## Galois Representations for the general symplectic group

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.

## The stabilized symplectic embedding problem

## The many forms of rigidity for symplectic embeddings

## On structure results for intertwining operators

## Applications of twisted technology

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.

## Homological mirror symmetry for the pair of pants

## Applications of monotone constraint satisfaction

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.

## Extremal problems in combinatorial geometry

## Applications of monotone constraint satisfaction

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.