Speculations about homological mirror symmetry for affine hypersurfaces

Denis Auroux
University of California, Berkeley; Member, School of Mathematics
March 31, 2017
The wrapped Fukaya category of an algebraic hypersurface $H$ in $(C*)^n$ is conjecturally related via homological mirror symmetry to the derived category of singularities of a toric Calabi-Yau manifold $X$, whose moment polytope is determined by the tropicalization of $H$. (The case of the pair of pants discussed in the first talk is a special case of this construction).

Rigid holomorphic curves are generically super-rigid

Chris Wendl
Humboldt-Universität zu Berlin
March 31, 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.

The many forms of rigidity for symplectic embeddings

Felix Schlenk
University of Neuchâtel
March 30, 2017
We look at the following chain of symplectic embedding problems in dimension four. \[E(1, a) \to Z_4(A),\ E(1, a) \to C_4(A),\ E(1, a) \to P(A, ba) (b \in {\mathbb N}_{\geq 2}),\ E(1, a) \to T_4(A).\] Here $E(1, a)$ is a symplectic ellipsoid, $Z_4(A)$ is the symplectic cylinder $D_2(A) \times R_2$, $C_4(A) = D_2(A) \times D_2(A)$ is the cube and $P(A, bA) = D_2(A) \times D_2(bA)$ the polydisc, and $T_4(A) = T_2(A) \times T_2(A)$, where $T_2(A)$ is the 2-torus of area $A$. In each problem we ask for the smallest $A$ for which $E(1, a)$ symplectically embeds.

Galois Representations for the general symplectic group

Arno Kret
University of Amsterdam
March 30, 2017

In a recent preprint with Sug Woo Shin (https://arxiv.org/abs/1609.04223) 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.

Homological mirror symmetry for the pair of pants

Denis Auroux
University of California, Berkeley; Member, School of Mathematics
March 29, 2017
Homological mirror symmetry postulates a derived equivalence between the wrapped Fukaya category of an exact symplectic manifold and a category of coherent sheaves or matrix factorizations on a mirror space. This talk will provide an introduction to the relevant concepts and illustrate the statement on one simple example: the pair of pants. We will describe explicitly the wrapped Fukaya category of the pair of pants, and relate it to algebraic geometry on the mirror. (This is based on joint work with Abouzaid, Efimov, Katzarkov and Orlov).

Applications of twisted technology

Christoph Thiele
University of California, Los Angeles
March 29, 2017

Recently we proved with Durcik, Kovac, Skreb variational estimates providing sharp quantitative norm convergence results for bilinear ergodic averages with respect to two commuting transformations. The proof uses so called twisted technology developed in recent years for estimating bi-parameter paraproducts. Another application of the technique is to cancellation results for simplex Hilbert transforms.

On structure results for intertwining operators

Wilhelm Schlag
University of Chicago
March 29, 2017
The intertwining wave operators are basic objects in the scattering theory of a Hamiltonian given as the sum of a Laplacian with a potential. These Hamiltonians are the classical Schroedinger operators of quantum mechanics. For the three dimensional case we will discuss a new representation of the wave operators as superpositions of reflections and translations. This is joint work with Marius Beceanu, Albany.

Applications of monotone constraint satisfaction

Robert Robere
University of Toronto
March 28, 2017

Recently, a certain "monotone" version of the constraint satisfaction problem has proved an extremely useful tool for attacking problems in circuit, communication, and proof complexity theory. In this talk we discuss this version of the constraint satisfaction problem and touch on its connection to fundamental lower-bounds problems in these areas. We also consider a recent and interesting application: the first exponential lower bounds on the length of cutting planes refutations of random CNF formulas.