**Title:** Notes on Real Rationally Connected Varieties and Fano Threefolds of Genus 12**Date: **Thursday, June 16, 2016**Thesis Advisor: **Brendan Hassett**Abstract: **

We show that a smooth projective geometrically rationally connected variety over the real numbers with at least one rational point admits a non-constant mapping from an a smooth projective curve. Additionally, we show that a real smooth Fano complete intersection admits a non-constant map from the real anisotropic conic. Furthermore, we compute the genus and degree of the singular locus of the locus of lines on a genus 12 Fano threefold. After blowing up this locus to obtain simple normal crossings divisor, we compute the cohomology of the complement, in which we see the genus of this curve appear in weight 5 of the third cohomology group.

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

Given a Riemann surface X = (Î£, 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, Â· Â· Â· , tÏ‰n). Moreover, we find an associated equivarient map from the universal cover (Î£Ëœ , JËœ) 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 Ï‰n.

**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**Abstract: **

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.

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

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.

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

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.

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

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 OzsvÃ¡th-SzabÃ³ tau invariant and the Rasmussen s invariant to differentiate between the p- and q-untwisting numbers for certain p and q.

**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 Oâ€™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 Oâ€™Donnolâ€™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.