# Open Neighborhood Seminar

### Harvard University Math Department

## Spring 2022

### February 2

**Speaker:**Noam Elkies (Harvard)**Title:**Making math on the margins**Abstract:**One way that mathematics grows is by finding new questions to study beyond the standard topics of serious mathematical research. We give three examples, ranging from the recreational (how many digits in 6561101970383!, and how did I find this curious factorial? What's the unusual feature of the "elementary" identity 12 + 21 + 27 + 38 + 44 + 53 = 2 + 5 + 11 + 22 + 33 + 48 + 74?) to a collaboration with faculty in the School of Public Health.

### February 16

**Speaker:**Hannah Larson (Stanford)**Title:**Lines in algebraic geometry**Abstract:**Suppose you write down a general polynomial in x, y, z and consider the surface of all points where it vanishes. What can you say about the family of lines contained in this surface? Are there no lines, a finite number of lines, infinitely many? We'll derive an expected dimension for the family of lines depending on the degree of the polynomial (and generalize this to more variables). In the case of cubic surfaces, we'll discuss some more subtle questions regarding the geometry of lines over the real numbers. This story motivates some results, joint with Isabel Vogt, about a closely related problem concerning bitangents (lines that are tangent twice) to a plane quartic. There will be many examples and "hands on demos."

### March 2

**Speaker:**Dylan Wilson (Harvard)**Title:**Counting shapes**Abstract:**We will explore the fascinating history of counting shapes, starting with counting dots (you can have no dots or one dot or two dots...) and then moving on to counting shapes of higher dimension. Hopefully you will see some connections to interesting questions in geometry, physics, homotopy theory, and number theory.

### March 23

**Speaker:**Anna Seigal (Harvard and Oxford)**Title:**Invariance, equivariance, and covariance**Abstract:**These are three concepts that examine how quantities vary: invariance and equivariance, from mathematics, and covariance, from statistics. How do they relate to each other? We will see how groups and symmetries are at the heart of various problems in statistics. I will also describe approaches to estimate parameters in statistical models using invariant theory, based on joint work with Carlos Amendola, Kathlen Kohn, and Philipp Reichenbach.

### April 6

**Speaker:**Noah Giansiracusa (Bentley)**Title:**The mathematics of misinformation**Abstract:**In this talk I'll gently survey various roles mathematics (often, but not always, in the form of machine learning) plays in our information ecosystem. I'll discuss the math behind YouTube's recommendation algorithm and Facebook's News Feed algorithm and the impact the choice of objective function has on what society sees and thinks. I'll explain how graph theory is used to quantitatively study the spread of news and misinformation on social media, and also how it is used to detect bot accounts. And I'll explain the math behind deepfake photos and videos and text generating AI. No prior knowledge of machine learning or data science will be assumed, and the math will be accessible to all undergraduates.

### April 20

**Speaker:**Bjorn Poonen (MIT)**Title:**Runge's method for solving diophantine equations**Abstract:**It has been proved that there is no general algorithm for finding all the integer solutions to a multivariable polynomial equation. Nevertheless, there are some methods that succeed if the equation has a certain form. I will explain one such method. If you want to try to rediscover it yourself, try to provably find all the integer solutions to y^2 = x^4 + 4x^3 - 2x^2 + 6x + 20.

**Organizers:**Ana Balibanu (ana@math.harvard.edu) and Dori Bejleri (bejleri@math.harvard.edu). Please drop us an email if you're curious about the seminar!