Rice University Math

2017 Ph.D Thesis Defenses

Anthony Bosman

Title: Shake Slice and Shake Concordant Links
Date: WThursday, April 13, 2017
Thesis Advisor: Shelly Harvey

The study of knots and links up to concordance has proved signicant for many problems in low dimensional topology. In the 1970s, Akbulut introduced the notion of shake concordance of knots, a generalization of the study of knot concordance.
Recent work of Cochran and Ray has progressed our understanding of how shake concordance rela tes to concordance, although fundamental question remain, especially for shake slice knots. We extend the notion of shake concordance to links, generalizing much of what is known for knots, and oer a characterization in terms of link concordance and the infection of a link by a string link. We also discuss a number of invariants and properties of link concordance which extend to shake concordance of links, as well as note several that do not. Finally, we give several obstructions to a link being shake slice.

Corey Bregman

Title:Automorphisms of nonpositively curved cube complexes, right-angled Artin groups, and homology
Date: Wednesday, April 12, 2017
Thesis Advisor: Michael Wolf

Recently, the geometry of CAT(0) cube complexes featured prominently in Agol's resolution of two longstanding conjectures of Thurston in low-dimensional topology: the virtually Haken and virtually fibered conjecture for hyperbolic 3-manifolds. A key step of the proof was to show that every hyperbolic 3-manifold group is virtually special, i.e. virtually the fundamental group of a special nonpositively curved (NPC) cube complexes. In this thesis, we study algebraic properties of special groups as they relate to the geometry of special cube complexes.

Tam Do

Title: Global Regularity and Finite-time Blow-up in Model Fluid Equations
Date: Monday, March 27, 2017
Thesis Advisor: Alexander Kiselev

Determining the long time behavior of many partial differential equations modeling fluids has been a challenge for many years. In particular, for many of these equations, the question of whether solutions exist for all time or form singularities is still open. The structure of the nonlinearity and non-locality in these equations makes their analysis difficult using classical methods. In recent years, many models have been proposed to study fluid equations. In this thesis, we will review some new results in regards to these models as well as give insigh t into the relation between these models and the true equations.

Carol Ann Downes

Title: A Mass Minimizing Flow for Real-Valued Flat Chains with Applications to Transport Networks
Date: Friday, April 14, 2017
Thesis Advisor: Robert Hardt

An oriented transportation network can be modeled by a 1-dimensional chain whose boundary is the difference between the demand and supply distributions, represented by weighted sums of point masses. To accom modate efficiencies of scale into the model, one uses a suitable Mα norm for transportation cost. One then finds that the minimal cost network has a branching structure since the norm favors higher multiplicity edges, representing shared transport. In this thesis, we construct a continuous flow that evolves some initial such network to reduce transport cost without altering its supply and demand distributions. Instead of limiting our scope to transport networks, we construct this Mα mass reducing flow for real-valued flat chains by finding a real current of locally finite mass with the property that its restrictions are flat chains; the slices of such a restriction dictate the flow. Keeping the boundary fixed, this flow reduces the Mα mass of the initial chain and is Lipschitz continuous under the flat-α norm. To complete the thesis, we apply this flow to transportation networks, showing that the flow indeed evolves branching transport networks to be more cost efficient.

Junghwan Park

Title: Derivatives of genus one and three knots
Date: Tuesday, April 11, 2017
Thesis Advisor: Shelly Harvey

A derivative L of an algebraically slice knot K is an oriented link disjointly embedded in a Seifert surface of K such that its homology class forms a basis for a metabolizer H of K. For genus one knots, we produce a new example of a smoothly slice knot with non-slice derivatives. Such examples were first discovered by Cochran and Davis. In order to do so, we define an operation on a homology B4 that we call an n-twist annulus modification. Further, we give a new construction of smoothly slice knots and exotically slice knots via n-twist annulus modifications. For genus three knots, we show that the set SK,H = {µ¯L(123) − µ¯LI (123) | L, LI ∈ dK/dH} contains n · Z, where dK/dH is the set of all the derivatives associated with a metabolizer H and n is an integer determined by a Seifert form of K and a metabolizer H. As a corollary, we show that it is possible to realize any integer as the Milnor’s triple linking number of a derivative of the unknot on a fixed Seifert surface.