Any compact Riemann surface is shown to have a canonical projective structure (which is different from the canonical one given by the uniformization theorem). Some properties of this projective structure are established. (Joint work with Elisabetta Colombo, Paola Frediani and Gian Pietro Pirola.)

Given a closed, oriented surface with genus greater that 2, we study quasi-Fuchsian hyperbolic 3-manifolds homeomorphic to this surface times the interval. Different properties of these manifolds have been carefully studied in previous important works on 3 manifold geometry and topology and some interesting questions about them still remain to be answered. In this talk, we will focus on a new geometric invariant associated to them which we call the measured foliations at infinity. These are horizontal measured foliations of a holomorphic quadratic differential ( the Schwarzian derivative ) associated canonically with each of the two connected component of the boundary at infinity of a quasi-Fuchsian manifold. We ask whether given any pair of measured foliations (F,G) on a surface, is there a quasi-Fuchsian manifold with F and G as it measured foliations at infinity. The answer is affirmative under certain assumptions; first, (F,G) satisfy the property of being an “arational filling pair” and second, the quasi-Fuchsian manifold should be very close to being “Fuchsian” . The goal of this talk would be introducing the concepts and outlining the proof idea.

By the resolution of the Poincare conjecture in 3D, we know that the only closed three-manifold with the trivial fundamental group is the three-sphere. In light of it, one can ask the following question: Suppose M is a closed three-manifold with the property that the only representation $\pi_1(M)\rightarrow SU(2)$ is the trivial one. Does this imply that $\pi_1(M)$ is trivial? The class of manifolds $M$ for which this question is interesting (and open) are integer homology spheres. We prove a result in this direction: the half-Dehn surgery on any non-trivial fibered knot $K$ in $S^3$ admits an irreducible representation. The proof uses instanton floer homology. I will give a brief introduction to instanton floer homology and sketch the strategy. This is based on work in progress, some jointly with Zhenkun Li and Fan Ye.

A “buckyball” or “fullerene” is a trivalent graph embedded in the sphere, all of whose rings have length 5 or 6. The term originates from the most famous buckyball, “Buckminsterfullerene,” a molecule composed of 60 carbon atoms. In this talk, I will explain why there are exactly 1203397779055806181762759 buckyballs with 10000 carbon atoms.

We shall discuss Reeb’s Theorem and basic differential topology of Morse functions. This was used by Milnor to prove the existence of exotic spheres in 7 dimensions. We shall propose a generalization of Reeb’s Theorem and discuss a proof of it. This is joint work with Sachchidanand Prasad.

The Asymptotic Plateau Problem is the problem of existence of submanifolds of vanishing mean curvature with prescribed boundary “at infinity”. It has been studied in the hyperbolic space, in the Anti-de Sitter space, and in several other contexts. In this talk, I will present the solution of the APP for complete spacelike maximal p-dimensional submanifolds in the pseudo-hyperbolic space of signature (p,q). In the second part of the talk, I will discuss applications of this result in Teichmüller theory and for the study of Anosov representations. This is joint work with Graham Smith and Jérémy Toulisse.

Minimal Lagrangian tori in CP^{2} are the expected local model for particular point singularities of Calabi-Yau 3-folds and numerous examples have been constructed. In stark contrast, very little is known about higher genus examples, with the only ones to date due to Haskins-Kapouleas and only in odd genus. Using loop group methods we construct new examples of minimal Lagrangian surfaces of genus 1/2(k-1)(k-2) for large k. In particular, we construct the first examples of such surfaces with even genus. This is joint work with Sebastian Heller and Franz Pedit.

This talk focuses on the recent resolutions of several well-known conjectures in studying the Einstein 4-manifolds with special holonomy. The main results include the following.

(1) Any volume collapsed limit of unit-diameter Einstein metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3D torus by an involution, a singular special Kaehler metric on the topological 2-sphere, or the unit interval.

(2) Any complete non-compact hyperkaehler 4-manifold with quadratically integrable curvature, namely gravitational instanton, must have an ALX model geometry with optimal asymptotic rate.

(3) Any gravitational instanton is biholomorphic to a dense open subset of some compact algebraic surface.

I will start by introducing contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. Tight contact structures have been classified on some 3 manifolds like S^3, R^3, Lens spaces, toric annuli, and almost all Seifert fibered manifolds with 3 exceptional fibers. We look at classification on one example of the Seifert fibered manifold with 4 exceptional fibers. I will explain the Legendrian surgery and convex surface theory which help us calculate the lower bound and upper bound of a number of tight contact structures. We will look at what more classification results can we hope to get using the same techniques and what is far-fetched.

Studying discrete subgroups of linear groups using a preserved geometric structure has a long tradition, for instance, using real hyperbolic geometry to study discrete subgroups of SO(n,1). Convex projective structures, a generalization of real hyperbolic structures, has recently received much attention in the context of studying discrete subgroups of PGL(n). In this talk, I will discuss convex projective structures and discuss results (joint with A. Zimmer) on relatively hyperbolic groups that preserve convex projective structures. In particular, I will discuss a complete characterization of relative hyperbolicity in terms of the geometry of the projective structure.

Hitchin’s theory of Higgs bundles associated holomorphic differentials on a Riemann surface to representations of the fundamental group of the surface into a Lie group. We study the geometry common to representations whose associated holomorphic differentials lie on a ray. In the setting of SL(3,R), we provide a formula for the asymptotic holonomy of the representations in terms of the local geometry of the differential. Alternatively, we show how the associated equivariant harmonic maps to a symmetric space converge to a harmonic map to a building, with geometry determined by the differential. All of this is joint work with John Loftin and Mike Wolf.

It is well known that solvability of the complex Monge- Ampere equation on compact Kaehler manifolds is related to the positivity of certain intersection numbers. In fact, this follows from combining Yau’s celebrated resolution of the Calabi conjecture, with Demailly and Paun’s generalization of the classical Nakai-Mozhesoin criteria. This correspondence was recently extended to a broad class of complex non-linear PDEs including the J-equation and the deformed Hermitian-Yang-Mills (dHYM) equations by the work of Gao Chen and others (including some at IISc). A natural question to ask is whether solutions (necessarily singular) exist in any reasonable sense if the Nakai criteria fails. Results of this nature are ubiquitous in Kaehler geometry - existence of weak Kaehler-Einstein metrics on normal varieties and Hermitian-Einstein metrics on reflexive sheaves to name a couple. Much closer to the present theme, is the work of Boucksom-Eyssidieux-Guedj-Zeriahi on solving the complex Monge-Ampere equation in big classes. In the talk, I will first speak about some joint and ongoing work with Ramesh Mete and Jian Song, that offers a reasonably complete resolution in complex dimension two, at least for the J-equation and the dHYM equations. Next, I will discuss some conjectures on what one can expect in higher dimensions.

The Thom conjecture, proven by Kronheimer and Mrowka in 1994, states that complex curves in $\mathbb{C}{\rm P}^2$ are genus minimizers in their homology class. We will show that an analogous statement does not hold for complex hypersurfaces in $\mathbb{C}{\rm P}^3$. This is joint work with Ruberman and Strle.

The intersection theory of the Grassmannian, known as Schubert calculus, is an important development in geometry, representation theory and combinatorics. The Quot scheme is a natural generalization of the Grassmannian. In particular, it provides a compactification of the space of morphisms from a smooth projective curve C to the Grassmannian. The intersection theory of the Quot scheme can be used to recover Vafa-Intriligator formulas, which calculate explicit expressions for the count of maps to the Grassmannian subject to incidence conditions with Schubert subvarieties.

The symplectic (or orthogonal) Grassmannian parameterizes isotropic subspaces of a vector space endowed with symplectic (or symmetric) bilinear form. I will present explicit formulas for certain intersection numbers of the symplectic and the orthogonal analogue of Quot schemes. Furthermore, I will compare these intersection numbers with the Gromov–Ruan–Witten invariants of the corresponding Grassmannians.

We construct a pointwise Boutet de Monvel-Sjostrand parametrix for the Szegő kernel of a weakly pseudoconvex three dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman’s boundary asymptotics of the Bergman kernel to weakly pseudo-convex domains in dimension two. Next we present an application where we prove that a weakly pseudoconvex two dimensional domain of finite type with a Kähler-Einstein Bergman metric is biholomorphic to the unit ball. This extends earlier work of Fu-Wong and Nemirovski-Shafikov. Based on joint works with C.Y. Hsiao and M. Xiao.

Abstract: A fundamental problem in complex geometry is to construct canonical metrics, such as Hermite-Einstein (HE) metrics on vector bundles and constant scalar curvature Kähler (cscK) metrics on Kahler manifolds. On a given vector bundle/manifold, such a metric may or may not exist, in general. The existence question for such metrics has been found to have deep connections to algebraic geometry. In the case of vector bundles, the Hitchin-Kobayashi correspondence proved by Uhlenbeck–Yau and Donaldson show that the existence of a HE metric is captured by the notion of slope stability for the vector bundle. In the case of manifolds, the still open Yau-Tian-Donaldson conjecture relates the existence of cscK metrics to K-stability of the underlying polarised variety.

Together with Ruadhaí Dervan, I started a research programme where we study canonical metrics, called Optimal Symplectic Connections, and a notion of stability, on fibrations. We proposed a Hitchin-Kobayashi/Yau-Tian-Donaldson type conjecture in this setting as well. In the case when the fibration is the projectivisation of a vector bundle, we recover the Hermite-Einstein and slope stability notions, respectively, and as such the theory can be seen as a generalisation of the classical bundle theory to more general fibrations. There has recently been great progress on this topic both on the differential and algebraic side, through works of Hallam, McCarthy, Ortu, Hattori, Spotti and Engberg, in addition to the joint works with Dervan. The aim of this talk is to give an introduction to and overview of the status of this programme.

Abstract: I will discuss the growth of the number of infinite dihedral subgroups of lattices G in PSL(2, R). Such subgroups exist whenever the lattice has 2-torsion and they are related to so-called reciprocal geodesics on the corresponding quotient orbifold. These are closed geodesics passing through an even order orbifold point, or equivalently, homotopy classes of closed curves having a representative in the fundamental group that’s conjugate to its own inverse. We obtain the asymptotic growth of the number of reciprocal geodesics (or infinite dihedral subgroups) in any orbifold, generalizing earlier work of Sarnak and Bourgain-Kontorivich on the growth of the number of reciprocal geodesics on the modular surface. Time allowing, I will explain how our methods also show that reciprocal geodesics are equidistributed in the unit tangent bundle. This is joint work with Juan Souto.

The Poincaré holonomy variety (or $sl(2, C)$-oper) is the set of holonomy representations of all complex projective structures on a Riemann surface. It is a complex analytic subvariety of the $PSL(2, C)$ character variety of the underlying topological surface. In this talk, we consider the intersection of such subvarieties for different Riemann surface structures, and we prove the discreteness of such an intersection. As a corollary, we reprove Bers’ simultaneous uniformization theorem, without any quasiconformal deformation theory.

In her thesis, Maryam Mirzakhani counted the number of simple closed geodesics of bounded length on a (real) hyperbolic surface. This breakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, symplectic geometry, surface topology, and homogeneous dynamics. In this talk, I’ll discuss some of these connections as well as a qualitative strengthening of her theorem that describes what these curves (and their complements) actually look like. This is joint work with Francisco Arana-Herrera.

We will discuss total mean curvatures, i.e., integrals of symmetric functions of the principle curvatures, of hypersurfaces in Riemannian manifolds. These quantities are fundamental in geometric variational problems as they appear in Steiner’s formula, Brunn-Minkowski theory, and Alexandrov-Fenchel inequalities. We will describe a number of new inequalities for these integrals in non positively curved spaces, which are obtained via Reilly’s identities, Chern’s formulas, and harmonic mean curvature flow. As applications we obtain several new isoperimetric inequalities, and Riemannian rigidity theorems. This is joint work with Joel Spruck.

In general, the equivalence of the stability and the solvability of an equation is an important problem in geometry. In this talk, we introduce the J-equation on holomorphic vector bundles over compact Kahler manifolds, as an extension of the line bundle case and the Hermitian-Einstein equation over Riemann surfaces. We investigate some fundamental properties as well as examples. In particular, we give algebraic obstructions called the (asymptotic) J-stability in terms of subbundles on compact Kahler surfaces, and a numerical criterion on vortex bundles via dimensional reduction. Also, we discuss an application for the vector bundle version of the deformed Hermitian-Yang-Mills equation in the small volume regime.

We will present some recent work on the classification of shrinking gradient Kähler-Ricci solitons on complex surfaces. In particular, we classify all non-compact examples, which together with previous work of Tian, Wang, Zhu, and others in the compact case gives the complete classification. This is joint work with R. Bamler, R. Conlon, and A. Deruelle.

We prove existence of twisted Kähler-Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when -K_X is big, we obtain a uniform Yau-Tian-Donaldson existence theorem for Kähler-Einstein metrics. To achieve this, we build up from scratch the theory of Fujita-Odaka type delta invariants in the transcendental big setting, using pluripotential theory. This is joint work with Kewei Zhang.

In 2006, Labourie defined a map from a bundle over Teichmuller space to the Hitchin component of the representation variety $Rep(\pi_1(S),PSL(n,R))$, and conjectured that it is a homeomorphism for every $n$ (it was known for $n =2,3$). I will describe some of the background to the Labourie conjecture, and then show that it does not hold for any $n >3$. Having shown that Labourie’s map is more interesting than a mere homeomorphism, I will describe some new questions and conjectures about how it might look.

The preservation of positive curvature conditions under the Ricci flow has been an important ingredient in applications of the flow to solving problems in geometry and topology. Works by Hamilton and others established that certain positive curvature conditions are preserved under the flow, culminating in Wilking’s unified, Lie algebraic approach to proving invariance of positive curvature conditions. Yet, some questions remain. In this talk, we describe positive sectional curvature metrics on $\mathbb{S}^4$ and $\mathbb{C}P^2$, which evolve under the Ricci flow to metrics with sectional curvature of mixed sign. This is joint work with Renato Bettiol.

The Thomas-Yau conjecture is an open-ended program to relate special Lagrangians to stability conditions in Floer theory, but the precise notion of stability is subject to many interpretations. I will focus on the exact case (Stein Calabi-Yau manifolds), and deal only with almost calibrated Lagrangians. We will discuss how the existence of destabilising exact triangles obstructs special Lagrangians, under some additional assumptions, using the technique of integration over moduli spaces.

I will describe the construction of an integer-valued symplectic invariant counting embedded pseudo-holomorphic curves in a Calabi–Yau 3-fold in certain cases. This may be seen as an analogue of the Gromov invariant defined by Taubes for symplectic 4-manifolds. The construction depends on a detailed bifurcation analysis of the moduli space of embedded curves along generic paths of almost complex structures. This is based on joint work with Shaoyun Bai.

We survey the recent progress on the fundamental group of open manifolds with nonnegative Ricci curvature. This includes finite generation and virtual abelianness/nilpotency of the fundamental groups.

The horofunction compactification of a metric space keeps track of the possible limits of balls whose centers go off to infinity. This construction was introduced by Gromov, and although it is usually hard to visualize, it has proved to be a useful tool for studying negatively curved spaces. In this talk I will explain how, under some metric assumptions, the horofunction compactification is a refinement of the significantly simpler visual compactification. I will then go over how this relation allows us to use the simplicity of the visual compactification to get geometric and topological properties of the horofunction compactification. Most of these applications will be in the context of Teichmüller spaces with respect to the Teichmüller metric, where the relation allows us to prove, among other things, that Busemann points are not dense within the horoboundary and that the horoboundary is path connected.

I will discuss a recent joint work with Olivier Biquard about conic Kähler-Einstein metrics with cone angle going to zero. We study two situations, one in negative curvature (toroidal compactifications of ball quotients) and one in positive curvature (Fano manifolds endowed with a smooth anticanonical divisor) leading up to the resolution of a folklore conjecture involving the Tian-Yau metric.

We present new obstructions to the existence of Lagrangian cobordisms in $\mathbb{R}^4$. The obstructions arise from studying moduli spaces of holomorphic disks with corners and boundaries on immersed objects called Lagrangian tangles. The obstructions boil down to area relations and sign conditions on disks bound by knot diagrams of the boundaries of the Lagrangian. We present examples of pairs of knots that cannot be Lagrangian cobordant and knots that cannot bound Lagrangian disks.

We consider certain degenerating families of complex manifolds, each carrying a canonical measure (for example, the Bergman measure on a compact Riemann surface of genus at least one). We show that the measure converges, in a suitable sense, to a measure on a non-Archimedean space, in the sense of Berkovich. No knowledge of non-Archimedean geometry will be assumed.

Recently, Markovic proved that there exists a maximal representation into (PSL(2,R))^3 such that the associated energy functional on Teichmuller space admits multiple critical points. In geometric terms, there is more than one minimal surface in the relevant homotopy class in the corresponding product of closed Riemann surfaces. This is related to an important question in Higher Teichmuller theory. In this talk, we explain that this non-uniqueness arises from non-uniqueness of minimal surfaces in products of trees. We plan to discuss energy minimizing properties for minimal maps into trees, as well as the geometry of the surfaces found in Markovic’s work. This is work in progress, joint with Vladimir Markovic.

A conjectural correspondence due to Yau, Tian and Donaldson relates the existence of certain canonical Kähler metrics (“constant scalar curvature Kähler metrics”) to an algebro-geometric notion of stability (“K-stability”). I will describe a general framework linking geometric PDEs (“Z-critical Kähler metrics”) to algebro-geometric stability conditions (“Z-stability”), in such a way that the Yau-Tian-Donaldson conjecture is the classical limit of these new broader conjectures. The main result will prove that a special case of the main conjecture: the existence of Z-critical Kähler metrics is equivalent to Z-stability.

In higher Teichmuller theory we study subsets of the character varieties of surface groups that are higher rank analogs of Teichmuller spaces, e.g. the Hitchin components, the spaces of maximal representations and the other spaces of positive representations. Fock-Goncharov generalized Thurston’s shear coordinates and Penner’s Lambda-lengths to the Hitchin components, showing that they have a beautiful structure of cluster variety. We applied a similar strategy to Maximal Representations and we found new coordinates on these spaces that give them a structure of non-commutative cluster varieties, in the sense defined by Berenstein-Rethak. This was joint work with Guichard, Rogozinnikov and Wienhard. In a project in progress we are generalizing these coordinates to the other sets of positive representations.

Up to biholomorphic change of variable, local invariants of a quadratic differential at some point of a Riemann surface are the order and the residue if the point is a pole of even order. Using the geometric interpretation in terms of flat surfaces, we solve the Riemann-Hilbert type problem of characterizing the sets of local invariants that can be realized by a pair (X,q) where X is a compact Riemann surface and q is a meromorphic quadratic differential. As an application to geometry of surfaces with positive curvature, we give a complete characterization of the distributions of conical angles that can be realized by a cone spherical metric with dihedral monodromy.

One interesting question in low-dimensional topology is to understand the structure of mapping class group of a given manifold. In dimension 2, this topic is very well studied. The structure of this group is known for various 3-manifolds as well (ref- Hatcher’s famous work on Smale conjecture). But virtually nothing is known in dimension 4. In this talk I will try to motivate why this problem in dimension 4 is interesting and how it is different from dimension 2 and 3. I will demonstrate some “exotic” phenomena and if time permits, I will talk a few words on my work with Jianfeng Lin where we used an idea motivated from this to disprove a long standing open problem about stabilizations of 4-manifolds.

Since the Calabi conjecture was proved in 1978 by S.T. Yau, there has been extensive studies into nonlinear PDEs on complex manifolds. In this talk, we consider a class of fully nonlinear elliptic PDEs involving symmetric functions of partial Laplacians on Hermitian manifolds. This is closely related to the equation considered by Székelyhidi-Tosatti-Weinkove in the proof of Gauduchon conjecture. Under fairly general assumptions, we derive apriori estimates and show the existence of solutions. In addition, we also consider the parabolic counterpart of this equation and prove the long-time existence and convergence of solutions.

Surprisingly there exist connected components of character varieties of fundamental groups of surfaces in semisimple Lie groups only consisting of injective representations with discrete image. Guichard and Wienhard introduced the notion of $\Theta$ positive representations as a conjectural framework to explain this phenomena. I will discuss joint work with Jonas Beyrer in which we establish several geometric properties of $\Theta$ positive representations in PO(p,q). As an application we deduce that they indeed form connected components of character varieties.

I will discuss some aspects of a singular version of the Donaldson-Uhlenbeck-Yau theorem for bundles and sheaves over normal complex varieties satisfying some conditions. Several applications follow, such as a characterization of the case of equality in the Bogomolov-Gieseker theorem. Such singular metrics also arise naturally under certain types of degenerations, and I will make some comments on the relationship between this result and the Mehta-Ramanathan restriction theorem.

Choose a homotopically non-trivial curve “at random” on a compact orientable surface- what properties is it likely to have? We address this when, by “choose at random”, one means running a random walk on the Cayley graph for the fundamental group with respect to the standard generating set. In particular, we focus on self-intersection number and the location of the metric in Teichmuller space minimizing the geodesic length of the curve. As an application, we show how to improve bounds (due to Dowdall) on dilatation of a point-pushing pseudo-Anosov homeomorphism in terms of the self-intersection number of the defining curve, for a “random” point-pushing map. This represents joint work with Jonah Gaster.

A celebrated theorem of Gromov-Lawson and Schoen-Yau states that a n-torus cannot admit metrics with positive scalar curvature. Thus, the torus is of vanishing Yamabe type. In this talk, we will discuss its extension to metrics with some singularity. This is a joint work with L.-F. Tam.

In the late 1980s, Nigel Hitchin and Michael Wolf independently discovered a parametrization of the Teichmüller space of a compact surface by holomorphic quadratic differentials. In this talk, I will describe work in progress on a generalization of their result. I will review the definition of the “enhanced Teichmüller space” which has been widely studied in the mathematical physics and cluster algebra literature. I will then describe a version of the result of Hitchin and Wolf which relates meromorphic quadratic differentials to the enhanced Teichmüller space. This builds on earlier work by a number of authors, including Wolf, Lohkamp, Gupta, and Biswas-Gastesi-Govindarajan.

We will discuss the $L^\infty$ estimates for a class of fully nonlinear partial differential equations on a compact Kahler manifold, which includes the complex Monge-Ampere and Hessian equations. Our approach is purely based on PDE methods, and is free of pluripotential theory. We will also talk about some generalizations to the stability of MA and Hessian equations. This is based on joint works with D.H. Phong and F. Tong.

The k-differentials are sections of the tensorial product of the canonical bundle of a complex algebraic curves. Fixing a partition (m_1,…,m_n) of k(2g-2), we can define the strata of k-differentials of type (m_1,…,m_n) to be the space of k-differentials on genus g curves with zeroes of orders m_i. After checking that the strata or not empty, the first interesting topological question about these strata is the classification of their connected component. In the case k=1, this was settled in an important paper of Kontsevich and Zorich. This result was extend to k=2 by Lanneau, with corrections of Chen-Möller. The classification is unknown for k greater or equal to 3 as soon as g is greater or equal to 2. In this talk, I will present partial results on this classification problem obtained together with Dawei Chen (arXiv:2101.01650) and in progress with Andrei Bogatyrev. In particular, I will highlight the way Pell-Abel equation appears in this problem.

Leon Simon showed that if an area minimizing hypersurface admits a cylindrical tangent cone of the form C x R, then this tangent cone is unique for a large class of minimal cones C. One of the hypotheses in this result is that C x R is integrable and this excludes the case when C is the Simons cone over S^3 x S^3. The main result in this talk is that the uniqueness of the tangent cone holds in this case too. The new difficulty in this non-integrable situation is to develop a version of the Lojasiewicz-Simon inequality that can be used in the setting of tangent cones with non-isolated singularities.

Triangulated surfaces are compact hyperbolic Riemann surfaces that admit a conformal triangulation by equilateral triangles. Brooks and Makover started the study of random triangulated surfaces in the large genus setting, and proved results about the systole, diameter and Cheeger constant of random triangulated surfaces. Subsequently Mirzakhani proved analogous results for random hyperbolic surfaces. These results, along with many others, suggest that the geometry of random triangulated surfaces mirrors the geometry of random hyperbolic surfaces in the case of large genus asymptotics. In this talk, I will describe an approach to show that triangulated surfaces are asymptotically well-distributed in moduli space.

In this talk we will discuss the geometry of Strominger connection of Hermitian manifolds, based on recent joint works with Quanting Zhao. We will focus on two special types of Hermitian manifolds: Strominger Kaehler-like (SKL) manifolds, and Strominger parallel torsion (SPT) manifolds. The first class means Hermitian manifolds whose Strominger connection (also known as Bismut connection) has curvature tensor obeying all Kaehler symmetries, and the second class means Hermitian manifolds whose Strominger conneciton has parallel torsion. We showed that any SKL manifold is SPT, which is known as (an equivalent form of) the AOUV Conjecture (namely, SKL implies pluriclosedness). We obtained a characterization theorem for SPT condition in terms of Strominger curvature, which generalizes the previous theorem. We will also discuss examples and some structural results for SKL and SPT manifolds.

Consider a proper geodesic metric space $(X,d)$ equipped with a Borel measure $\mu.$ We establish a family of uniform Poincare inequalities on $(X,d,\mu)$ if it satisfies a local Poincare inequality ($P_{loc}$), and a condition on the growth of volume. Consequently, if $\mu$ is doubling and supports $(P_{loc})$ then it satisfies a uniform $(\sigma,\beta,\sigma)$-Poincare inequality. If $(X,d,\mu)$ is a Gromov-hyperbolic space, then using the volume comparison theorem introduced by Besson, Courtoise, Gallot, and Sambusetti, we obtain a uniform Poincare inequality with the exponential growth of the Poincare constant. Next, we relate the growth of Poincare constants to the growth of discrete subgroups of isometries of $X$, which act on it properly. This is Joint work with Gautam Nilakantan.

Homological stability is an interesting phenomenon exhibited by many natural sequences of classifying spaces and moduli spaces like the moduli spaces of curves M_g and the moduli spaces of principally polarized abelian varieties A_g. In this talk I will explain some efforts to find similar phenomena in the cohomology of discrimination complements.

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension n ≥ 4, we find a family of Z2 × O(n − 1)-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operators. We show that these solitons are non-collapsed.

In the last decades there have been many connections made between the analysis of a manifold M and the geometry of M. Said correctly, there are now many ways to make precise that well-behaved analysis on M is ’equivalent’ to the existence of lower bounds on Ricci curvature. Such ideas are the starting point for regularity theories and more abstract settings for analysis, including analysis on metric-measure spaces. We will begin this talk with an elementary review of these ideas. More recently it has become apparent analysis on the path space PM of a manifold is closely connected to two sided bounds on Ricci curvature. Again, said correctly one can make an equivalence that the analysis on PM is well behaved iff M has a two sided Ricci curvature bound. As a general phenomena, one see’s that analytic estimates on M lift to estimates on PM in the presence of two sided Ricci bounds. Our talk will mainly focus on explaining all the words in this abstract and giving some rough understanding of the broad ideas involved. Time allowing, we will briefly explain newer results with Haslhofer/Kopfer on differential harnack inequalities on path space.

The Hitchin-Simpson equations defined over a Kaehler manifold are first order, non-linear equations for a pair of a connection on a Hermitian vector bundle and a 1-form with values in the endomorphism bundle. We will describe the behavior of solutions to the Hitchin–Simpson equations with norms of these 1-forms unbounded. We will talk about two applications of this compactness theorem, one is the realization problem of the Taubes’ Z2 harmonic 1-form and another is the Hitchin’s WKB problem in higher dimensional. We will also discuss some open questions related to this question.

Let K(n, V) be the space of n-dimensional compact Kahler-Einstein manifolds with negative scalar curvature and volume bounded above by V. We prove that any sequence in K(n, V) converges in pointed Gromov-Hausdorff topology to a finite union of complete Kahler-Einstein metric spaces without loss of volume, which is biholomorphic to an algebraic semi-log canonical model with its non-log terminal locus removed. We further show that the Weil-Petersson metric extends uniquely to a Kahler current with continuous local potentials on the KSB compactification of the moduli space of canonically polarized manifolds. In particular, the Weil-Petersson volume of the KSB moduli space is finite.

I will discuss the geometry of Kaehler manifolds with a lower bound on the holomorphic bisectional curvature, along with their pointed Gromov-Hausdorff limits. Some of the proofs use Ricci flow.

Projective geometry provides a common framework for the study of classical Euclidean, spherical, and hyperbolic geometry. A major difference with the classical case is that a projective structure is not completely determined by its holonomy representation. In general, a complete description of the space of structures with the same holonomy is still missing. We will consider certain structures on punctured surfaces, and we will discuss how to describe all of those with a given holonomy in the case of the thrice-punctured sphere. This is done in terms of a certain geometric surgery known as grafting. Our approach involves a study of the Möbius completion, and of certain meromorphic differentials on Riemann surfaces. This is joint work with Sam Ballas, Phil Bowers, and Alex Casella.

‘Growth’ is a geometrically defined property of a group that can reveal algebraic aspects of the group. For instance, Gromov showed that a group has polynomial growth if and only if it is virtually nilpotent. In this talk, we will focus on growth of groups that act on a CAT(0) cube complex. Such spaces are combinatorial versions of the more general CAT(0) (negatively curved) spaces. For instance, the fundamental group of a closed hyperbolic 3-manifold acts non-trivially on a CAT(0) cube complex. Kar and Sageev showed that if a group acts freely on a CAT(0) square complex, then it either has ‘uniform exponential growth’ or it is virtually abelian. I will present some generalizations of their theorem. This is joint work with Kasia Jankiewicz and Thomas Ng.

We develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion-free Fuchsian groups that can be thus mated are punctured sphere groups. We describe a new class of maps that are topologically orbit-equivalent to Fuchsian punctured sphere groups. We call these higher Bowen-Series maps. The existence of this class ensures that the Teichmüller space of matings is disconnected. Further, they also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers’ boundary groups that are mateable in our framework. Broadly speaking, the first talk (by Sabyasachi Mukherjee) will emphasize the aspects to do with complex dynamics more, and the second (by Mahan Mj) will lay emphasis on the 1-dimensional dynamics aspects of the problem.

We develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion-free Fuchsian groups that can be thus mated are punctured sphere groups. We describe a new class of maps that are topologically orbit-equivalent to Fuchsian punctured sphere groups. We call these higher Bowen-Series maps. The existence of this class ensures that the Teichmüller space of matings is disconnected. Further, they also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers’ boundary groups that are mateable in our framework. Broadly speaking, the first talk (by Sabyasachi Mukherjee) will emphasize the aspects to do with complex dynamics more, and the second (by Mahan Mj) will lay emphasis on the 1-dimensional dynamics aspects of the problem.

The Khovanov chain complex is a categorification of the Jones polynomial and is built using a functor from Kauffman’s cube of resolutions to Abelian groups. By lifting this functor to the Burnside category, one can construct a CW complex whose reduced cellular chain complex agrees with the Khovanov complex. This produces a Khovanov homotopy type whose reduced homology is Khovanov homology. I will present a general outline of this construction, starting with Khovanov’s functor. This work is joint with Tyler Lawson and Robert Lipshitz.

Anosov representations are representations of word hyperbolic groups into semisimple Lie groups with many good geometric properties. In this talk I will develop a theory of Anosov representation of geometrically finite Fuchsian groups (a special class of relatively hyperbolic groups). I will only discuss the case of representations into the special linear group and avoid general Lie groups. This is joint work with Canary and Zhang.

In this talk I will present a quantization approach which directly relates Fujita-Odaka’s delta-invariant to the optimal exponent of certain Moser-Trudinger type inequality on polarized manifolds. As a consequence we obtain new criterions for the existence of twisted Kahler-Einstein metrics or constant scalar curvature Kahler metrics on possibly non-Fano manifolds.

We describe a general context, related to metric spaces, in which a weak version of the celebrated Bishop-Gromov inequality is valid and suggest that this could serve of a synthetic version of a lower bound on the Ricci curvature.

On a Riemann surface with a holomorphic $r$-differential, one can naturally define a Toda equation and a cyclic Higgs bundle with a grading. A solution of the Toda equation is equivalent to a harmonic metric of the Higgs bundle for which the grading is orthogonal. In this talk, we focus on a general non-compact Riemann surface with an $r$-differential which is not necessarily meromorphic at infinity. We introduce the notion of complete solution of the Toda equation, and we prove the existence and uniqueness of a complete solution by using techniques for both Toda equations and harmonic bundles. Moreover, we show some quantitative estimates of the complete solution. This is joint work with Takuro Mochizuki (RIMS).

Yau’s solution of the Calabi conjecture provided the first nontrivial examples of Ricci-flat Riemannian metrics on compact manifolds. Attempts to understand the degeneration patterns of these metrics in families have revealed many remarkable phenomena over the years. I will review some aspects of this story and present recent joint work with Valentino Tosatti where we obtain a complete asymptotic expansion (locally uniformly away from the singular fibers) of Calabi-Yau metrics collapsing along a holomorphic fibration of a fixed compact Calabi-Yau manifold. This relies on a new analytic method where each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.

William Thurston proposed regarding the map induced from two circle packings with the same tangency pattern as a discrete holomorphic function. A discrete analogue of the Riemann mapping is deduced from Koebe-Andreev-Thurston theorem. A natural question is how to extend this theory to Riemann surfaces and relate classical conformal structures to discrete conformal structures. Since circles are preserved under complex projective transformations, one can consider circle packings on surfaces with complex projective structures. Kojima, Mizushima and Tan conjectured that for a given combinatorics the deformation space of circle packings is diffeomorphic to the Teichmueller space via a natural projection. In this talk, we will report progress on the torus case.

The deformed Hermitian-Yang-Mills (dHYM) equation is the mirror equation for the special Lagrangian equation.
The “small radius limit” of the dHYM equation is the J-equation, which is closely related to the constant scalar curvature K"ahler (cscK) metrics.
In this talk, I will explain my recent result that the solvability of the J-equation is equivalent to a notion of stability.

I will also explain my similar result on the supercritical dHYM equation.

Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths – and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled. Another outcome of the approach is a new concept of volume for vector bundles.

I will discuss some aspects of SYZ mirror symmetry for pairs $(X,D)$ where $X$ is a del Pezzo surface or a rational elliptic surface
and $D$ is an anti-canonical divisor. In particular I will explain the existence of special Lagrangian fibrations, mirror symmetry
for (suitably interpreted) Hodge numbers and, if time permits, I will describe a proof of SYZ mirror symmetry conjecture for del Pezzo surfaces.

This is joint work with Adam Jacob and Yu-Shen Lin.

The pseudo-hyperbolic space $H^{2,n}$ is the pseudo-Riemannian analogue of the classical hyperbolic space. In this talk, I will explain how to solve an asymptotic Plateau problem in this space: given a topological circle in the boundary at infinity of $H^{2,n}$, we construct a unique complete maximal surface bounded by this circle. This construction relies on Gromov’s theory of pseudo-holomorphic curves. This is a joint work with François Labourie and Mike Wolf.

We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds.

As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.

Let $(M,g)$ be a Riemannian manifold and ‘$c$’ be some homology class of $M$. The systole of $c$ is the minimum of the $k$-volume over all possible representatives of $c$. We will use combine recent works of Gromov and Zhu to show an upper bound for the systole of $S^2 \times \{*\}$ under the assumption that $S^2 \times \{*\}$ contains two representatives which are far enough from each other.

The analysis for Yang-Mills functional and in general, problems related to higher dimensional gauge theory, often requires one to work with weak notions of principal G-bundles and connections on them. The bundle transition functions for such bundles are not continuous and thus there is no obvious notion of a topological isomorphism class.

In this talk, we shall discuss a few natural classes of weak bundles with connections which can be approximated in the appropriate norm topology by smooth connections on smooth bundles. We also show how we can associate a topological isomorphism class to such bundle-connection pairs, which is invariant under weak gauge changes. In stark contrast to classical notions, this topological isomorphism class is not independent of the connection.

The quintic threefold (the zero set of a homogeneous degree 5 polynomial on CP^4) is one of the most famous examples of a Calabi Yau manifold. It is one of the most studied in the field of Enumerative Geometry. For example, how many lines are there on a Quintic threefold? In this talk we will explain some approaches to count curves on the Quintic threefold. In particular, we will try to explain the following idea: If Y is a submanifold of X, and we understand the Enumerative Geometry of X, how can we answer questions about the Enumerative Geometry of Y? We will try to explain the idea used by Andreas Gathman to compute all the genus zero Gromov-Witten invariants of the Quintic Threefold.

The talk will be self contained and will not assume any prior knowledge of Enumerative Geometry or Gromov-Witten Invariants.

We will discuss some work on the Ricci flow on manifolds with symmetries. In particular, cohomogeneity one manifolds, i.e. a Riemannian manifold M with an isometric action by a Lie group G such that the orbit space M/G is one-dimensional. We will also explain how this relates to diagonalizing the Ricci tensor on Lie groups and homogeneous spaces.

We consider the natural embedding for SO(r) into SL(r) and study the corresponding map between the moduli spaces of principal bundles on smooth projective curves. We compare the spaces of global sections of natural line bundles (non-abelian theta functions) for these moduli spaces and their twisted analogues with the space of theta functions. We will discuss how these results can be applied to obtain an alternate proof of a result of Pauly-Ramanan. If time permits, we will also discuss some applications to the monodromy of the Hitchin/WZW connections. This is a joint work with H. Zelaci.

We will define an invariant for annular links using the combinatorial link Floer complex that gives genus bounds for annular cobordisms. The celebrated slice-Bennequin inequality relates slice genus of a knot with its contact geometric invariants. We investigate similar relations in our context. In particular, we will define an invariant of transverse knots that refines the transverse invariant $\theta$ in knot Floer homology.

This talk deals with (generalized) holomorphic Cartan geometries on compact complex manifols. The concept of holomorphic Cartan geometry encapsulates many interesting geometric structures including holomorphic parallelisms, holomorphic Riemannian metrics, holomorphic affine connections or holomorphic projective connections. A more flexible notion is that of a generalized Cartan geometry which allows some degeneracy of the geometric structure. This encapsulates for example some interesting rational parallelisms. We discuss classification and uniformization results for compact complex manifolds bearing (generalized) holomorphic Cartan geometries.

In the first part of this talk I shall recall what the Hot spots conjecture is. Putting it in mathematical terms, I shall provide a brief history of the conjecture. If time permits I shall explain a proof of the conjecture for Euclidean triangles.

I will begin by reviewing the relationship between Hitchin’s Integrable System and 4d N=2 Supersymmetric Quantum Field Theories. I will then discuss two classes of deformations of the Hitchin system which correspond, in the physical context, to relevant and marginal deformations of a conformal theory. The study of relevant deformations turns out to be related to the theory of sheets in a complex Lie algebra and their classification leads to a surprising duality between sheets in a Lie algebra and Slodowy slices in the Langlands dual Lie algebra (work done with J. Distler) . If there is time, I will discuss marginal deformations which are related to studying the Hitchin system as a family over the moduli space of curves including over nodal curves (ongoing project with J. Distler and R. Donagi) .

Given a closed orientable surface $S$, a $(G,X)-$structure on $S$ is the datum of a maximal atlas whose charts take values on $X$ and transition functions are restrictions of elements in $G$. Any such structure induces a holonomy representation $\rho:\pi_1(\widetilde{S})\to X$ which encodes geometric data of the structure. Conversely, can we recover a geometric structure from a given representation? Is such a structure unique? In this talk we answer these questions by providing old and new results.

Counting holomorphic curves in a symplectic manifold has been an area of research since Gromov’s work on this subject in the 1980s. Symplectic manifolds naturally allow a ‘cut’ operation. We explore what happens to curve count-based invariants when a collection of cuts is applied to a symplectic manifold. An interesting feature of curves in a multiply-cut manifold is that they have an underlying ‘tropical graph’, which is a graph that lives in the polytope associated to the cut.

A celebrated theorem of Margulis characterizes arithmetic lattices in terms of density of their commensurators. A question going back to Shalom asks the analogous question for thin subgroups. We shall report on work during the last decade or so and conclude with a recent development. In recent work with Thomas Koberda, we were able to show that for a large class of normal subgroups of rank one arithmetic lattices, the commensurator is discrete.

Let $\{ M_k \}_{k=1}^{\infty}$ be a sequence of closed, singular, minimal hypersurfaces in a closed Riemannian manifold $(N^{n+1},g), n+1 \geq 3$. Suppose, the volumes of $M_k$ are uniformly bounded from above and the $p$-th Jacobi eigenvalues of $M_k$ are uniformly bounded from below. Then, there exists a closed, singular, minimal hypersurface $M$ in $N$ with the above mentioned volume and eigenvalue bounds such that possibly after passing to a subsequence, $M_k$ weakly converges (in the sense of varifolds) to $M$, possibly with multiplicities. Moreover, the convergence is smooth and graphical over the compact subsets of $reg(M) \setminus Y$ where $Y$ is a finite subset of $reg(M)$ with $|Y|\leq p-1$. This result generalizes the previous results of Sharp and Ambrozio-Carlotto-Sharp in higher dimensions.

We shall present the circle of ideas concerning the coarse geometric approach to the Baum-Connes conjecture, via the analytic surgery sequence of N. Higson and J. Roe. In joint work with M.-T. Benameur, we elicit its relations with some secondary invariants of Dirac operators on co-compact coverings, which generalize classical deep results originally due to N. Keswani. We have now partially extended this program further to the context of non-compact, complete spin foliations, which provide new obstructions to the existence of leafwise metrics of (uniformly) positive scalar curvature via the coarse index, generalizing the results of Connes for smooth foliations of compact manifolds. If time permits, we shall also outline the connections of this framework with the work of Gromov-Lawson.

A theorem of J. Jones relates cohomology of free loop space of a simply connected manifold to Hochschild homology of its singular cochain algebra. In this talk I will discuss a potential Hochschild chain model for a product on cohomology of free loop space known as the Goresky-Hingston product.

In this talk, we consider tilings on surfaces (2-dimensional manifolds without boundary). If the face-cycle at all the vertices in a tiling are of same type then the tiling is called semi-regular. In general, semi-regular tilings on a surface M form a bigger class than vertex-transitive and regular tilings on M. This gives us little freedom to work on and construct semi-regular tilings. We present a combinatorial criterion for tilings to decide whether a tiling is elliptic, flat or hyperbolic. We also present classifications of semi-regular tilings on simply-connected surfaces (i.e., on 2-sphere and plane) in the first two cases. At the beginning, we present some examples and brief history on semi-regular tilings.

Consider Riemannian functionals defined by L^2-norms of Ricci curvature, scalar curvature, Weyl curvature and Riemannian curvature. I will talk about rigidity, stability and local minimizing properties of Einstein metrics and their products as critical metrics of these quadratic functionals. We prove that the product of a spherical space form and a compact hyperbolic manifold is unstable for certain quadratic functionals if the first eigenvalue of the Laplacian of the hyperbolic manifold is sufficiently small. We also prove the stability of L^{n/2}-norm of Weyl curvature at compact quotients of Sn × Hm.

It is open question if the standard round metric on $S^4$ is the unique positive Einstein metric, up to isometry and scaling. In this talk, I will discuss a compactness theorem which rules out nonstandard unit-volume Einstein metrics whose scalar curvatures lie within a certain range.

Motivated by mirror symmetry considerations (the deformed Hermitian-Yang-Mills equation due to Jacobs-Yau) and a desire to study stability conditions involving higher Chern forms, a vector bundle version of the usual complex Monge-Ampere equation (studied by Calabi, Aubin, Yau, etc) will be discussed. I shall also discuss a Kobayashi-Hitchin type correspondence for a special case (a dimensional reduction to Riemann surfaces).

Given a group G, we will define conjugacy invariant norm and discuss the boundedness of such norms. We will relate this problem with the existence of homogeneous quasimorphisms on groups. If time permits, we will discuss bounded cohomology and its relation to the above. This talk will be largely self-contained. No background of any of the mentioned objects will be required.

Spectral networks are certain decorated graphs drawn on a Riemann surface. I will describe a conjectural picture in which spectral networks can be viewed as analogues of Hermitian-Einstein metrics on vector bundles, and in which holomorphic differentials on the Riemann surface arise as stability conditions on certain Fukaya-type categories. This talk is based on various joint projects with Fabian Haiden, Ludmil Katzarkov, Maxim Kontsevich, and Carlos Simpson.

Two groups are considered ‘elementarily equivalent’, if they have the same ‘elementary’ theory. Classification of various families of groups based on elementary equivalence, has been of long standing interest to both group theorists and model theorists, the most celebrated example of which was the elementary equivalence in free groups posed by Tarski. By studying examples of groups with different elementary theories, we can gain insight into the nature of expressibility of properties of groups. In this talk, I shall elementary equivalence in Artin groups of finite type, which forms a generalization of braid groups and are of interest in geometric group theory. This was a part of joint work with Arpan Kabiraj and Rishi Vyas. The talk should be largely self-contained. No background in logic, model theory or braid groups shall be assumed.

Last updated: 01 Oct 2023