Seminars and Colloquia by Series

A dynamic view on the probabilistic method: random graph processes

Series
School of Mathematics Colloquium
Time
Thursday, May 7, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
https://gatech.bluejeans.com/344615810
Speaker
Lutz Warnke – Georgia Tech

 Random graphs are the basic mathematical models for large-scale disordered networks in many different fields (e.g., physics, biology, sociology).
Since many real world networks evolve over time, it is natural to study various random graph processes which arise by adding edges (or vertices) step-by-step in some random way.

The analysis of such random processes typically brings together tools and techniques from seemingly different areas (combinatorial enumeration, differential equations, discrete martingales, branching processes, etc), with connections to the analysis of randomized algorithms.
Furthermore, such processes provide a systematic way to construct graphs with "surprising" properties, leading to some of the best known bounds in extremal combinatorics (Ramsey and Turan Theory).

In this talk I shall survey several random graph processes of interest (in the context of the probabilistic method), and give a glimpse of their analysis.
If time permits, we shall also illustrate one of the main proof techniques (the "differential equation method") using a simple toy example.

Every surface is a leaf

Series
Geometry Topology Student Seminar
Time
Wednesday, May 6, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Justin Lanier – Georgia Tech –

Every closed 3-manifold admits foliations, where the leaves are surfaces. For a given 3-manifold, which surfaces can appear as leaves? Kerékjártó and Richards gave a classification up to homeomorphism of noncompact surfaces, which includes surfaces with infinite genus and infinitely many punctures. In their 1985 paper "Every surface is a leaf", Cantwell--Conlon prove that for every orientable noncompact surface L and every closed 3-manifold M, M has a foliation where L appears as a leaf. We will discuss their paper and construction and the surrounding context.

Constructive methods in KAM theory- from numerics to regularity

Series
CDSNS Colloquium
Time
Wednesday, May 6, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Attendee link: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Rafael de la Llave – Georgia Tech –

Please Note: This is the first installment of our CDSNS virtual colloquium, which will be held in a Bluejeans event space on Wednesdays at 9AM (EST).

We will present the "a-posteriori" approach to KAM theory.

We formulate an invariance equation and show that an approximate-enough solution which verifies some non-degeneracy conditions leads to a solution.  Note that this does not have any reference to integrable systems and that the non-degeneracy conditions are not global properties of the system, but only properties of the solution. The "automatic reducibility" allows to take advantage of the geometry to develop very efficient Newton methods and show that they converge.

This leads to very efficient numerical  algorithms (which moreover can be proved to lead to correct solutions), to validate formal expansions. From a more theoretical point of view, it can be applied to other geometric contexts (conformally symplectic, presymplectic) and other geometric objects such as whiskered tori. One can deal well with degenerate systems, singular perturbation theory and obtain simple proofs of monogenicity and Whitney regularity.

This is joint work with many people.

Embedded eigenvalues of the Neumann Poincaré operator

Series
Math Physics Seminar
Time
Thursday, April 23, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/730205379
Speaker
Wei Li – Louisiana State University –

The Neumann-Poincaré (NP) operator arises in boundary value problems, and plays an important role in material design, signal amplification, particle detection, etc. The spectrum of the NP operator on domains with corners was studied by Carleman before tools for rigorous discussion were created, and received a lot of attention in the past ten years. In this talk, I will present our discovery and verification of eigenvalues embedded in the continuous spectrum of this operator. The main ideas are decoupling of spaces by symmetry and construction of approximate eigenvalues. This is based on two works with Stephen Shipman and Karl-Mikael Perfekt.

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 Li – School 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. 

Bordered Floer Homology via Immersed Curves

Series
Geometry Topology Student Seminar
Time
Wednesday, April 22, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Sally Collins – Georgia Tech

In the setting of manifolds with connected torus boundary, we can reinterpret bordered invariants as immersed curves in the once punctured torus. This machinery, due to Hanselman, Rasmussen, and Watson, is particularly useful in the context of knot complements. We will show how a type D structure can be viewed as a multicurve in the boundary of a manifold, and we will consider how the operation of cabling acts on this new invariant. If time permits, we will discuss how to extract concordance invariants from the curves.

Cancelled

Series
Analysis Seminar
Time
Wednesday, April 22, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker

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 Lee – Georgia 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.

 

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