Dan Freed: Complex Chern-Simons invariants of 3-manifolds and abelianization

28th April, 2017

A hyperbolic 3-manifold M carries a flat PSL2(ℂ)-connection whose Chern-Simons invariant has been much studied since the early 1980s. For example, its real part is the volume of M. Explicit formulas in terms of a triangulation involve the dilogarithm. In joint work with Andy Neitzke we use 3-dimensional spectral networks to abelianize the computation of complex Chern-Simons invariants. The locality of the classical Chern-Simons invariant, expressed in the language of topological field theory, plays an important role.

Chiu-Chu Melissa Liu: Mirror Symmetry and Topological Recursion

28th April, 2017

The Remodelling Conjecture proposed by Bouchard-Klemm-Mariño-Pasquetti provides a precise correspondence between open-closed Gromov-Witten invariants of a symplectic toric Calabi-Yau threefold and the invariants of the mirror curve defined by Eynard-Orantin topological recursion. It can be viewed as a version of all genus open-closed mirror symmetry. I will present a proof of the conjecture and describe its implications on the structure of higher genus Gromov-Witten invariants, based on joint work with Bohan Fang and Zhengyu Zong.

Cliff Taubes: The behaviour of sequence of solutions to the Vafa-Witten equations

28th April, 2017

The Vafa-Witten equations on an oriented Riemannian 4-manifold are first order, non-linear equations for a pair of connection on a principal SO3 bundle over a 4-manifold and a self-dual 2-form with values in the associated Lie algebra bundle. This talk will describe a theorem about the behaviour of
sequences of solutions to the Vafa-Witten equations which have no convergent subsequence. This theorem says in part that a renormalization of a subsequence of the self-dual 2-form components of any given solution sequence converges on the complement of a closed set with Hausdorff dimension at most 2; and the limit defines a harmonic 2-form with values in a real line bundle. This behaviour generalizes Karen Uhlenbeck's compactness theorem for the self-dual Yang-Mills equations; it is similar to what happens in other first order generalizations of the Seiberg-Witten/self-duality equations.

Douglas Schultz: Lagrangian Floer theory in symplectic fibrations

27th April, 2017

Given a fibration of compact symplectic manifolds and an induced fibration of Lagrangians, one can ask if we can compute the Floer cohomology of the total Lagrangian from information about the base and fibre Lagrangians. The primary example that we have in mind is the manifold of full flags in ℂ3 which fibres as P1→Flag(ℂ3)→P2, and a Lagrangian T3 that fibres over the Clifford torus in P2. It turns out that one can prove the usual transversality and compactness results when the base is a rational symplectic manifold and the fibres are monotone. Assuming that we have a solution to the Maurer-Cartan equation, we then write down a Leray-Serre type spectral sequence which computes the Floer cohomology of the fibered Lagrangian. In the special case that the fibers are Kähler, we derive a formula for the leading-order disk potential.

Richard Siefring: Symplectic field theory and codimension-2 stable Hamiltonian submanifolds

20th April, 2017

Motivated by the goal of establishing a 'symplectic sum formula' in symplectic field theory, we will discuss the intersection behavior between punctured pseudoholomorphic curves and symplectic hypersurfaces in a symplectization. In particular we will show that the count of such intersections is always bounded from above by a finite, topologically determined quantity even though the curve, the target manifold, and the symplectic hypersurface in question are all non-compact.

Olivier Fouquet: Congruences between motives and congruences between values of L-functions

13th April, 2017

If two motives are congruent, is it the case that the special values of their respective L-functions are congruent? More precisely, can the formula predicting special values of motivic L-functions be interpolated in p-adic families of motives? I will explain how the formalism of the Weight-Monodromy filtration for p-adic families of Galois representations sheds light on this question (and suggests a perhaps surprising answer).

Sheng-Fu Chiu: Sheaves and contact non-squeezing in ℝ2n × S1

13th April, 2017

In this talk I will introduce a way to associate a triangulated category of sheaves with a domain of ℝ2n×S1. The cohomological information on the category side helps to detect the contact non-squeezing property of the domain on the topology side.

Xuhua He: Basic loci of Shimura varieties

6th April, 2017

In mod-p reductions of modular curves, there is a finite set of supersingular points and its open complement corresponding to ordinary elliptic curves. In the study of mod-p reductions of more general Shimura varieties, there is a "Newton stratification" decomposing the reduction into finitely many locally closed subsets, of which exactly one is closed. This closed set is called the basic locus; it recovers the supersingular locus in the classical case of modular curves. In certain cases, the basic locus admits a simple description as a union of classical Deligne-Lusztig varieties. The precise description in these case has proved to be useful for several purposes: to compute intersection numbers of special cycles and to prove the Tate conjecture for certain Shimura varieties. We will describe a group-theoretic approach to understand this phenomenon. We will show that this phenomenon is closely related to the Hodge-Newton decomposition, and many other nice properties on the Shimura varieties. This talk is based on the joint work with Ulrich Gortz and Sian Nie.

Sebastian Hurtado-Salazar: On Zimmer’s conjecture

6th April, 2017

The group SLn(ℤ) (when n > 2) is very rigid, for example, Margulis proved all its linear representations come from representations of SLn(ℝ) and are as simple as one can imagine. Zimmer's conjecture states that certain 'non-linear' representations ( group actions by diffeomorphisms on a closed manifold) come also from simple algebraic constructions.

For example, conjecturally the only action on SLn(ℤ) on an (n−1) dimensional manifold (up to some trivialities) is the one on the (n−1) sphere coming projectivizing natural action of SLn(ℝ) on ℝn. I'll describe some recent progress on these questions due to A. Brown, D. Fisher and myself.

Chris Wendl: Rigid holomorphic curves are generically super-rigid

31st March, 2017

I will explain the main ideas of a proof that for generic compatible almost complex structures in symplectic manifolds of dimension at least 6, closed embedded J-holomorphic curves of index 0 are always 'super-rigid', implying that their multiple covers are never limits of sequences of curves with distinct images. This condition is especially interesting in Calabi-Yau 3-folds, where it follows that the Gromov-Witten invariants can be 'localized' and computed in terms of Euler classes of obstruction bundles for a finite set of disjoint embedded curves. By the same techniques, we can also show that unbranched covers of simple J-holomorphic curves are generically regular. These results are based on a decomposition of the space of branched covers into smooth strata on which certain twisted Cauchy-Riemann operators have kernel and cokernel of constant dimension.

Arno Kret: Galois Representations for the general symplectic group

30th March, 2017

In a recent preprint with Sug Woo Shin I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. In this talk I will explain some parts of this construction that involve the eigenvariety.

Dusa McDuff: The stabilized symplectic embedding problem

30th March, 2017

I will discuss some recent work (mostly joint with Dan Cristofaro-Gardiner and Richard Hind) on the stabilized symplectic embedding problem for ellipsoids into balls. The main tools come from embedded contact homology.

Felix Schlenk: The many forms of rigidity for symplectic embeddings

30th March, 2017

We look at the following chain of symplectic embedding problems in dimension four.

E(1,a)→Z4(A), E(1,a)→C4(A), E(1,a)→P(A,ba)(b ∈ ℕ≥2), E(1,a)→T4(A).

Here E(1,a) is a symplectic ellipsoid, Z4(A) is the symplectic cylinder D2(A)×R2, C4(A)=D2(A)×D2(A) is the cube and P(A,bA)=D2(A)×D2(bA) the polydisc, and T4(A)=T2(A)×T2(A), where T2(A) is the 2-torus of area A. In each problem we ask for the smallest A for which E(1,a) symplectically embeds. The answer is very different in each case: total rigidity, total flexibility with a hidden rigidity, and a two-fold subtle transition between them.

François Lalonde: Continuous covers on symplectic manifolds

24th March, 2017

In this talk, we first introduce the notion of a continuous cover of a manifold parametrised by any compact manifold endowed with a mass 1 volume-form. We prove that any such cover admits a partition of unity where the usual sum is replaced by integrals. We then generalize Polterovich's notion of Poisson non-commutativity to such a context in order to get a richer definition of non-commutativity and to be in a position where one can compare various invariants of symplectic manifolds, for instance the relation between critical values of phase transitions of symplectic balls and eventual critical values of the Poisson non-commutativity. Our first main theorem states that our generalisation of Poisson non-commutativity depends only on real one-parameter spaces since intuitively the Hilbert curve in any high dimensional parameter space fills out the entire manifold and preserves the measure. Our second main theorem states that the Poisson non-commutativity is a (not necessarily strictly) decreasing function of the size of the symplectic balls used to cover continuously any given symplectic manifold. This function has other nice properties as well that do not prevent it from undergoing singularities similar to phase transitions.

Daniel Alvarez-Gavela: The simplification of caustics

23rd March, 2017

We present a full h-principle (relative, parametric, C0-close) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy-theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of an ambient Hamiltonian isotopy. The main ingredients in the proof are a refinement of the holonomic approximation lemma and the construction of a local wrinkling model for Lagrangian and Legendrian submanifolds. We give sample applications of our h-principle, including an Igusa-type theorem which states that higher singularities are unnecessary for the homotopy-theoretic study of the space of Legendrian knots in the standard contact Euclidean 3-space. This last result can be understood as a generalization of the Reidemeister theorem for families of Legendrian knots parametrized by a space of arbitrarily high dimension.

Philipp Habegger: On small sums of roots of unity

9th March, 2017

Let k be a fixed positive integer. Myerson (and others) asked how small the modulus of a non-zero sum of k roots of unity can be. If the roots of unity have order dividing N, then an elementary argument shows that the modulus decreases at most exponentially in N (for fixed k). Moreover it is known that the decay is at worst polynomial if k=5. In this talk I will present evidence that the modulus decreases at most polynomially for prime values of N by showing that counterexamples must be very sparse. We do this by counting rational points that approximate a set that is definable in an o-minimal structure. This is motivated by the counting results of Bombieri-Pila and Pila-Wilkie. I will also discuss progress on Myerson's related conjecture on Gaussian periods, as well as strong equidistribution properties of tuples of roots of unity, and connections to an ergodic result of Lind-Schmidt-Verbitskiy.