Starting with a surface with negative constant Gauss curvature in the Euclidean 3-space, it is possible to (nontrivially) generate an infinitude of such surfaces by solving ODE systems alone. This fact, established by Bäcklund in the late 19th century, is one of the earliest-known examples of a Bäcklund transformation. The past century has seen rich interplay between Bäcklund transformations, integrable systems and soliton theory. However, a classification of Bäcklund transformations remains largely unknown. In this talk, I will discuss some recent progress on the classification of certain types Bäcklund transformations.

High-dimensional data can be difficult to analyze. Assume data are distributed on a low-dimensional manifold. The Vector Diffusion Mapping (VDM), introduced by Singer-Wu, is a non-linear dimension reduction technique and is shown robust to noise. It has applications in cryo-electron microscopy and image denoising and has potential application in time-frequency analysis. In this talk, I will present a theoretical analysis of the effectiveness of the VDM. Specifically, I will discuss parametrisation of the manifold and an embedding which is equivalent to the truncated VDM. In the differential geometry language, I use eigen-vector fields of the connection Laplacian operator to construct local coordinate charts that depend only on geometric properties of the manifold. Next, I use the coordinate charts to embed the entire manifold into a finite-dimensional Euclidean space. The proof of the results relies on solving the elliptic system and provide estimates for eigenvector fields and the heat kernel and their gradients.

We can describe the point-pushing subgroup of the mapping class group topologically as the set of maps that push a puncture around loops in the surface. However, we can characterize this topological subgroup in purely algebraic terms. Using group theoretic tools and a classical theorem of Burnside, we can recover a result of Ivanov-McCarthy establishing the triviality of Out(Mod±). To this end, we’ll demonstrate that the point-pushing subgroup is “unique” in the mapping class group.

We consider an isoperimetric inequality for area minimizing submanifolds $R$ lying outside a convex body $K$ in $\mathbb{R}^{n+1}$. Here $R$ is an $(m+1)$-dimensional submanifold whose boundary consists of a submanifold $T$ in $\mathbb{R}^{n+1} \setminus K$ and a free boundary (possibly not rectifiable) along $\partial K$. An isoperimetric inequality outside a convex body was previously proven by Choe, Ghomi, and Ritore in the codimension one setting where $m = n$. We extend their result to higher codimension. A key aspect of the proof are estimates on the concentration of mass of $T$ and $R$ near $\partial K$.

Every four-dimensional Stein domain has a height function whose regular level sets are contact three-manifolds. This allows us to study complex curves in the Stein domain via their intersection with these contact level sets, where we can comfortably apply three-dimensional tools. We use this perspective to characterize the links in Stein-fillable contact manifolds that bound complex curves in their Stein fillings. (Some of this is joint work with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)

In contact topology, invariants of Legendrian submanifolds in 1-jet spaces have been obtained through a variety of techniques. I will discuss how I am enriching one Morse-theoretic invariant, Generating Family Cohomology, to an A-infinity algebra by constructing product maps. The construction uses moduli spaces of Morse flow trees: spaces of intersecting gradient trajectories of functions whose critical points encode Reeb chords of the Legendrian submanifold. I will focus my talk on the construction of a 2-to-1 product and discuss how it lays the foundation for the A-infinity algebra.

We introduce a construction that associates a Riemannian metric $g_F$ (called the \emph{Binet-Legendre} metric) to a given Finsler metric $F$ on a smooth manifold $M$. The transformation $F \mapsto g_F$ is $C^0$-stable and has good smoothness properties, in contrast to previously considered constructions. The Riemannian metric $g_F$ also behaves nicely under conformal or isometric transformations of the Finsler metric $F$ that makes it a powerful tool in Finsler geometry. We illustrate that by solving a number of named problems in Finsler geometry. In particular, we extend a classical result of Wang to all dimensions. We answer a question of Matsumoto about local conformal mapping between two Berwaldian spaces and use it to investigate essentially conformally Berwaldian manifolds. We describe all possible conformal self maps and all self similarities on a Finsler manifold, generalizing the famous result of Obata to Finslerian manifolds. We also classify all compact conformally flat Finsler manifolds. We solve a conjecture of Deng and Hou on locally symmetric Finsler spaces. We prove smoothness of isometries of Holder-continuous Finsler metrics. We construct new `easy to calculate' conformal and metric invariants of Finsler manifolds. The results are based on the papers arXiv:1104.1647, arXiv:1409.5611, arXiv:1408.6401, arXiv:1506.08935, arXiv:1406.2924 partially joint with M. Troyanov (EPF Lausanne) and Yu. Nikolayevsky (Melbourne)

    An algebraic cycle homologous to zero on a variety leads to an extension of Hodge-theoretic data. In a variational context, the resulting section of a bundle of complex tori is called a normal function, and is used to study cycles modulo rational or algebraic equivalence.
    The archetype for interesting normal functions arises from the Ceresa cycle, consisting of the difference of two copies of a curve in its Jacobian. The profound geometric consequences of its existence are evidenced in work of Nori, Hain and (most recently) Totaro. In contrast, a theorem of Green and Voisin demonstrates the *absence* of normal functions arising from cycles on very general projective hypersurfaces of large enough degree.
    Inspired by recent work of Friedman-Laza on Hermitian variation of Hodge structure and Oort's conjecture on special subvarieties in the Torelli locus, R. Keast and I wondered about the existence of normal functions over etale neighborhoods of Shimura varieties. In this talk I will explain our classification of the cases where a Green-Voisin analogue does *not* hold, and where one expects interesting cycles (and generalized cycles) to occur. I will also give evidence that these predictions might be "sharp", and draw some geometric consequences.

This is joint work with Hyam Rubinstein. Matveev and Piergallini independently show that the set of triangulations of a three-manifold is connected under 2-3 and 3-2 Pachner moves, excepting triangulations with only one tetrahedron. We give a more direct proof of their result which (in work in progress) allows us to extend the result to triangulations of four-manifolds.

The lecture includes the main part (to be chosen on the spot) and a few mini-talks with just definitions, motivations, some ideas of proofs, and open problems. I will discuss some (hardly all) of the following topics. 1. “A survival guide for feeble fish”. How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water flow. This is related to homogenization of G-equation which is believed to govern many combustion processes. Based on a joint work with S. Ivanov and A. Novikov. 2. One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that a small perturbation of a non-degenerate completely integrable system still has an overwhelming measure of invariant tori with quasi-periodic dynamics. What happens outside KAM tori has been remaining a great mystery. The main quantitate invariants so far are entropies. It is easy, by modern standards, to show that topological entropy can be positive. It lives, however, on a zero measure set. We are now able to show that metric entropy can become infinite too, under arbitrarily small C^{infty} perturbations, answering an old-standing problem of Kolmogorov.. Furthermore, a slightly modified construction resolves another long–standing problem of the existence of entropy non-expansive systems. In these modified examples positive positive metric entropy is generated in arbitrarily small tubular neighborhood of one trajectory. Join with S. Ivanov and Dong. Chen. 3. “What is inside?” Imagine a body with some intrinsic structure, which, as usual, can be thought of as a metric. One knows distances between boundary points (say, by sending waves and measuring how long it takes them to reach specific points on the boundary). One may think of medical imaging or geophysics. This topic is related to minimal fillings and surfaces in normed spaces. Joint work with S. Ivanov. 4. How well can we approximate an (unbounded) space by a metric graph whose parameters (degree of vertices, length of edges, density of vertices etc) are uniformly bounded? We want to control the ADDITIVE error. Some answers (the most difficult one is for $\R^2$) are given using dynamics and Fourier series. Joint with Ivanov. 5.How can one discretize elliptic PDEs without using finite elements, triangulations and such? On manifolds and even reasonably “nice” mm–spaces. A notion of \rho-Laplacian and its stability. Joint with S. Ivanov and Kurylev. 6. A solution of Busemann’s problem on minimality of surface area in normed spaces for 2-D surfaces (including a new formula for the area of a convex polygon). Joint with S. Ivanov.

There is a natural relationship between moduli spaces of Riemann surfaces, mapping class groups of surfaces, and intersection patterns of simple closed curves on surfaces. In this talk, we describe an analogous relationship between moduli spaces of real algebraic curves, mapping class groups of surfaces with orientation reversing involutions, and intersection patterns of involution invariant simple closed curves on surfaces. After establishing these relationships, we obtain that the homology and cohomology groups of mapping class groups of surfaces with orientation reversing involutions satisfy duality relationships analogous to those for compact manifolds. We also obtain that higher homotopy groups associated to the moduli spaces of real algebraic curves relative to its boundary vanish in all degrees less than a determinable constant.

TBA

The central axis of the famous DNA double helix is often topologically constrained or even circular. The topology of this axis can influence which proteins interact with the underlying DNA. Subsequently, in all cells there are proteins whose primary function is to change the DNA axis topology -- for example converting a torus link into an unknot. Additionally, there are several protein families that change the axis topology as a by-product of their interaction with DNA. This talk will describe typical DNA conformations, and the families of proteins that change these conformations. I'll present a few examples illustrating how Dehn surgery and other low-dimensional topological methods have been useful in understanding certain DNA-protein interactions, and discuss the most common topological techniques used to attack these problems.

One of the central conjectures in Kähler geometry is the Yau-Tian-Donaldson conjecture relating the existence of canonical Kähler metrics to algebro-geometric stability. A natural question is to ask what happens when such a metric does not exist, and here Kähler metrics of Poincaré type are expected to play an important role. These metrics are Kähler metrics defined on the complement of a divisor in a compact complex manifold and have a cusp-like singularity near the divisor. The blow-up theorem of Arezzo-Pacard and its generalizations give sufficient conditions for the blow-up of a compact Kähler manifold admitting a canonical metric to also carry such a metric. I will describe an extension of this result to the Poincaré type setting.

I will discuss a generalization of the Seiberg-Witten equations on 3-manifolds and its relation to higher-dimensional gauge theory. The main new feature is the non-compactness of the moduli space of solutions. I will explain how to tackle this problem and count the solutions when the 3-manifold is the product of a surface and a circle. In this case, the problem of compactness reduces to studying degenerations of solutions to a non-linear scalar PDE resembling the Kazdan-Warner equation.

I will present a natural method for producing unstable critical points of the Ginzburg-Landau functionals on an arbitrary manifold, and describe results showing that a nontrivial portion of the energy must concentrate on a generalized minimal submanifold of codimension two.

Weinstein manifolds are an important class of symplectic manifolds with convex ends/boundary. These 2n dimensional manifolds come with a retraction onto a core n-dimensional stratified complex called the skeleton, which generally has singularities. The topology of the skeleton does not generally determine the smooth or symplectic structure of the 2n dimensional Weinstein manifold. However, if the singularities fall into a simple enough class (Nadler’s arboreal singularities), the whole Weinstein manifold can be recovered just from the data of the n-dimensional complex. We discuss work in progress showing that every Weinstein manifold can be homotoped to have a skeleton with only arboreal singularities (focusing in low-dimensions). Then we will discuss some of the expectations and hopes for what might be done with these ideas in the future.

In this talk, I will show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector, extending Lev Rozansky's work with infinite torus braids. I will also show a similar result for the limiting Lipshitz-Sarkar-Khovanov homotopy types of the closures of such braids. Extensions to more general infinite braids will also be considered. This is joint work with Gabriel Islambouli.

I will discuss recent advances in the theory of Fano varieties over nonclosed fields (joint with B. Hassett, A. Kresch, and A. Pirutka).

An SU(p, q)-flag domain is an open orbit of the real Lie group SU(p, q) acting on the complex flag manifold associated to its complexification SL(p + q, C). Any such flag domain contains certain compact complex submanifolds, called cycles, which encode much of the topological, complex geometric and repre- sentation theoretical properties of the flag domain. This talk is concerned with the description of these cycles in homology using a specific type of Schubert varieties. They are defined by the condition that the fixed point of the Borel group in question is in the closed SU(p,q)-orbit in the ambient manifold. We consider the Schubert varieties of this type which are of com- plementary dimension to the cycles. It is known that if such a variety has non-empty intersection with a certain base cycle, then it does so transversally (in finitely many points). With the goal of understanding this duality, we describe these points of intersection in terms of flags as well as in terms of fixed points of a given maximal torus. The relevant Schubert varieties are described in terms of Weyl group elements.

To a Legendrian knot, one can associate an $A_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.

Developments in high energy physics, specifically the theory of mirror symmetry, have led to deep conjectures regarding the geometry of a special class of complex manifolds called Calabi-Yau manifolds. One of the most intriguing of these conjectures states that various geometric invariants, some classical and some more homological in nature, agree for any two Calabi-Yau manifolds which are "birationally equivalent" to one another. I will discuss how new methods in equivariant geometry have shed light on this conjecture over the past few years, leading to the first substantial progress for compact Calabi-Yau manifolds of dimension greater than three. The key technique is the new theory of "Theta-stratifications," which allows one to bring ideas from equivariant Morse theory into the setting of algebraic geometry.

In this talk we will describe some results about Betti, Hodge, and characteristic numbers of compact hyperkahler manifolds. In (complex) dimension four one can find universal bounds for all of these invariants (Beauville, Guan); in higher dimensions it is still possible to find some bounds. We also describe how these bounds are related to the question: are there finitely many hyperkahler manifolds in each dimension, up to deformation?

Knot concordance concerns the classification of knots in the 3-sphere that occur as the boundaries of embedded disks in the 4-ball. Unlike in higher dimensions, one obtains vastly different results depending on whether the disks are required to be smoothly embedded or merely locally flat (i.e. continuously embedded with a topological normal bundle); many tools arising from gauge theory and symplectic geometry can be used to illustrate this distinction. After surveying some of the recent progress in this area, I will discuss the extension of these questions to knots in 3-manifolds other than S^3. I will show how to use invariants coming from Heegaard Floer homology to obstruct not only smoothly embedded disks but also non-locally-flat piecewise-linear disks; this answers questions from the 1970s posed by Akbulut and Matsumoto. I will also discuss more recent results (joint with Jennifer Hom and Tye Lidman) giving infinitely many knots that are distinct up to non-locally-flat piecewise-linear concordance.

In this talk, we will describe foliations of high dimensional ''iterated planar" contact manifolds by J-holomorphic curves and show that, by using this kind of technology, one can prove the long-standing Weinstein conjecture for iterated planar contact manifolds.

Contact geometry is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability. The associated one form is called a contact form and uniquely determines a vector field called the Reeb vector field on the manifold. I will explain how to make use of J-holomorphic curves to obtain a Floer theoretic contact invariant whose chain complex is generated by closed Reeb orbits. In particular, I will explain the pitfalls in defining contact homology and discuss my work which gives a rigorous construction of cylindrical contact homology via geometric methods. This talk will feature numerous graphics to acclimate people to the realm of contact geometry.

We will prove that certain simply-connected four-manifolds with positive-definite intersection forms cannot admit symplectic structures. This is related to the existence of so-called perfect Morse functions. This is joint work with Jennifer Hom.

We first review the relation between wrapped Floer cohomology of co-core disks after Lagrangian handle attachment and the Legendrian DGA of the corresponding attaching spheres. Then we discuss a generalization of this result to the partially wrapped setting where the Legendrian dga should be enriched with loop space coefficients, and describe several cases when explicit calculations are possible via parallel copies or local coefficient systems. We also discuss applications of these ideas to the topology of Lagrangian fillings of Legendrian submanifolds. The talk reports on joint work with Y. Lekili.

I will discuss joint work with Hutchings which gives a rigorous construction of cylindrical contact homology via geometric methods. This talk will highlight our use of non-equivariant constructions, automatic transversality, and obstruction bundle gluing. Together these yield a nonequivariant homological contact invariant which is expected to be isomorphic to SH^+ under suitable assumptions. By making use of family Floer theory we obtain an S^1-equivariant theory defined over Z coefficients, which when tensored with Q recovers the classical cylindrical contact homology, now with the guarantee of well-definedness and invariance. This integral lift of contact homology also contains interesting torsion information.

Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4-manifold X with the homology of S^1 x S^3. Specifically, we show that for any smoothly embedded 3-manifold Y representing a generator of H_3(X), a suitable version of the Heegaard Floer d invariant of Y, defined using twisted coefficients, is a diffeomorphism invariant of X. We show how this invariant can be used to obstruct embeddings of certain types of 3-manifolds, including those obtained as a connected sum of a rational homology 3-sphere and any number of copies of S^1 x S^2. We also give similar obstructions to embeddings in certain open 4-manifolds, including exotic R^4s. This is joint work with Danny Ruberman.