Selected Talks at Penn

The Not-So-Random Graph

Penn Graduate Student Lunch Seminar, Fall 2024

Consider the following experiment: take a pair of natural numbers and flip a fair coin. If the coin lands heads, connect the pair with an edge; if tails, do nothing. Repeat this for every distinct pair. What kind of graph emerges from this process? What structural properties might it have? Wanna bet on it?

This talk begins with a betting game. For any graph property you name, I'll tell you the exact probability that the resulting countable random graph satisfies it. If I'm right, you donate your wager to charity. Remember: the house always wins.

In the second half of the talk, I'll reveal how I managed to win every single bet. The secret involves one of my favorite theorems in all of mathematics.

IRS Table 2000CM

Penn Graduate Student Lunch Seminar, Fall 2023

Internal Revenue Service Table 2000CM is a statistical triumph. Table 2000CM uses the enormous dataset collected in the 2000 US census to answer a single morbid question: given a population of 100,000 Americans born at the same time, how many do we expect to die each year? Unfortunately, the answer is pretty grim. After an initial spike for infant mortality, the year-by-year chance of dying grows exponentially, approximately doubling every eight years. Aging is central to mortality risk.

In this talk, we will contemplate our own mortality and examine Table 2000CM. We will use 2000CM to shed light on various models of aging, why the body is not like a car, and why we should be skeptical of treating illness as a mechanism for increasing human longevity. Despite the macabre content, I promise we will end on a happy note.

Perfect Sets and the Continuum Hypothesis

Penn Graduate Student Logic Seminar, Fall 2023

The Continuum Hypothesis was the first deep problem in cardinal arithmetic. In short, it asks whether every uncountable subset of the real numbers has the same cardinality as the full continuum. Though we now know the hypothesis is independent of the standard axioms of set theory, it inspired many failed proof attempts.

This talk explores one such attempt, often called the “Perfect Set Program.” Since perfect sets are known to have the cardinality of the continuum, the idea was simple: prove that every uncountable set of reals contains a perfect subset. While this strategy ultimately couldn't succeed, it led to remarkable partial results. Mathematicians discovered that many rich and natural families of uncountable sets do, in fact, contain perfect subsets.

No background in set theory or logic will be assumed. Bring $5 – there may be a chance to place a small wager...

Induction and Coinduction

Penn Undergraduate Math Society, Winter 2023

Everyone is familiar with defining and proving things by induction. With a little thought, you might realize that inductive reasoning can be applied to structures beyond just the natural numbers. This talk begins by exploring a general perspective on induction through the language of algebras.

In mathematics, there's a familiar trick: sometimes we “turn all the arrows around” and attach a “co-” prefix to every noun. But what happens when we do this to induction? What are the co-natural numbers? The second part of the talk introduces the basics of co-algebra and shows how it can be used to model the behavior of “black box” systems. We'll conclude by discussing stream differential equations as a concrete example of how co-algebraic thinking can simplify mathematical argumentation.

So You Think You Know ℝ

Penn Graduate Student Lunch Seminar, Fall 2022

In an undergraduate mathematics education, what object gets the most attention? For most students, I'd wager it's the real numbers. When someone says “number,” they usually mean “real number.” The reals are the foundation of calculus. In algebra, they're examples of a group (in two ways), a ring, and a field. In linear algebra, they're the default field over which vector spaces are defined. They're constructed and scrutinized in analysis, and the geometry of the reals motivates the entire subject of manifold theory. In nearly every area of math, the real numbers are a go-to example of an “extremely basic example.”

With all this attention, surely we all have a deep and nuanced understanding of the real numbers. Let's put that to the test. In this talk, audience members will compete for fabulous prizes (including Amazon gift cards) by demonstrating their superior knowledge of the real numbers. Who will be crowned The Realest? Who will be exposed as a faker? Throw your hat in the ring and prove yourself.

Why Homotopy Theorists Should Care About Homotopy Type Theory

Penn Graduate Geometry/Topology Seminar, Fall 2022

While Homotopy Type Theory has generated much buzz from logicians, computer scientists, and cranks on the internet, many pure homotopy theorists remain uninterested. In this talk, I attempt to change this (at least on a small scale within Penn) by making a philosophical case for why homotopy theorists should care about HoTT.

Models of Dependent Type Theory, I and II

Penn Graduate Student Logic Seminar, Fall 2022

In set-theoretic foundations, models are taken to be sets (in some meta-set theory). In type theoretic foundations, the categorical semantics are a little more complicated. In the first talk, we discuss the syntactic category of a dependent type theory. This category, which directly translates types and substitutions into objects and morphisms, forms the most basic model of a dependent type theory. Type constructors naturally correspond to categorical constructions. Next, we discuss functorial semantics, where a model is a functor from the syntactic category to some other suitable category.

In the second talk, we discuss why functorial semantics have fallen out of fashion, and more contemporary notions of models such as display map categories, categories with families, and comprehension categories.

Model Theory and Tame Topology, I and II

Graduate Research Seminar in Applied Topology, Spring 2021

In his Esquisse d'un Programme, Grothendieck expressed a desire for a new set of "tame" topological foundations that would better accommodate stratified spaces defined using attaching maps satisfying certain regularity conditions. The o-minimal structures of model theory are the leading contender for tame foundations today.

The first talk introduces ideas from first-order model theory, such as models, compactness, categoricity, and decidability. Special attention is paid to quantifier elimination and the Tarski-Seidenberg theorem for semi-algebraic sets. The second talk introduces o-minimality and discusses how o-minimal structures can serve as a suitable framework for "tame" (or definable) algebraic topology. We conclude with some new results from tame homotopy theory.

Infinite Ethics

Penn Graduate Student Lunch Seminar, Fall 2020

Aggregative Consequentialism describes a broad class of ethical frameworks, including the most familiar forms of utilitarianism. Even today, many philosophers and thinkers continue to advocate for such theories. But these frameworks, at least in their standard form, rely on implicit assumptions about physics and cosmology. In particular, if we allow for a universe with infinitely many morally considerable beings, aggregative consequentialism begins to break down.

Can these theories be revised to remain coherent in an infinite universe? In this talk, I'll explore several proposed adaptations and assess their strengths and weaknesses. Along the way, we'll encounter some fascinating mathematical tools, such as ultrafilters and ordered over-fields of the real numbers.

Basic Personal Finance for PhD Students

Penn Graduate Student Lunch Seminar, Winter 2020

Dirty secret: the financial services industry often hides simple ideas behind layers of jargon, making people think they can't handle their own finances. I've spent a couple of years in the industry, and I'm here to cut through the noise and share some practical basics that might help you get started investing.

In this talk, we'll cover the essentials: how markets work, what mutual funds and ETFs are, how to open a brokerage account, how to think about credit cards, buying a home, and when to pay off debt. Content will be tailored toward the financial situation of graduate students, so no 401(k) talk. Come for the free lunch, stay for your financial future.