# Open Neighborhood Seminar

### Harvard University Math Department

Welcome to Open Neighborhood Seminar! ONS is a general-audience colloquium series for all members of the Harvard math community, including undergraduates at any level. It meets every other Wednesday at 4:30pm in SC507 for an hour-long talk, followed by snacks and a social hour with the speaker. (We alternate with Math Table.) You can subscribe to our mailing list here.

## Next talk: November 15

**Speaker:**Joshua Wang (MIT)**Title:**Knots, tangles, braids, and rational numbers**Abstract:**The subject of knot theory concerns the phenomena of "knotting", "tangling", and "braiding", investigated from a mathematical point of view. I'll discuss a family of tangles called rational tangles and the related concepts of 2-bridge knots and plat closures of 4-braids. Rational tangles are named after John Conway's remarkable classification of them: they admit a natural one-to-one correspondence with the rational numbers (and infinity).

## Past talks

### November 1

**Speaker:**Scott Sheffield (MIT)**Title:**How to Build a Random Surface**Abstract:**The theory of "random surfaces" has emerged in recent decades as a significant field of mathematics, lying somehow at the interface between geometry, probability, combinatorics, analysis and mathematical physics. Just as "Brownian motion" is a special kind of random path, there is a similarly special kind of random surface.

Random surfaces are often motivated by physics: statistical physics, string theory, quantum field theory, and so forth. They have also been independently studied by mathematicians working in random matrix theory and enumerative graph theory. But even without that motivation, one may be drawn to wonder what a "typical" two-dimensional manifold looks like, or how one can make sense of that question.

I will give an overview of what this theory is about, including many computer simulations and illustrations. In particular, I will discuss the so-called Liouville quantum gravity surfaces, and explain how they are approximated by discrete random surfaces called random planar maps.

### October 18

**Speaker:**Hana Jia Kong (Harvard)**Title:**How to add things up without "="**Abstract:**In many mathematical scenarios, the conventional notion of strict equality doesn't hold. When we work with algebraic operations in such contexts, we need special notions to encode addition, multiplication, and other mathematical operations. In this talk, I will introduce the concept of operads, which serves as an effective tool for this purpose and finds applications across various mathematical fields.

### October 4

**Speaker:**Colin Defant (Harvard)**Title:**Stack-Sorting and Beyond**Abstract:**In 1990, West introduced the stack-sorting map, a combinatorially-defined operator on the set of permutations of size n that serves as a deterministic analogue of Knuth's stack-sorting machine. I will discuss a method for computing the fertility of an arbitrary permutation, which is simply the number of preimages of the permutation under the stack-sorting map. This method uses combinatorial objects called valid hook configurations. Very surprisingly, valid hook configurations also appear in a formula that converts from free to classical cumulants in free probability theory. This allows us to use tools from free probability theory to prove deep facts about the stack-sorting map. On the other hand, we can also leverage the stack-sorting map to prove a new theorem that relates cumulants with special families of binary plane trees called troupes. This talk will be very elementary and combinatorial, and I will probably mention some open problems at the end.

### September 20

**Speaker:**Dan Freed (Harvard)**Title:**A moduli space problem in condensed matter physics**Abstract:**This talk is an elementary account of what is at first a surprising application of stable homotopy theory to theoretical physics. No specialized background is required; the storyline is the focus, not technical details. The main result is a classification of invertible gapped phases in quantum mechanical systems. We will take a journey to get there, along the way meeting moduli problems in geometry and topological field theory. This is joint work with Mike Hopkins.

### September 6 (MT/ONS joint opening talk)

**Speaker:**Cliff Taubes (Harvard)**Title:**Do you know a proof of the fundamental theorem of algebra?**Abstract:**The fundamental theorem of algebra (d’Alembert’s theorem) says that any monic polynomial of degree n has n roots counting multiplicity. There are lots of proofs—but which is the simplest? And, what do you need to know to prove it?

**Organizers:**Benjamin Gammage (gammage@math.harvard.edu), Dori Bejleri (bejleri@math.harvard.edu), and Gage Martin (gagemartin@math.harvard.edu). Please drop us an email if you're curious about the seminar!