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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
Thursday, May 9, 2019 - 11:00 ,
Location: ISyE Main 228 ,
Evdokia Nikolova ,
ECE, UT Austin ,
Nikolova@austin.utexas.edu ,
Organizer:
Friday, May 3, 2019 - 10:00 ,
Location: Skiles 005 ,
Sergio Mayorga ,
Graduate student ,
smayorga3@gatech.edu ,
Organizer: Sergio Mayorga
Thursday, May 2, 2019 - 13:30 ,
Location: Skiles 006 ,
Bounghun Bock ,
Georgia Tech ,
bbock3@gatech.edu ,
Organizer: Bounghun Bock

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.

Series: ACO Student Seminar

In network routing users often tradeoff different objectives in selecting their best route. An example is transportation networks, where due to uncertainty of travel times, drivers may tradeoff the average travel time versus the variance of a route. Or they might tradeoff time and cost, such as the cost paid in tolls.

We wish to understand the effect of two conflicting criteria in route selection, by studying the resulting traffic assignment (equilibrium) in the network. We investigate two perspectives of this topic: (1) How does the equilibrium cost of a risk-averse population compare to that of a risk-neutral population? (i.e., how much longer do we spend in traffic due to being risk-averse) (2) How does the equilibrium cost of a heterogeneous population compare to that of a comparable homogeneous user population?

We provide characterizations to both questions above.

Based on joint work with Richard Cole, Thanasis Lianeas and Nicolas Stier-Moses.

At the end I will mention current work of my research group on algorithms and mechanism design for power systems.

**Biography: ** Evdokia Nikolova is an Assistant Professor in the Department of Electrical and Computer Engineering at the University of Texas at Austin, where she is a member of the Wireless Networking & Communications Group. Previously she was an Assistant Professor in Computer Science and Engineering at Texas A&M University. She graduated with a BA in Applied Mathematics with Economics from Harvard University, MS in Mathematics from Cambridge University, U.K. and Ph.D. in Computer Science from MIT.

Nikolova's research aims to improve the design and efficiency of complex systems (such as networks and electronic markets), by integrating stochastic, dynamic and economic analysis. Her recent work examines how human risk aversion transforms traditional computational models and solutions. One of her algorithms has been adapted in the MIT CarTel project for traffic-aware routing. She currently focuses on developing algorithms for risk mitigation in networks, with applications to transportation and energy. She is a recipient of an NSF CAREER award and a Google Faculty Research Award. Her research group has been recognized with a best student paper award and a best paper award runner-up. She currently serves on the editorial board of the journal Mathematics of Operations Research.

Series: Dissertation Defense

For a first order (deterministic) mean-field game with non-local running and initial couplings, a classical solution is constructed for the associated, so-called master equation, a partial differential equation in infinite-dimensional space with a non-local term, assuming the time horizon is sufficiently small and the coefficients are smooth enough, without convexity conditions on the Hamiltonian.

Series: Dissertation Defense

In independent bond percolation with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite cluster? Grimmett-Holroyd-Kozma used the triangle condition to show that for d > 18, the set of such p contains values strictly larger than the percolation threshold pc. With the work of Fitzner-van der Hofstad, this has been reduced to d > 10. We reprove this result by showing that for d > 10 and some p>pc, there are infinite paths consisting of "shielded"' vertices --- vertices all whose adjacent edges are closed --- which must be in the complement of the infinite cluster. Using numerical values of pc, this bound can be reduced to d > 7. Our methods are elementary and do not require the triangle condition.

Invasion percolation is a stochastic growth model that follows a greedy algorithm. After assigning i.i.d. uniform random variables (weights) to all edges of d-dimensional space, the growth starts at the origin. At each step, we adjoin to the current cluster the edge of minimal weight from its boundary. In '85, Chayes-Chayes-Newman studied the "acceptance profile"' of the invasion: for a given p in [0,1], it is the ratio of the expected number of invaded edges until time n with weight in [p,p+dp] to the expected number of observed edges (those in the cluster or its boundary) with weight in the same interval. They showed that in all dimensions, the acceptance profile an(p) converges to one for p<pc and to zero for p>pc. In this paper, we consider an(p) at the critical point p=pc in two dimensions and show that it is bounded away from zero and one as n goes to infinity.