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Series: Dissertation Defense

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.

Tuesday, June 25, 2019 - 14:00 ,
Location: Skiles 006 ,
Igor Belykh ,
Georgia State ,
ibelykh@gsu.edu ,
Organizer: Rachel Kuske

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.

Friday, June 21, 2019 - 14:00 ,
Location: Skiles 317 ,
Joan Gimeno ,
Universitat de Barcelona (BGSMath) ,
joan@maia.ub.es ,
Organizer: Yian Yao

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.

Series: Dissertation Defense

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.

Series: Geometry Topology Seminar

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.
Wednesday, May 29, 2019 - 14:00 ,
Location: Skiles 006 ,
Paolo Aceto ,
University of Oxford ,
paoloaceto@gmail.com ,
Organizer: JungHwan Park
Wednesday, May 15, 2019 - 14:00 ,
Location: Skile 005 ,
Roger Casals ,
UC Davis ,
Organizer:
Monday, May 13, 2019 - 14:00 ,
Location: Skiles 006 ,
Inna Zakharevich ,
Cornell ,
Organizer: Kirsten Wickelgren

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.

Thursday, May 30, 2019 - 14:00 ,
Location: Skiles 005 ,
Rafael de la Llave ,
Georigia Inst. of Technology ,
Organizer:

he KAM (Kolmogorov Arnold and Moser) theory studies

the persistence of quasi-periodic solutions under perturbations.

It started from a basic set of theorems and it has grown

into a systematic theory that settles many questions.

The basic theorem is rather surprising since it involves delicate

regularity properties of the functions considered, rather

subtle number theoretic properties of the frequency as well

as geometric properties of the dynamical systems considered.

In these lectures, we plan to cover a complete proof of

a particularly representative theorem in KAM theory.

In the first lecture we will cover all the prerequisites

(analysis, number theory and geometry). In the second lecture

we will present a complete proof of Moser's twist map theorem

(indeed a generalization to more dimensions).

The proof also lends itself to very efficient numerical algorithms.

If there is interest and energy, we will devote a third lecture

to numerical implementations.

Series: Geometry Topology Seminar

We prove that every rational homology cobordism class in the subgroup generated by lens spaces contains a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As a consequence we show that several natural maps to the rational homology cobordism group have infinite rank cokernels, and obtain a divisibility condition between the determinants of certain 2-bridge knots and other knots in the same concordance class. This is joint work with Daniele Celoria and JungHwan Park.

Wednesday, May 29, 2019 - 14:00 ,
Location: Skiles 005 ,
Rafael de la Llave ,
Georgia Institute of Technology ,
Organizer: Yian Yao

The KAM (Kolmogorov Arnold and Moser) theory studies

the persistence of quasi-periodic solutions under perturbations.

It started from a basic set of theorems and it has grown

into a systematic theory that settles many questions.

The basic theorem is rather surprising since it involves delicate

regularity properties of the functions considered, rather

subtle number theoretic properties of the frequency as well

as geometric properties of the dynamical systems considered.

In these lectures, we plan to cover a complete proof of

a particularly representative theorem in KAM theory.

In the first lecture we will cover all the prerequisites

(analysis, number theory and geometry). In the second lecture

we will present a complete proof of Moser's twist map theorem

(indeed a generalization to more dimensions).

The proof also lends itself to very efficient numerical algorithms.

If there is interest and energy, we will devote a third lecture

to numerical implementations.

Series: Geometry Topology Seminar

In this talk, I will discuss progress in our understanding of Legendrian surfaces. First, I will present a new construction of Legendrian surfaces and a direct description for their moduli space of microlocal sheaves. This Legendrian invariant will connect to classical incidence problems in algebraic geometry and the study of flag varieties, which we will study in detail. There will be several examples during the talk and, in the end, I will indicate the relation of this theory to the study of framed local systems on a surface. This talk is based on my work with E. Zaslow.

Series: Geometry Topology Seminar

One of the classical problems in scissors congruence is

this: given two polytopes in $n$-dimensional Euclidean space, when is

it possible to decompose them into finitely many pieces which are

pairwise congruent via translations? A complete set of invariants is

provided by the Hadwiger invariants, which measure "how much area is

pointing in each direction." Proving that these give a complete set

of invariants is relatively straightforward, but determining the

relations between them is much more difficult. This was done by

Dupont, in a 1982 paper. Unfortunately, this result is difficult to

describe and work with: it uses group homological techniques which

produce a highly opaque formula involving twisted coefficients and

relations in terms of uncountable sums. In this talk we will discuss

a new perspective on Dupont's proof which, together with more

topological simplicial techniques, simplifies and clarifies the

classical results. This talk is partially intended to be an

advertisement for simplicial techniques, and will be suitable for

graduate students and others unfamiliar with the approach.