Date  Speaker  Title (click for abstract) 
Jan. 22nd  No Seminar 

Jan. 29th 


Feb. 5th 


Feb. 12th 


Feb. 19th  Khalid BouRabee (CUNY) 

Feb. 26th  Guillem Cazassus (Indiana) 

March 5th 


March 12th 


March 19th  No Seminar  Spring break

March 26th 


April 2nd  Franco Vargas Pallete (IAS) 

April 9th 


April 16th 


April 23rd 


April 30rd 

The Inequalities of HitchinThorpe and ThorpeJ. A. Thorpe and N. Hitchin independently discovered that the Euler characteristic and signature of a compact, oriented, 4dimensional Einstein manifold must satisfy a remarkable inequality. As a consequence there are infinitely many simplyconnected topological manifolds that cannot support an Einstein metric. An underemphasized aspect of the story of this socalled HitchinThorpe inequality is that it is only a special case of Thorpe's original and more general inequality. In this talk, we give some background on the classical HitchinThorpe inequality as well as Thorpe's inequality before discussing our recent Generalized Thorpe Inequality, which identifies the classical inequalities as resulting from a pure result in ChernWeil theory. 
Normal generation and fibered conesThis is joint work with Harry Baik, Eiko Kin and Hyunshik Shin. We showed that for any 2d slice in the Thurston's fibered cone, all but finitely many primitive points correspond to pseudo anosov maps that normally generate the mapping class group. 
The Price twist and trisectionsLet S be an RP^2 embedded in a smooth 4manifold X^4. With some mild conditions, the Price twist is a surgery operation on S that yields a 4manifold homeomorphic (but not necessarily diffeomorphic) to X^4. In particular, for every RP^2 embedded in S^4, this operation yields a homotopy 4sphere. In this talk, we will understand the Price twist via the theory of trisections. In particular, I will show how to produce an explicit trisection diagram of the surgered 4manifold. This is joint work with Seungwon Kim. 
Continuous sections of families of complex algebraic varietiesFamilies of algebraic varieties exhibit a wide range of fascinating topological phenomena. Even families of zerodimensional varieties (configurations of points on the Riemann sphere) and onedimensional varieties (Riemann surfaces) have a rich theory closely related to the theory of braid groups and mapping class groups. In this talk, I will survey some recent work aimed at understanding one aspect of the topology of such families: the problem of (non)existence of continuous sections of "universal" families. Informally, these results give answers to the following sorts of questions: is it possible to choose a distinguished point on every Riemann surface of genus g in a continuous way? What if some extra data (e.g. a level structure) is specified? Can one instead specify a collection of n distinct points for some larger n? Or, in a different direction, if one is given a collection of n distinct points on CP^1, is there a rule to continuously assign an additional m distinct points? In this last case there is a remarkable relationship between n and m. For instance, we will see that there is a rule which produces 6 new points given 4 distinct points on CP^1, but there is no rule that produces 5 or 7, and when n is at least 6, m must be divisible by n(n1)(n2). These results are joint work with Lei Chen. 
Schwartz functions on subanalytic manifoldsSchwartz functions are classically defined on R^n as smooth functions such that they, and all their (partial) derivatives, decay at infinity faster than the inverse of any polynomial. The space of Schwartz functions is a Frechet space, and its continuous dual space is called the space of tempered distributions. A third space that plays a key role in the Schwartz theory is the space of tempered functions – a function is said to be tempered if pointwise multiplication by it preserves the space of Schwartz functions. This theory was formulated on R^n by Laurent Schwartz, later on Nash manifolds (smooth semialgebraic varieties) by Fokko du Cloux and by Avraham Aizenbud and Dmitry Gourevitch, and on singular algebraic varieties by Boaz Elazar and myself. The goal of this talk is to present the recently developed Schwartz theory on subanalytic manifolds. I will first explain how one can attach a Schwartz space to an arbitrary open subset of R^n. Then, I will define (globally) subanalytic manifolds – loosely speaking these are manifolds that locally look like subanalytic open subsets of R^n (I will explain what are these too) and have some ”finiteness” property. Model theorists may think of definable manifolds in R^{an}. I will prove that one can intrinsically define the space of Schwartz functions (as well as the spaces of tempered functions and of tempered distributions) on these manifolds, and prove that these spaces are ”well behaved” (in the sense that they form sheaves and cosheaves on the Grothendieck subanalytic topology). Along the way we will see where subanalyticity is used, and why this theory is illdefined in the category of smooth (not necessarily subanalytic) manifolds. Mainly, some ”polynomially bounded behaviour” (that holds in the subanalytic case thanks to Lojasiewicz’s inequality) is required. As time permits I will describe some possible applications. 
Geometry of biperiodic alternating linksIn this talk we will study the hyperbolic geometry of alternating link complements in the thickened torus. We will give conditions which imply the hyperbolicity of alternating link complements in the thickened torus. We show that these complements admit a positively oriented, unimodular geometric ideal triangulation, and determine sharp upper and lower volume bounds. For links which arise from semiregular Euclidean tilings, called semiregular links, we explicitly determine the complete hyperbolic structure on their complements and will discuss nice consequences like determination of exact volumes, arithmeticity and commensurability for this class of links. We will also discuss the Volume Density Conjecture and examples. 
The SeibergWitten equations and the length spectrum of hyperbolic threemanifoldsWhile both hyperbolic geometry and Floer homology have both been tremendously successful tools when studying threedimensional topology, their relationship is still very mysterious. In this talk, we provide sufficient conditions for a hyperbolic rational homology sphere not to admit irreducible solutions to the SeibergWitten equations in terms of its volume and the length spectrum (i.e. the set of lengths of closed geodesics). We discus explicit examples in which this criterion can be applied. This is joint work with Michael Lipnowski. 
Mahler measure and the VolDet ConjectureA basic open problem is to understand how the hyperbolic volume of knots and links is related to diagrammatic knot invariants. The VolDet Conjecture relates the volume and determinant of alternating links. We prove the VolDet Conjecture for infinite families of alternating links using the dimer model, the Mahler measure of 2variable polynomials, and the hyperbolic geometry of biperiodic alternating links. This is joint work with Abhijit Champanerkar and Matilde Lalin. 
Compactness for generalized SeibergWitten equationsThe SeibergWitten equation can be generalized to higherrank vector bundles with a hyperKahler moment map. Examples among these are the KapustinWitten equations, VafaWitten equations, SeibergWitten equations with multiple spinors, and the ADHM equations. Witten, Haydys, and DoanWalpuski have proposed several conjectures about the gauge theories from these equations, and the conjectures suggest that they will reflect topological information of the base manifold beyond the YangMills and SeibergWitten equations. The analytic difficulty of establishing gauge theories based on generalized SeibergWitten equations is the lack of compactness. It was proved by Taubes and HaydysWalpuski that solutions to many of these equations satisfy certain compactness properties described by Z/2harmonic spinors. In this talk, we will discuss some recent progress on the compactness problems for generalized SeibergWitten equations. Some of the works presented in the talk are in collaboration with Thomas Walpuski. 
Application of extremal length in elliptic estimatesExtremal length is a conformal invariant which is useful in a wide variety of areas. Discrete extremal length on graphs is also useful in discrete potential theory and circle packing problems. In this talk we will report their new applications in gradient estimates for 2nd order (discrete) elliptic PDEs in divergence form. We will also give one motivating example, which is about synchronization conditions for coupled oscillators on lattices. 
Circle packings and Delaunay circle patterns for complex projective structuresAt the interface of discrete conformal geometry and the study of Riemann surfaces lies the KoebeAndreevThurston theorem. Given a triangulation of a surface S, this theorem produces a unique hyperbolic structure on S and a geometric circle packing whose dual is the given triangulation. In this talk, we explore an extension of this theorem to the space of complex projective structures  the family of maximal CP^1atlases on S up to Möbius equivalence. Our goal is to understand the space of all circle packings on complex projective structures with a fixed dual triangulation. As it turns out, this space is no longer a unique point and evidence suggests that it is homeomorphic to Teichmüller space via uniformization  a conjecture by Kojima, Mizushima, and Tan. In joint work with JeanMarc Schlenker, we show that this projection is proper, giving partial support for the conjectured result. Our proof relies on geometric arguments in hyperbolic ends and allows us to work with the more general notion of Delaunay circle patterns, which may be of separate interest. I will give an introductory overview of the definitions and results and demonstrate some software used to motive the conjecture. 
A characterization of separable subgroups of 3manifold groupsThe subgroup separability is a property in group theory that is closely related to low dimensional topology, especially lifting \pi_1injective immersed objects in a space to be embedded in some finite cover and the virtual Haken conjecture of 3manifolds resolved by Agol. We give a complete characterization on which finitely generated subgroups of finitely generated 3manifold groups are separable. Our characterization generalizes Liu's spirality character on \pi_1injective immersed surface subgroups of closed 3manifold groups. A consequence of our characterization is that, for any compact, orientable, irreducible and boundaryirreducible 3manifold M with nontrivial torus decomposition, \pi_1(M) is LERF if and only if for any two adjacent pieces in the torus decomposition of M, at least one of them has a boundary component with genus at least 2. 