2016 Ph.D Thesis Defenses


Jorge Acosta

Title: Holonomy Limits of Cyclic Opers
Date: Thursday, April 07, 2016
Thesis Advisor: Michael Wolf

Given a Riemann surface X = (Σ, J ) we find an expression for the dominant term for the asymptotics of the holonomy of opers over that Riemann surface corresponding to rays in the Hitchin base of the form (0, 0, · · · , tωn). Moreover, we find an associated equivarient map from the universal cover (Σ˜ , J˜) to the symmetric space SLn(C)/SU(n) and show that limits of these maps tend to a sub-building in the asymptotic cone. That sub-building is explicitly constructed from the local data of ωn.

Natalie Durgin

Title: Geometric Invariant Theory Quotient of the Hilbert Scheme of Six Points on the Projective Plane
Date: Friday, May 29, 2015
Thesis Advisor: Brendan Hassett

We provide an asymptotic stability portrait for the Hilbert scheme of six points on the complex projective plane, and provide a description of
its geometric invariant theory (GIT) quotient.

Quentin Funk

Title:Two Variants on the Plateau Problem
Date: Thursday, March 10, 2016
Thesis Advisor: Robert Hassett

In this thesis, we approach two generalizations to the classical Plateau problem. First, we prove a Homological Plateau problem in the singular setting of semi-algebraic geometry using the tools of geometric measure theory. We obtain similar results to those of Federer and Fleming even in this singular case. Second we generalize the minimal mapping problem solved independantly by Douglas and Rado to so-called “multiple-valued” mapping of the disk. Multiple-valued maps are a cornerstone of the regularity theorems of Almgren and are interesting in their own right for many problems in the geometric calculus of variations. We prove existence and regularity for these Plateau solutions under fairly general conditions and we also produce a class of examples and analyze a degenerate case.


Andy Huang

Title: Handle crushing harmonic maps between surfaces
Date: Thursday, March 24, 2016
Thesis Advisor: Michael Wolf

In this thesis, we construct polynomial growth harmonic maps from once-punctured Riemann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyperbolic plane. We also establish their uniqueness within a class of exponentially decaying variations. Previously, harmonic maps from the complex plane have been parameterized by holomorphic quadratic differentials. Our harmonic maps, mapping a genus g>1 punctured surface to a k-sided polygon, correspond to meromorphic quadratic differentials with one pole of order (k+2) at the puncture and (4g +k-2) zeros (counting multiplicity) on the Riemann surface domain. As an example, we explore a special case of our theorems: the unique harmonic map from a punctured square torus to an ideal square. We use the symmetries of the map to deduce the three possibilities for its Hopf differential.

Kenan Ince

Title:The untwisting number of a knot
Date: Tuesday, April 12, 2016
Thesis Advisor: Andrew Putman

The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander polynomial-one knot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The p-untwisting number is the minimum number (over all diagrams of a knot) of full twists on at most 2p strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot into the unknot. First, we show that the algebraic untwisting number is equal to the algebraic unknotting number. However, we also exhibit several families of knots for which the difference between the unknotting and untwisting numbers is arbitrarily large, even when we only allow twists on a fixed number of strands or fewer. Second, we show that a common route for obstructing low unknotting number, the Montesinos trick, does not generalize to the untwisting number. However, we use a different approach to get conditions on the Heegaard Floer correction terms of the branched double cover of a knot with untwisting number one. This allows us to show that several $10$-crossing knots cannot be unknotted by a single positive or negative generalized crossing change. We also use the Ozsváth-Szabó tau invariant and the Rasmussen s invariant to differentiate between the p- and q-untwisting numbers for certain p and q.

Katherine Vance

Title:Tau invariants of spatial graphs
Date: Wednesday, April 13, 2016
Thesis Advisor: Shelly Harvey

In 2003, Ozsvath and Szabo defined the concordance invariant Tau for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of Tau for knots in S^3 and a combinatorial proof that Tau gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in S^3 which extends knot Floer homology. We define a Z-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined Tau invariant for balanced spatial graphs generalizing the Tau knot concordance invariant. In particular, this defines a Tau invariant for links in S^3. Using techniques similar to those of Sarkar, we show that our Tau invariant gives an obstruction to a link being slice.


Professional information

Research Interests
B.S., Peking University; Ph.D., University of Chicago, 1996. Nonlinear partial differential equations from fluid mechanics, geophysics, astrophysics and meteorology. Numerical linear Algebra.  He is interested in the analysis, computations and applications of these partial differential equations. One issue he has been working on is whether or not these partial differential equations are globally well-posed.

Research Areas
Nonlinear partial differential equations, mathematical fluid mechanics, numerical computation and analysis

Full Professor


Personal information

Last Name

First name




Pierre-Emmanuel  Jabin
University Professor,
Department of Mathematics
Center for Scientific Computation And Mathematical Modeling, CSCAMM
e-mail : pjabin@cscamm.umd.edu
Phone : (CSCAMM)  301 405 1647
(Math)        301 405 5122
Fax : 301 314 6674

Office hours : Monday and Friday 10-11:30 and by appointment (Fall 2014)
Office : CSCAMM 4117        Math  2307

Mail (CSCAMM) :   CSCAMM, 4149 CSIC Building #406
Paint Branch Drive
University of Maryland
College Park, MD 20742-3289

Mail (Math) :           Department of Mathematics, Math Building #084
Campus Drive
University of Maryland
College Park, MD 20742-4015

Various useful links :  The Applied Math RIT webpage in Maryland
The department of Mathematics at Maryland