from the point of view of tropical geometry. Given an oriented matroid, we describe, in detail, a polyhedral complex which plays the role of the Bergman complex for ordinary matroids. We show how this complex can be used to give a new proof of the celebrated Bohne-Dress theorem on tilings of zonotopes by zonotopes with an approach which relies on a novel interpretation of the chirotope of an oriented matroid.

On the Synchronization Myth for Lateral Pedestrian-Instability of Suspension Bridges

Series
Applied and Computational Mathematics Seminar
Time
Tuesday, June 25, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Igor Belykh – Georgia State –

The pedestrian-induced lateral oscillation of London's Millennium bridge on the day it opened in 2000 has become a much cited paradigm of an instability caused by phase synchronization of coupled oscillators. However, a closer examination of subsequent theoretical studies and experimental observations have brought this interpretation into question. 

To elucidate the true cause of instability, we study a model in which each pedestrian is represented by a simplified biomechanically-inspired two-legged inverted pendulum. The key finding is that synchronization between individual pedestrians is not a necessary ingredient of instability onset. Instead, the side-to-side pedestrian motion should on average lag that of the bridge oscillation by a fraction of a cycle. Using a multi-scale asymptotic analysis, we derive a mathematically rigorous general criterion for bridge instability based on the notion of effective negative damping. This criterion suggests that the initiation of wobbling is not accompanied by crowd synchrony and crowd synchrony is a consequence but not the cause of bridge instability.

On numerical integrators for state-dependent delay equations

Series
Dynamical Systems Working Seminar
Time
Friday, June 21, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 317
Speaker
Joan Gimeno – Universitat de Barcelona (BGSMath) –
Abstract: Many real-life phenomena in science can be modeled by an Initial Value Problem (IVP) for ODE's. To make the model more consistent with real phenomenon, it sometimes needs to include the dependence on past values of the state. Such models are given by retarded functional differential equations. When the past values depend on the state, the IVP is not always defined. Several examples illustrating the problems and methods to integrate IVP of these kind of differential equations are going to be explained in this talk.

Topics in Dynamical Systems

Series
Dissertation Defense
Time
Wednesday, June 5, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Longmei Shu – Georgia Inst. of Technology –

Isospectral reductions is a network/graph reduction that preserves the
eigenvalues and the eigenvectors of the adjacency matrix. We analyze the
conditions under which the generalized eigenvectors would be preserved and
simplify the proof of the preservation of eigenvectors. Isospectral reductions
are associative and form a dynamical system on the set of all matrices/graphs.
We study the spectral equivalence relation defined by specific characteristics
of nodes under isospectral reductions and show some examples of the attractors.
Cooperation among antigens, cross-immunoreactivity (CR) has been observed in
various diseases. The complex viral population dynamics couldn't be explained
by traditional math models. A new math model was constructed recently with
promising numerical simulations. In particular, the numerical results recreated
local immunodeficiency (LI), the phenomenon where some viruses sacrifice
themselves while others are not attacked by the immune system. Here we analyze
small CR networks to find the minimal network with a stable LI. We also
demonstrate that you can build larger CR networks with stable LI using this
minimal network as a building block.

Factorization homology: sigma-models as state-sum TQFTs.

Series
Geometry Topology Seminar
Time
Friday, May 31, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
David Ayala – Montana State University
Roughly, factorization homology pairs an n-category and an n-manifold to produce a vector space.  Factorization homology is to state-sum TQFTs as singular homology is to simplicial homology: the former is manifestly well-defined (ie, independent of auxiliary choices), continuous (ie, beholds a continuous action of diffeomorphisms), and functorial; the latter is easier to compute.  
 
Examples of n-categories to input into this pairing arise, through deformation theory, from perturbative sigma-models.  For such n-categories, this state sum expression agrees with the observables of the sigma-model — this is a form of Poincare’ duality, which yields some surprising dualities among TQFTs.  A host of familiar TQFTs are instances of factorization homology; many others are speculatively so.  
 
The first part of this talk will tour through some essential definitions in what’s described above.  The second part of the talk will focus on familiar manifold invariants, such as the Jones polynomial, as instances of factorization homology, highlighting the Poincare’/Koszul duality result.  The last part of the talk will speculate on more such instances.  

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