## Infinite Generaton of Non-Cocompact Lattices on Right-Angled Buildings

SPECIAL LECTURE

## "We know that God exists because mathematics is consistent and we know that the devil exists because we cannot prove the consist

**MATHEMATICAL CONVERSATIONS**

"We know that God exists because mathematics is consistent and we know that the devil exists because we cannot prove the consistency." -- Andre Weil

## Symplectic Dynamics of Integrable Hamiltonian Systems

## Improved Bounds for the Randomized Decision Tree Complexity of Recursive Majority

Recursive Majority-of-three (3-Maj) is a deceptively simple problem in the study of randomized decision tree complexity. The precise complexity of this problem is unknown, while that of the similarly defined Recursive NAND tree is completely understood.

## Local Universal Lifting Rings When | \neq p

## Expander Graphs: Why Number Theorists Might Care About Network Optimization

## Reductions of Local Galois Representations Arising From Hilbert Modular Forms

## General Hardness Amplification of Predicates and Puzzles

In this talk, I will give new proofs for the hardness amplification of fficiently samplable predicates and of weakly verifiable puzzles. More oncretely, in the first part of the talk, I will give a new proof of Yao's XOR-Lemma as well as related theorems in the cryptographic setting. This proof seems simpler than previous ones, yet immediately generalizes to statements similar in spirit such as the extraction lemma used to obtain pseudo-random generators from one-way functions [Hastad, Impagliazzo, Levin, Luby, SIAM J. on Comp. 1999].