# Opening Day: May 3, 2021

**Venue:** online

**Organizers:** Valentin Blomer, Farrell Brumley, Philip Gressman, Marina Iliopoulou, Lillian B. Pierce

Given the current situation with COVID-19, the ongoing hygienic restrictions, difficulties in obtaining visa and further obstacles for travelers, the opening day will be available for viewing live online.

## Program

15:50 |
Welcome by the HIM director, Christoph Thiele |

16:00 |
Larry Guth (MIT): Sharp examples for decoupling and related questions |

17:30 |
Terence Tao (UCLA): Pseudorandomness of the Liouville function |

#### Larry Guth (MIT): Sharp examples for decoupling and related questions

This talk will be about some open questions related to decoupling. The starting question will be, what are the known sharp examples for decoupling? This question can be made precise in various ways. Depending on how exactly we phrase the question, there can be many sharp examples or very few known sharp examples. For some versions of the question, the known sharp examples are very rare and are based on number theoretic structure. Is it true that all nearly sharp examples have some sort of number theoretic structure? I have no idea. In the talk, we will make this question precise, see how it connects to some problems in harmonic analysis, and describe an obstacle to understanding it using current harmonic analysis techniques.

#### Terence Tao (UCLA): Pseudorandomness of the Liouville function

The Liouville pseudorandomness principle (a close cousin of the Mobius pseudorandomness principle) asserts that the Liouville function $\lambda(n)$, which is the completely multiplicative function that equals $-1$ at every prime, should be "pseudorandom" in the sense that it behaves statistically like a random function taking values in $-1,+1$. Various formalizations of this principle include the Chowla conjecture, the Sarnak conjecture, the local uniformity conjecture, and the Riemann hypothesis. In this talk we survey some recent progress on some of these conjectures.

## Video recordings and slides

**Larry Guth (MIT): Sharp examples for decoupling and related questions**

**Terence Tao (UCLA): Pseudorandomness of the Liouville function**

### Online Registration: Opening Day

Please fill out the online form to participate in the Opening Day. All registered participants will receive the Zoom data in due time.

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