## Period 1

**Title:**How do we find planets around other stars?

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__Professor:__David W. Hogg

Professor of Physics and Data Science, New York University

Adjunct Senior Staff Scientist, MPI für Astronomie

Group Leader, Astronomical Data Group, Flatiron Institute

__Abstract:__In the last 20 years, the astronomical community has found thousands of planets around other stars, and we now know that many or even most stars in our Galaxy host planets. These planets have been found by making exceedingly precise measurements of stars. Here we will look at how these measurements are made, and how planets are found in the data. The data analysis—the search for the planets in the mountains of data—involves cutting-edge ideas from data science and machine learning. These technologies are transforming our capabilities in astronomy. Some of the planets we find are extremely strange; most known planetary systems are very different from our own Solar System.

**Difficulty:**Easy

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**Title:****A Hands-On Introduction to Low-Dimensional Topology**

**Professor:**Corrin Clarkson

Clinical Assistant Professor of Mathematics, CIMS

**Abstract:**We will begin by comparing topology to geometry. Both areas of mathematics are interested in shapes, but they focus on very different characteristics of those shapes. We will then explore how the flexibility of topological shapes can result in objects that look very different actually being topologically equivalent. All of this will be done with many examples, pictures and models.

**Difficulty:**Easy

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**Title:**Money Matters

**Professor:**Mithul Roy, Junior, CAS

Mathematics and Economics

**Abstract:**Most of the fiscal decisions we make every day are determined by mathematics, as are fiscal policies made by governments and firms. How do we choose what to invest in when there are multiple variables and risks? This talk will explore the basics of optimisation in the context of economics and finance to provide an insight into how economies and firms try to achieve the optimum in terms of profit, portfolio returns or budgets. Concepts covered include Lagrangian and constrained optimisation, discounting, projecting future income streams using calculus and statistics, and how we can account for risks in our projections.

**Difficulty:**Easy

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**Title:**An Englishman, a Russian Woman, and a Pakistani walk into a bar…

**Professor:**Gautam Kirshan Luhna

MS Scientific Computing (2nd Year)

Courant Institute

**Abstract:**Alan Turing was a brilliant English mathematician who is now considered the father of theoretical computer science. Sofia Kovalevskaya was a Russian mathematician who was the first woman to receive a doctorate in Mathematics and advanced the fields of Analysis, Partial differential equations and Mechanics. Abdus Salam was a Nobel Prize winning Pakistani Physicist who made significant contributions to our understanding of the nature of our physical reality.

What these people also had in common , apart from delivering groundbreaking contributions to their respective fields, was the significant social injustice they faced in their lives.

Turing’s crime was being a homosexual. Salam’s, an Ahmadi Muslim. Sofia’s, a woman.

This talk is as much about mathematics and science, as it is about the social hurdles that existed in the past, and may yet exist today in different forms. Hurdles that stand in the way of brilliant minds and hinder our progress as an intelligent species.

**Difficulty:**Medium

## Period 2

**Title:**Introduction to mathematical modeling through difference equations

**Professor:**Jennifer Crodelle,

PhD, Postdoctoral Associate, Courant Institute

**Abstract:**Many biological systems can be studied by writing rules that govern how the system changes from one moment in time to another. Difference equations are used to model that change and are essential components of many mathematical models. In this course, we will define a difference equation and explore several rules governing changes in different biological systems. Specifically, we will use difference equations to study and analyze population growth models (logistic), models of species competing for resources (predator-prey), and models in epidemiology (SIR).

**Difficulty:**Medium

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**Title:**The Math of a Beach Vacation

**Professor:**Tristan Leger,

4th year Mathematics PhD student, Courant Institute

__In this talk, I will present a simple model for the behavior of beachgoers.__

**Abstract:**My main goal will be to use this problem to introduce general ideas from various scientific fields.

- -First I will explain how the model is derived. This part of the talk will be an introduction to typical undergraduate physics techniques.
- -Then I will show how the equations that arise can be studied. This will be an introduction to partial differential equations.
- -Finally I will present numerical simulations for the problem. This part of the talk will introduce ideas from linear algebra and numerical analysis.

**Difficulty:**Hard

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**Title:**Upstart Puzzles

**Professor:**Dennis Shasha,

Professor of Computer Science, CIMS

**Abstract:**The writer of puzzles often invents puzzles to illustrate a principle. The puzzles, however, sometimes have other ideas. They speak up and say that they would be so much prettier as slight variants of their original selves.

The dilemma is that the puzzle inventor sometimes can't solve those variants. Sometimes he finds out that his colleagues can't solve them either, because there is no existing theory for solving them. At that point, these sassy variants deserve to be called upstarts.

We discuss a few upstarts inspired originally from the Falklands/Malvinas Wars, zero-knowledge proofs, and hikers in Colorado, and city planning.

They have given a good deal of trouble to a certain mathematical detective whom I know well.

**Difficulty:**Medium

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**Title:**How Many M&Ms Are in That Jar and Why Does it Matter: Experiments with Colloids and Candies

**Professor:**Paul Chaikin

Silver Professor of Physics; Center for Soft Matter Research, NYU

**Abstract:**Packing problems, how densely objects can fill a volume, are among the most ancient and persistent problems in mathematics and science. For equal spheres, Kepler conjectured in 1611 that the face-centered cubic crystal has the highest possible packing fraction ~ 0.74, but the proof by Hales only came in 1998. Grocers have known the answer for millennia from the way they stack oranges.

Non-spherical particles are another matter. Ellipsoids, M&M’s, can pack more densely than spheres. The higher density relates directly to the fact that rotating an ellipsoid makes a difference while rotating a sphere doesn't (no surprise). The number of contacts per particle Z is ~10 for M&M's as compared to Z ~ 6 for spheres. These results provide insights into melting, granular materials, rigidity, jamming, and why amorphous glasses don't flow.

__Medium__

**Difficulty:**

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## Period 3

__Title:__Reasoning with modern believers of Flat Earth Theory

__Professor:__Han Wang, 2020/Phd candidate, NYU/ Mathematics

__Abstract:__Believe it or not, today there are still many people who are convinced that the earth is flat, and some of them are actually intelligent and creative. I will give an introduction on what are the main points of a modern, popular version of Flat Earth Theory, and give some rebuttals that I find compelling, which are not as easy to find as one would assume.

**Prerequisites:**Most parts, and the general picture, of the talk, can be grasped well with a good command of high-school physics (Newtonian laws of motion, gravity, refraction of light). The Coriolis force will be mentioned, and to understand that part, familiarity with linear algebra or vector calculus is desirable.

**Difficulty:**Easy

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__Title:__Biology for math nerds

__Professor:__Alex Rubinsteyn

Instructor at Mount Sinai Medical School, Courant CS PhD (2014)

__Abstract:__The inner workings of the cell are often presented as many unrelated facts which must be memorized. I will try to instead present the cell's basic functions through simple mathematical abstractions. DNA, RNA, and proteins are all different kinds of sequences over simple alphabets. The flow of information from DNA to RNA to proteins depends on molecular machines that recognize patterns in these sequences.

**Difficulty:**Medium

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__Title:__Introduction to Recommender System and Crowdsourcing

__Professor:__Xi Chen

2014- Assistant Professor, Department of Technology, Operations, and Statistics. Stern School of Business, New York University

__Abstract:__You see recommendations every day on Amazon, eBay, Yelp and almost everywhere. In this talk, we will provide an introduction to recommender systems. From the talk, you will learn how Amazon recommends the product to you. If time permits, we will also talk about a new platform called “crowdsourcing", which is a new technique for collecting ratings with a little bit payment. Prerequisites: basic math (Trigonometric function)

**Difficulty:**Medium

## Period 4

**Title:**Ramsey Theory: Order in Disorder

**Professor:**Vindya Bhat

Clinical Assistant Professor, Mathematics, Courant Institute of Mathematical Sciences

**Abstract:**Ramsey theory answers the question, "under what circumstances can we find order in disorder?" For example, suppose six people are at a party and each pair of guests is either friends or strangers. Can we always find three people at the party who are mutual friends (or strangers)? How many people must we invite to the party so that four people are mutual friends? Five? The Ramsey number is the smallest number of people we must invite to the party to ensure a certain number of guests are mutual friends. We will generalize The Party Problem in the context of Ramsey's Theorem for graphs and discuss bounds for Ramsey numbers. Join us for one. BIG. party!

**Difficulty:**Medium

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**Title:**PolyPasswordHasher

**Professor:**Justin Cappos,

Associate Professor, Computer Science and Engineering, Tandon School of Engineering

**Abstract:**Password file disclosures are a frequent problem for many companies, which makes their users the target of identity theft and similar attacks. This work provides a new general cryptographic technique to prevent an attacker from efficiently cracking individual passwords from a stolen password database. PolyPasswordHasher employs a threshold cryptosystem to protect password hashes so that they cannot be verified unless a threshold of them are known. (This is conceptually similar to encrypting the passwords with a key that is only recoverable when a threshold of passwords are known.) Even if the password file and all other data on disk is obtained by a malicious party, the attacker cannot crack any individual password without simultaneously guessing a large number of them correctly. PolyPasswordHasher is the first single server, software-only technique that increases the attacker's search space exponentially. The result is that even cracking small numbers of weak passwords is infeasible for an attacker.

PolyPasswordHasher achieves these properties with similar efficiency, storage, and memory requirements to existing salted hash schemes, performing tens of thousands of account authentications per second. When using the current best practice (of salting and hashing), cracking three passwords that are comprised of 6 random characters on a modern laptop would take under an hour. However, when protected with PolyPasswordHasher, cracking these passwords when using every computer in existence would take longer than the estimated age of the universe.

**Difficulty:**Medium

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__An Introduction to Evolutionary Game Theory__

**Title:**

**Professor:**Olivia Chu,

4th Year Ph.D. Student, Princeton University, Quantitative and Computational Biology

**Abstract:**Classical game theory is the study of strategic interactions between rational decision makers and the optimal strategies that these individuals employ in order to maximize their payoffs. But what if we want to study evolving populations of humans or animals, who aren’t necessarily behaving rationally? Evolutionary game theory (EGT) adapts ideas from classical game theory to the study of biological and cultural evolution. In EGT, payoffs are interpreted as fitness and rationality is replaced by natural selection. In this talk, we will discuss the fundamental concepts of EGT, including evolutionarily stable strategies, and look at different evolutionary games, including the famous Prisoner’s Dilemma.

**Difficulty:**Easy

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**Title:**Learning from Data using Probability

**Professor:**Carlos Fernandez-Granda,

Assistant professor, Courant Math and Center for Data Science

**Abstract:**In this talk we will introduce some basic concepts in probability and show how they can be used to learn from data. We will use a simple running example based on a real-world dataset of voting records.

**Difficulty:**Medium