Seminars Sorted by Series

A very affine hypersurface is the vanishing locus of a Laurent polynomial in a complex torus; its complement is also a very affine hypersurface, but in two subtly-different ways. The (partially) wrapped Fukaya categories of the hypersurface and its...

ECH capacities have found many applications to symplectic embedding problems, most of which in the toric setting. I will discuss a new application of ECH to studying optimal embeddings for non-toric rational surfaces. The key convex geometric...

The contact connected sum is a well-understood operation for contact manifolds. I will focus on the 3-dimensional case and the Weinstein 1-handle model for the contact connected sum. I will discuss how pseudo-holomorphic curves in the...

I will first give an overview of ECH. Then I will describe how to compute ECH in the Morse-Bott setting a la Bourgeois. I will discuss some classes of examples where this approach works. Finally I will sketch the gluing results that allow us to...

Since their introduction in the early 1960s, Anosov flows have defined an important class of dynamics, thanks to their many interesting chaotic features and rigidity properties. Moreover, their topological aspects have been deeply explored, in...

I will show that in symplectic space smooth strongly convex bodies with all characteristics planar or all outer billiard trajectories planar are affine symplectic images of a ball.

The four dimensional ellipsoid embedding function of a toric symplectic manifold M measures when a symplectic ellipsoid embeds into M. It generalizes the Gromov width and ball packing numbers. This function can have a property called an infinite...

I will discuss work in progress with Morgan Weiler on knot filtered embedded contact homology (ECH) of open book decompositions of S^3 along T(2,q) torus knots to deduce information about the dynamics of symplectomorphisms of the genus (q-1)/2 pages...

In this talk I will discuss a Bennequin type inequality for symplectic caps of $S^3$ with standard contact structure. This has interesting applications which can help us to understand the smooth topology of symplectic caps and smoothly embedded...

Hutchings used Embedded Contact Homology to show the following for area-preserving disc diffeomorphisms that are a rotation near the boundary of the disc: if the asymptotic mean action on the boundary is greater than the Calabi invariant, then the...

Embedded contact homology (ECH) is a diffeomorphism invariant of three-manifolds due to Hutchings, defined using a contact form. This very diffeomorphism invariance makes it quite useful when studying contact dynamics, because it is possible to...

I will discuss coproduct structures and why we care about them. Then I will define a new coproduct structure on symplectic cohomology and indicate how to compute it in an example.

Given an area-preserving surface diffeomorphism, what can one say about the topological properties of its periodic orbits? In particular, a finite set of periodic orbits gives rise to a braid in the mapping torus, and one can ask which isotopy...

While much is known about the algebraic structure of the group of volume preserving homeomorphisms of a manifold when the dimension of the manifold is at least three, the structure of these groups in the two-dimensional case has long been mysterious...

Contact homology is a Floer-type invariant for contact manifolds, and is a part of Symplectic Field Theory. One of its first applications was the existence of exotic contact structures on spheres. Originally, contact homology was defined only for...

A Lagrangian cobordism between Legendrian knots is an important notion in symplectic geometry. Many questions, including basic structural questions about these surfaces are yet unanswered. For instance, while it is known that these cobordisms form a...

In the 1960's, Burgess proved a subconvexity bound for Dirichlet L-functions. However, the quality of this bound was not as strong, in terms of the conductor, as the classical Weyl bound for the Riemann zeta function. In a major breakthrough, Conrey...

Given a Shimura variety, we can construct two kinds of automorphic local systems, i.e., local systems attached to algebraic representations of certain associated algebraic group. The first is based on the classical complex analytic construction...

Machine learning algorithms are ubiquitous in most scientific, industrial and personal domains, with many successful applications. As a scientific field, machine learning has always been characterized by the constant exchanges between theory and...

Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. These obstructions are known to be sharp in several interesting cases, for example for symplectic embeddings of one...

ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain is a "concave toric domain" and...

We will discuss some recent constructions of "exotic" Lagrangian tori in simple symplectic manifolds such as $\mathbb{CP}^2$ (work of Renato Vianna) and $\mathbb R^6$ that are not Hamiltonian isotopic to previously known examples, inspired by wall...

We study symplectic invariants of the open symplectic manifolds $X$ obtained by plumbing cotangent bundles of spheres according to a plumbing tree. We prove that certain models for the Fukaya category $\mathcal F(X)$ of closed exact Lagrangians in...

I will describe how the notion of Lagrangian cobordism, introduced by Arnold in 1980, offers a systematic perspective on the study of Lagrangian topology. There are three aspects that will be emphasized: the relations with the triangulated structure...

We produce examples of non-trivial Hamiltonian fibrations that are not detected by previous methods (the characteristic classes of Reznikov for example), and improve theorems of Reznikov and Spacil on cohomology-surjectivity to the level of...

A point $q$ in a contact manifold $(M,\xi)$ is said to be a translated point of a contactomorphism $\phi$, with respect to a contact form $\alpha$ for $\xi$, if it is a "fixed point modulo the Reeb flow", i.e. if $q$ and $\phi(q)$ are in the same...

The Fukaya category of a Landau-Ginzburg (LG) model $W: E \to C$, denoted $F(E,W)$, enlarges the Fukaya category of $E$ to include certain non-compact Lagrangians determined by $W$ (for instance, Lefschetz fibrations and their thimbles). I will...

The log canonical threshold of a hypersurface singularity is an important invariant which appears in many areas of algebraic geometry. For instance it is used in the minimal model program, has been used to prove vanishing theorems, find Kahler...

We will describe how to get almost toric fibrations for all del Pezzo surfaces (endowed with monotone symplectic form), in particular for $\\mathbb{CP}^2\\#k\\overline{\\mathbb{CP}}^2$ for $4\\le k \\le 8$, where there is no toric fibrations. From...

There is growing interest in looking at operations on quantum cohomology that take into account symmetries in the holomorphic spheres (such as the quantum Steenrod powers, using a Z/p-symmetry). In order to prove relations between them, one needs to...

Given a smooth knot K in the 3-sphere, a classic question in knot theory is: What surfaces in the 4-ball have boundary equal to K? One can also consider immersed surfaces and ask a “geography” question: What combinations of genus and double points...

A central question from systolic geometry is to find upper bounds for the systolic ratio of a Riemannian metric on a closed $n$-dimensional manifold, i.e. the ratio of the $n$-th power of the shortest length of closed geodesics by the volume...

In this talk, I will address a geometric inverse problem from contact geometry: is it possible to recognize whether all orbits of a given Reeb flow are closed from the knowledge of the action spectrum? Borrowing the terminology from Riemannian...

This talk beings with a light introduction, including some historical anecdotes to motivate the development of this Floer theoretic machinery for contact manifolds some 25 years ago. I will discuss joint work with Hutchings which constructs...

Homological mirror symmetry predicts that the derived category of coherent sheaves on a curve has a symplectic counterpart as the Fukaya category of a mirror space. However, with the exception of elliptic curves, this mirror is usually a symplectic...

I will explain a duality theorem with products in Rabinowitz-Floer homology. This has a bearing on string topology and explains a number of dualities that have been observed in that setting. Joint work in progress with Kai Cieliebak and Nancy...

Morgan Weiler, Rice University:Infinite staircases of symplectic embeddings of ellipsoids into Hirzebruch surfaces

Gromov nonsqueezing tells us that symplectic embeddings are governed by more complex obstructions than volume. In particular, in...

Let f be a polynomial over the complex numbers with an isolated singular point at the origin and let d be a positive integer. To such a polynomial we can assign a variety called the dth contact locus of f. Morally, this corresponds to the space of d...

An exotic symplectomorphism is a symplectomorphism that is not isotopic to the identity through compactly supported symplectomorphisms.Using Floer-theoretic methods, we prove that the non-existence of an exotic symplectomorphism on the standard...

We present techniques for constructing families of compact, monotone (including exact) Lagrangians in certain affine varieties, starting with Brieskorn-Pham hypersurfaces. We will focus on dimensions 2 and 3. In particular, we'll explain how to set...

A classic result, due to McDuff and Schlenk, asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corners...