We will discuss the moduli space of genus 0 stable maps to a low degree hypersurface X inside an Orthogonal Grassmannian OG(k,n). In particular we show that the space of lines in such a general hypersurface is irreducible of the expected dimension. Next, we study the space of chains of lines connecting two general points and obtain the result that such a space is rationally connected. Through a smoothing argument, we also obtain rationally connectedness for a component of the stable map space. Finally we construct a very twisting surface in a general such X and conclude rationally simply connectedness.
Thesis Defense - Shrijan Gosh, May 7, 2026 11:00am
Rational Curves in Low Degree Hypersurfaces in Orthogonal Grassmannians
Thesis Defense - Yu Xiao, May 5, 2026 10:00am
Fibrations of Large Genera on Threefolds
We study fibrations by curves on smooth projective threefolds. Our main result shows that given a smooth threefold , there exists an integer g0 = g0(X ) such that for every g ≥ g0, there is a birational model X admitting a morphism X′ → ℙ2 whose general fibre is a smooth curve of genus .
Thesis Defense - Bowen Zhang
Blowing Up Scalar-Flat Asymptotically Conical Kahler Manifolds
Abstract:
This dissertation constructs a new family of non-compact scalar-flat Kahler manifolds asymptotic to Calabi-Yau cones using a gluing method. The main result shows that if a manifold admits a scalar-flat Kahler metric, then its blow-up also admits a scalar-flat Kahler metric. The method follows from the work of Claudio Arezzo and Frank Pacard, who established analogous results in the compact setting. We extend their approach to the non-compact case. The key analytic ingredient is the bijectivity of the Lichnerowicz operator on weighted Holder spaces.
Congratulations to Evita Nestoridi
for receiving the 2026 Godfrey Excellence in Teaching Award!
Congratulations to Lynne Barnett
for receiving the 2026 College of Arts and Sciences Staff Excellence Award!
Thesis Defense - Prabhat Devkota, April 21st, 2026
Moduli Spaces of Abelian Differentials in Genus
Speaker: Prabhat Devkota
Abstract:
We explore the geometry and topology of the moduli spaces of meromorphic differentials on curves of genus 0. In particular, we compute the cohomology with rational coefficients of the smooth orbifold compactification of these moduli spaces given by the multi-scale differentials. Additionally, we also determine all the cases where these moduli spaces are smooth varieties, and in these cases, we also compute their integral cohomology. In the case of special signature (0n , −2), we prove that the moduli space of multi-scale differentials is isomorphic to the classically studied space called the wonderful variety. Furthermore, in this special case, we relate it to a similar moduli space parameterizing meromorphic 1-forms on genus 0 curves, introduced by Halpern-Leistner and Robotis, called the moduli space of multiscale lines with collision.
Congratulations to Dima Dudko on winning the 2026 Brin Prize in Dynamical Systems.
The fifteenth Michael Brin Prize in Dynamical Systems has been awarded to Dzmitry Dudko recognizing his important contributions to renormalization techniques in complex dynamics with applications to the conjecture of local connectivity of the Mandelbrot set (MLC), and for his work on Thurston Maps. Congratulations to Dima for this well-deserved award!
Congratulations to Spencer Cattalani
for receiving the President's Award to Distinguished Doctoral Students!
Congratulations to Spencer Cattalani for receiving the President's Award to Distinguished Doctoral Students!
Thesis Defense - Spencer Cattalani, April 16th, 2026
Complex Cycles and Symplectic Geometry
Speaker: Spencer Cattalani
Location: Math Tower 5-127
Abstract:
Gromov revolutionized symplectic geometry by connecting it with (almost) complex geometry. This connection, centered on the notion of pseudoholomorphic curve, has proven to be incredibly fruitful for both fields. We aim to expand it through a study of complex cycles, a generalization of pseudoholomorphic curves due to Sullivan. In particular, we extend the positivity of intersection between pseudoholmorphic curves to include complex cycles and approximate complex cycles by ``coarsely'' holomorphic curves. Using these results, we prove two geometric criteria proposed by Gromov for an almost complex manifold to be tamed by a symplectic form. We also study a special class of complex cycles, called Ahlfors currents. We construct Ahlfors currents using a continuity method, show that they control the asymptotic behavior of pseudoholomorphic curves, and prove that the set of Ahlfors currents is convex. Using these results, we are able to characterize when a product of surfaces contains a complex line, generalizing a theorem of Bangert and answering a question of Ivashkovich and Rosay. These results indicate a parallel to the theory of rational curves and provide tools for the nascent study of symplectic non-hyperbolicity envisioned by Gromov.
Thesis Defense - Hanbing Fang, April 17th, 2026
Ricci flow limit space and its singular set
Speaker: Hanbing Fang
Location: Math 5-127
Abstract:
In the first part, we establish a weak compactness theorem for the moduli space of closed Ricci flows with uniformly bounded entropy, each equipped with a natural spacetime distance, under pointed Gromov--Hausdorff convergence. Furthermore, we develop a structure theory for the corresponding Ricci flow limit spaces, showing that the regular part, where convergence is smooth, admits the structure of a Ricci flow spacetime, while the singular set has codimension at least four.
In the second part, we study the singular sets of Ricci flow limit spaces. Firstly, we establish a Lojasiewicz inequality for the pointed W-entropy in the Ricci flow, under the assumption that the geometry near the base point is close to a standard cylinder or the quotient thereof. As a consequence, we prove the strong uniqueness of the cylindrical tangent flow at the first singular time of the Ricci flow.