# ESP Biography

## JASON GROSS, MIT PhD student studying computer science

Major: comp sci, math, physics

College/Employer: MIT

Year of Graduation: G

Not Available.

## Past Classes

(Clicking a class title will bring you to the course's section of the corresponding course catalog)

H14169: Making deep friendships - Circling in Splash 2020 (Nov. 14 - 15, 2020)
Most of your Splash classes will be about objects and things. Some of your conversations will involve personal history, where you grew up, what you like and dislike. This class will be a third kind of conversation, about what our present experience is, as we're having it. Access to this level of conversation has a way of facilitating deep connections where you can feel deeply seen and welcomed. Circling is a practice about getting others' worlds, sharing what it's really like to be you, and having that be seen and reflected. Come experience the magic.

H14170: Circling - Intersubjective Mindfulness Meditation in Splash 2020 (Nov. 14 - 15, 2020)
Most of your Splash classes will be about objects and things. Some of your conversations will involve personal history, where you grew up, what you like and dislike. This class will be a third kind of conversation, about what our present experience is, as we're having it. There's a kind of magic to being deeply seen, and to being welcomed as you are. Circling is a practice about getting others' worlds, and sharing what it's really like to be you, and having that be seen and reflected. Come experience the magic

M14172: Linear Logic in Splash 2020 (Nov. 14 - 15, 2020)
It's a well-known fact of logic that if from $$P$$ you can get $$Q$$, then from $$P$$ you can also get $$Q$$ and $$Q$$.* So since you can get two dimes and a nickel from a quarter, you can get two dimes and a nickel and two dimes and a nickel from a single quarter. Come to learn about linear logic, which is a version of logic which doesn't claim that you can get infinite amounts of money from a quarter. *For example, since $$n = 2$$ implies that $$n = 1 + 1$$ then, $$n = 2$$ implies that $$n = 1 + 1$$ and also $$n = 1 + 1$$.

M14173: Proving Löb's Theorem in Splash 2020 (Nov. 14 - 15, 2020)
Löb's theorem is a beautiful theorem with a deceptively short proof. It states that $$\square (\square P \to P) \to \square P$$ for all $$P$$---that if you can show that proving $$P$$ is sufficient to make $$P$$ true, then you can prove $$P$$. Löb's theorem has a variety of applications, from enabling robust cooperation in the prisoner's dilemma, to curing social anxiety, from proving Gödel's incompleteness theorem, to proving that the halting problem is undecidable. I will present a few proofs of Löb's theorem, all of which are twisty in subtly different ways. We will spend the rest of the time working on wrapping our minds around these proofs, and discussing related topics.

H13791: Circling - Intersubjective Mindfulness Meditation in Spark 2020 (Mar. 14 - 15, 2020)
Most of your Spark classes will be about objects and things. Some of your conversations will involve personal history, where you grew up, what you like and dislike. This class will be a third kind of conversation, about what our present experience is, as we're having it. There's a kind of magic to being deeply seen, and to being welcomed as you are. Circling is a practice about getting others' worlds, and sharing what it's really like to be you, and having that be seen and reflected. Come experience the magic

H13792: Making deep friendships - Circling in Spark 2020 (Mar. 14 - 15, 2020)
Most of your Spark classes will be about objects and things. Some of your conversations will involve personal history, where you grew up, what you like and dislike. This class will be a third kind of conversation, about what our present experience is, as we're having it. Access to this level of conversation has a way of facilitating deep connections where you can feel deeply seen and welcomed. Circling is a practice about getting others' worlds, sharing what it's really like to be you, and having that be seen and reflected. Come experience the magic.

P13793: Your classical intuition is wrong! in Spark 2020 (Mar. 14 - 15, 2020)
Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe!

Z13869: Economic Games(?!) in Spark 2020 (Mar. 14 - 15, 2020)
We will attempt to generate some key concepts in economics through puzzles. These concepts will enhance your ability to engage and reason about social phenomena in the world. In class, the puzzles will stimulate your logical and rigorous selves, and occasionally, your social selves!

H12110: Facilitating Emotional Awareness and Conversations - Circling in HSSP Spring 2018 (Feb. 24, 2018)
Have you ever felt overwhelmed, and not known how to deal with it? Have you ever wanted to connect more deeply with someone, and not known how? Have you ever found yourself in the middle of a situation with multiple people who felt strongly, and wished for the grace to navigate the social dynamics well? Have you ever wondered what it would be like to look out at the world through another person's eyes? Welcome to circling, a practice of sharing our in-the-moment experiences with each other, in real time, in service of connection, presence, and awareness. How this class looks depends on what you want. We might simply spend an hour or two each week in guided meditation, giving you a refuge from the obligations of everyday life and a place to develop your awareness of yourself. We might circle each week---have a group conversation about what it's like to be you, what it's like to be me, and what it's like to be us. If you're interested, we might guide you through the experience of facilitating a circle, of facilitating deeper emotional connections and conversations.

P12098: Your classical intuition is wrong! in Spark 2018 (Mar. 17 - 18, 2018)
Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe!

H12099: Communication in Navigating Relationships, Friendships, Family in Spark 2018 (Mar. 17 - 18, 2018)
Do you have questions or uncertainties about how to deal with dating, relationships, friendships, or family interactions? Do you have unmet desires for deeper connections, being more understood, or understanding others better? Bring your questions, thoughts, and insights, and let's discuss these topics! We may also practice some exercises that I've found especially useful in my ability to communicate, foster emotional safety for myself and others, and resolve conflicts.

H12100: Making deep friendships - Circling in Spark 2018 (Mar. 17 - 18, 2018)
Most of your Spark classes will be about objects and things. Some of your conversations will involve personal history, where you grew up, what you like and dislike. This class will be a third kind of conversation, about what our present experience is, as we're having it. Access to this level of conversation has a way of facilitating deep connections where you can feel deeply seen and welcomed. Circling is a practice about getting others' worlds, sharing what it's really like to be you, and having that be seen and reflected. Come experience the magic.

H12101: Circling - Intersubjective Mindfulness Meditation in Spark 2018 (Mar. 17 - 18, 2018)
Most of your Spark classes will be about objects and things. Some of your conversations will involve personal history, where you grew up, what you like and dislike. This class will be a third kind of conversation, about what our present experience is, as we're having it. There's a kind of magic to being deeply seen, and to being welcomed as you are. Circling is a practice about getting others' worlds, and sharing what it's really like to be you, and having that be seen and reflected. Come experience the magic

X12158: Winning Arguments with Grownups in Spark 2018 (Mar. 17 - 18, 2018)
Something like half of the disagreements between kids, teens, and their parents and teachers comes from just being on different communication wavelengths, rather than *actually* having incompatible goals. If you can fix the *way* that you're arguing, you can make the argument go away—and often in a way that leaves the grownup understanding what you want, and willing to help you get it.

S11951: Your classical intuition is wrong! in Splash 2017 (Nov. 18 - 19, 2017)
Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe!

H11953: Circling - Intersubjective Mindfulness Meditation in Splash 2017 (Nov. 18 - 19, 2017)
Most of your Splash classes will be about objects and things. Some of your conversations will involve personal history, where you grew up, what you like and dislike. This class will be a third kind of conversation, about what our present experience is, as we're having it. There's a kind of magic to being deeply seen, and to being welcomed as you are. Circling is a practice about getting others' worlds, and sharing what it's really like to be you, and having that be seen and reflected. Come experience the magic

H11959: Emotional Sensitivity Training in Splash 2017 (Nov. 18 - 19, 2017)
One of the most essential ingredients to successful emotional communication is awareness of what you're feeling, what others are feeling, and how these feelings evolve moment-by-moment. Come practice listening and sharing emotions and sensations as they arise moment-to-moment.

C11960: What is a "what if"? in Splash 2017 (Nov. 18 - 19, 2017)
Say there's a $5 bill and a$10 bill on a table. You can take either one of them. Which would you take? The $10 bill, right? What if you took the$5 bill? This is a surprisingly subtle question, especially when you consider the fact that you are a good decision-maker, and so must have had a good reason to take the $5 bill if you took it. It's subtle enough that many decision theories get it wrong. Come learn about four decision theories (evidential, causal, timeless, and updateless), and about the decision puzzles that demonstrate why none of these is an entirely satisfactory answer to the question "What is a 'what if'?" H11963: Making deep friendships - Circling in Splash 2017 (Nov. 18 - 19, 2017) Most of your Splash classes will be about objects and things. Some of your conversations will involve personal history, where you grew up, what you like and dislike. This class will be a third kind of conversation, about what our present experience is, as we're having it. Access to this level of conversation has a way of facilitating deep connections where you can feel deeply seen and welcomed. Circling is a practice about getting others' worlds, sharing what it's really like to be you, and having that be seen and reflected. Come experience the magic. H11971: Communication in Navigating Relationships, Friendships, Family in Splash 2017 (Nov. 18 - 19, 2017) Do you have questions or uncertainties about how to deal with dating, relationships, friendships, or family interactions? Do you have unmet desires for deeper connections, being more understood, or understanding others better? Bring your questions, thoughts, and insights, and let's discuss these topics! We may also practice some exercises that I've found especially useful in my ability to communicate, foster emotional safety for myself and others, and resolve conflicts. M11326: Counting Past Infinity in Spark 2017 (Mar. 11 - 12, 2017) Did you ever have arguments about whether or not $$\infty + 1 = \infty$$? Come learn how to count past infinity, graphically! P11327: Your classical intuition is wrong! in Spark 2017 (Mar. 11 - 12, 2017) Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe! C11328: What is a "what if"? in Spark 2017 (Mar. 11 - 12, 2017) Say there's a$5 bill and a $10 bill on a table. You can take either one of them. Which would you take? The$10 bill, right? What if you took the $5 bill? This is a surprisingly subtle question, especially when you consider the fact that you are a good decision-maker, and so must have had a good reason to take the$5 bill if you took it. It's subtle enough that many decision theories get it wrong. Come learn about four decision theories (evidential, causal, timeless, and updateless), and about the decision puzzles that demonstrate why none of these is an entirely satisfactory answer to the question "What is a 'what if'?"

M11347: Dutch Books, Probability, and Logical Induction in Spark 2017 (Mar. 11 - 12, 2017)
How likely is it that the first digit of $$\pi$$ is 3? It's a certainty. How likely is it that the $$10^{100}$$th digit of $$\pi$$ is 3? Probably $$1/10$$. What's the difference between these two questions, and what does it mean to ask about the likelihood of something that's either certainly true or certainly false? Come to this class to find out!

H11348: Authentic Relating Games in Spark 2017 (Mar. 11 - 12, 2017)
Do you detest small talk? Are you eager to discover new ways to interact with people? Do you crave deep and interesting and vulnerable connections with people, but aren't sure how to get them? Come play authentic relating games, short games that are designed to get you in touch with what's true and alive for you, and for your conversational partners! Example games we might play: - Curiosity - ask an individual any question you want (they can choose whether or not to answer), and the only rule is that you must be genuinely curious about your question - Noticing - take turns sharing back and forth with someone what you notice about your physical experience or emotions - Context conversations - e.g., I'd write an enthusiastic explanation of context conversations, because I want to share my excitement about this game. - Expression conducting - for two minutes, use your hands to "conduct" your partner's facial expressions

H11349: Circling - Intersubjective Meditation in Spark 2017 (Mar. 11 - 12, 2017)
Most of your Spark classes will be about objects and things. Some of your conversations will involve personal history, where you grew up, what you like and dislike. This class will be a third kind of conversation, about what our present experience is, as we're having it. There's a kind of magic to being deeply seen, and to being welcomed as you are. Circling is a practice about getting others' worlds, and sharing what it's really like to be you, and having that be seen and reflected. Come experience the magic

C10301: What is a "what if"? in Spark 2016 (Mar. 12 - 13, 2016)
Say there's a $5 bill and a$10 bill on a table. You can take either one of them. Which would you take? The $10 bill, right? What if you took the$5 bill? This is a surprisingly subtle question, especially when you consider the fact that you are a good decision-maker, and so must have had a good reason to take the $5 bill if you took it. It's subtle enough that many decision theories get it wrong. Come learn about four decision theories (evidential, causal, timeless, and updateless), and about the decision puzzles that demonstrate why none of these is an entirely satisfactory answer to the question "What is a 'what if'?" C10302: Robust Cooperation in the Prisoner's Dilemma: Program Equilibrium via Provability Logic in Spark 2016 (Mar. 12 - 13, 2016) The Prisoner's Dilemma is a standard example in game theory of a situation in which two "rational" players will choose not to cooperate, even though they would both be better off if they cooperated. However, if both players are mind readers, then it is possible to construct rational agents that cooperate without going in to infinite loops. P10303: Your classical intuition is wrong! in Spark 2016 (Mar. 12 - 13, 2016) Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe! M10304: Uncountability of the reals in Spark 2016 (Mar. 12 - 13, 2016) The set of real numbers is uncountable. No matter how you order them, if you try to count them one by one, you will miss some. M10305: Counting Past Infinity in Spark 2016 (Mar. 12 - 13, 2016) Did you ever have arguments about whether or not $$\infty + 1 = \infty$$? Come learn how to count past infinity, graphically! S9586: Your classical intuition is wrong! in Splash 2015 (Nov. 21 - 22, 2015) Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe! M9587: Linear Logic in Splash 2015 (Nov. 21 - 22, 2015) It's a well-known fact of logic that if from $$P$$ you can get $$Q$$, then from $$P$$ you can also get $$Q$$ AND $$Q$$ (see example below). So since you can get two dimes and a nickel from a quarter, you can get two dimes and a nickel and two dimes and a nickel from a single quarter. Come learn about linear logic, which is a version of logic which doesn't claim that you can get infinite amounts of money from a quarter. EXAMPLE: For example, since $$n = 2$$ implies that $$n = 1 + 1$$ then, $$n = 2$$ implies that $$n = 1 + 1$$ and also $$n = 1 + 1$$. M9588: Counting Past Infinity with Ordinal Arithmetic in Splash 2015 (Nov. 21 - 22, 2015) Did you ever have arguments about whether or not $$\infty + 1 = \infty$$? Have you ever wondered how to define numbers, rigorously? Come learn how to count past infinity, graphically! We'll then go over the formal definition of numbers as sets, and talk a little bit about ordinal arithmetic. C9589: Robust Cooperation in the Prisoner's Dilemma: Program Equilibrium via Provability Logic in Splash 2015 (Nov. 21 - 22, 2015) The Prisoner's Dilemma is a standard example in game theory of a situation in which two "rational" players will choose not to cooperate, even though they would both be better off if they cooperated. However, if both players are mind readers, then it is possible to construct rational agents that cooperate without going in to infinite loops. C9590: What is a "what if"? in Splash 2015 (Nov. 21 - 22, 2015) Say there's a$5 bill and a $10 bill on a table. You can take either one of them. Which would you take? The$10 bill, right? What if you took the $5 bill? This is a surprisingly subtle question, especially when you consider the fact that you are a good decision-maker, and so must have had a good reason to take the$5 bill if you took it. It's subtle enough that many decision theories get it wrong. Come learn about four decision theories (evidential, causal, timeless, and updateless), and about the decision puzzles that demonstrate why none of these is an entirely satisfactory answer to the question "What is a 'what if'?"

M9593: Proving Löb's Theorem in Splash 2015 (Nov. 21 - 22, 2015)
Löb's theorem is a beautiful theorem with a deceptively short proof. It states that $$\square (\square P \to P) \to \square P$$ for all $$P$$---that if you can show that proving $$P$$ is sufficient to make $$P$$ true, then you can prove $$P$$. Löb's theorem has a variety of applications, from enabling robust cooperation in the prisoner's dilemma, to curing social anxiety, from proving Gödel's incompleteness theorem, to proving that the halting problem is undecidable. I will present a few proofs of Löb's theorem, all of which are twisty in subtly different ways. We will spend the rest of the time working on wrapping our minds around these proofs, and discussing related topics.

C9599: Löb's Theorem: From the prisoner's dilemma to the halting problem, and back again in Splash 2015 (Nov. 21 - 22, 2015)
What do the prisoner's dilemma, the halting problem, and Gödel's incompleteness theorem have in common? Löb's Theorem! Come learn how Löb's theorem robust enables cooperation in the prisoner's dilemma! Come learn how, via the Curry-Howard isomorphism, the proof of Löb's theorem is essentially the same as the proof that the halting problem is undecidable! Come learn how Löb's theorem trivially proves Gödel's incompleteness theorem!

M9625: Homotopy Type Theory in Splash 2015 (Nov. 21 - 22, 2015)
What does it mean for two things to be equal? What does this have to do with loops of string and with the foundations of mathematics? Come to this class to find out! In the past few years, type theory has emerged as a possible replacement for set theory as a foundation of mathematics. Homotopy type theory is an exciting new way to base math on homotopy theory, in the setting of type theory.

C9626: Interactive Theorem Proving: The Law of Excluded Middle, Double Negation Elimination, and Peirce's Law in Splash 2015 (Nov. 21 - 22, 2015)
Are you interested in getting computers to do your math homework for you? Are you confused about what constitutes a "valid" proof? Are you interested in seeing computers check your proofs? Come experiment with the interactive proof assistant, Coq! I'll begin by talking a bit about the history of computer-assisted proofs, including the first proof of the four color theorem and the recent formalization of the odd order theorem. Then, you'll experiment with simple logic proofs in the Coq proof assistant, while I walk around and answer questions and give help. The central puzzle of the class will be proving the following three statements equivalent: 1. $$\forall P, P \vee \neg P$$ 2. $$\forall P, \neg\neg P \to P$$ 3. $$\forall P\ Q, ((P \to Q) \to P) \to P)$$ Near the end of class, I'll tell a story about how to use a time machine to solve P vs. NP (or to solve the Riemann Hypothesis); this story will prove that $$3 \Longrightarrow 1$$.

P9380: Your classical intuition is wrong! in Spark 2015 (Mar. 14 - 15, 2015)
Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe!

M9381: Counting Past Infinity in Spark 2015 (Mar. 14 - 15, 2015)
Did you ever have arguments about whether or not $$\infty + 1 = \infty$$? Come learn how to count past infinity, graphically!

C9382: Interactive Theorem Proving in Spark 2015 (Mar. 14 - 15, 2015)
Are you interested in getting computers to do your math homework for you? Are you confused about what constitutes a "valid" proof? Are you interested in seeing computers check your proofs? Come experiment with the interactive proof assistant, Coq! I'll begin by talking a bit about the history of computer-assisted proofs, including the first proof of the four color theorem and the recent formalization of the odd order theorem. Then, you'll experiment with simple arithmetic and logic proofs in the Coq proof assistant, while I walk around and answer questions and give help.

M9383: Homotopy Type Theory in Spark 2015 (Mar. 14 - 15, 2015)
What does it mean for two things to be equal? What does this have to do with loops of string and with the foundations of mathematics? Come to this class to find out! In the past few years, type theory has emerged as a possible replacement for set theory as a foundation of mathematics. Homotopy type theory is an exciting new way to base math on homotopy theory, in the setting of type theory. During the first hour, we'll talk about paths between points, and paths between paths, and I'll teach about the basics of homotopy theory. During the second hour, I'll explain how equality and isomorphism can be taken to be the same thing, and how equality looks a lot like paths, when you define it the right way.

M9384: Uncountability of the reals in Spark 2015 (Mar. 14 - 15, 2015)
The set of real numbers is uncountable. No matter how you order them, if you try to count them one by one, you will miss some.

C9392: Robust Cooperation in the Prisoner's Dilemma: Program Equilibrium via Provability Logic in Spark 2015 (Mar. 14 - 15, 2015)
The Prisoner's Dilemma is a standard example in game theory of a situation in which two "rational" players will choose not to cooperate, even though they would both be better off if they cooperated. However, if both players are mind readers, then it is possible to construct rational agents that cooperate without going in to infinite loops.

S8683: Your classical intuition is wrong! in Splash 2014 (Nov. 22 - 23, 2014)
Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe!

M8684: What is equality? in Splash 2014 (Nov. 22 - 23, 2014)
What does it mean for two things to be equal? What does this have to do with string and the foundations of mathematics? When are two proofs of equality themselves equal? Come to this class to discover these things for yourself! We'll begin with a discussion of the nature of equality, seguing with the presentation of a particular definition of equality that turns out to be surprisingly powerful and interestingly structured. Following that, you'll learn how to reason about equality, using this definition, in the proof assistant Coq, and explore the nature of equality.

M8685: Counting Past Infinity with Ordinal Arithmetic in Splash 2014 (Nov. 22 - 23, 2014)
Did you ever have arguments about whether or not $$\infty + 1 = \infty$$? Have you ever wondered how to define numbers, rigorously? Come learn how to count past infinity, graphically! We'll then go over the formal definition of numbers as sets, and talk a little bit about ordinal arithmetic.

M8686: Linear Logic in Splash 2014 (Nov. 22 - 23, 2014)
It's a well-known fact of logic that if from $$P$$ you can get $$Q$$, then from $$P$$ you can also get $$Q$$ AND $$Q$$ (see example below). So since you can get two dimes and a nickel from a quarter, you can get two dimes and a nickel and two dimes and a nickel from a single quarter. Come learn about linear logic, which is a version of logic which doesn't claim that you can get infinite amounts of money from a quarter. EXAMPLE: For example, since $$n = 2$$ implies that $$n = 1 + 1$$ then, $$n = 2$$ implies that $$n = 1 + 1$$ and also $$n = 1 + 1$$.

M8279: Counting Past Infinity in Spark 2014 (Mar. 15 - 16, 2014)
Did you ever have arguments about whether or not $$\infty + 1 = \infty$$? Come learn how to count past infinity, graphically!

S8280: Your classical intuition is wrong! in Spark 2014 (Mar. 15 - 16, 2014)
Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe!

C7435: Automated Theorem Proving in Splash! 2013 (Nov. 23 - 24, 2013)
Are you interested in getting computers to do your math homework for you? Are you confused about what constitutes a "valid" proof? Are you interested in seeing computers check your proofs? Come experiment with the interactive proof assistant, Coq! I'll begin by talking a bit about the history of computer-assisted proofs, including the first proof of the four color theorem and the recent formalization of the odd order theorem. Then, you'll experiment with simple arithmetic and logic proofs in the Coq proof assistant, while I walk around and answer questions and give help.

M7436: Thinking Categorically and the Curry-Howard Isomorphism in Splash! 2013 (Nov. 23 - 24, 2013)
Did you know that logic and set theory are, in some sense, the same thing? Did you know that proofs and programs are, in some sense the same thing? Come learn about category theory, a beautiful and abstract mathematical language which is useful for unifying various areas of math.

M7437: Counting Past Infinity with Ordinal Arithmetic in Splash! 2013 (Nov. 23 - 24, 2013)
Did you ever have arguments about whether or not $$\infty + 1 = \infty$$? Have you ever wondered how to define numbers, rigorously? Come learn how to count past infinity, graphically! We'll then go over the formal definition of numbers as sets, and talk a little bit about ordinal arithmetic.

M7438: Linear Logic in Splash! 2013 (Nov. 23 - 24, 2013)
It's a well-known fact of logic that if from $$P$$ you can get $$Q$$, then from $$P$$ you can also get $$Q$$ and $$Q$$.* So since you can get two dimes and a nickel from a quarter, you can get two dimes and a nickel and two dimes and a nickel from a single quarter. Come to learn about linear logic, which is a version of logic which doesn't claim that you can get infinite amounts of money from a quarter. *For example, since $$n = 2$$ implies that $$n = 1 + 1$$ then, $$n = 2$$ implies that $$n = 1 + 1$$ and also $$n = 1 + 1$$.

M7441: Homotopy Type Theory in Splash! 2013 (Nov. 23 - 24, 2013)
What does it mean for two things to be equal? What does this have to do with string and the foundations of mathematics? Come to this class to find out! In the past few years, type theory has emerged as a possible replacement for set theory as a foundation of mathematics. Homotopy type theory is an exciting new way to base math on homotopy theory, in the setting of type theory. During the first hour, we'll talk about paths between points, and paths between paths, and I'll teach about the basics of homotopy theory. During the second hour, I'll explain how equality and isomorphism can be taken to be the same thing, and how equality looks a lot like paths, when you define it the right way.

M7886: Ask anything about math! in Splash! 2013 (Nov. 23 - 24, 2013)
Come ask all the math questions you've been dying to have answered to a panel of MIT math majors! We'll answer anything from conceptual questions (what are Lagrange multipliers?) to computational questions (how do I compute this integral?) to philosophical questions (what is math?)! We'll have teachers studying all areas of math, so hopefully we can answer any (reasonable) questions you throw at us.

M7070: Thinking Categorically in Spark! 2013 (Mar. 16, 2013)
Did you know that logic and set theory are, in some sense, the same thing? Did you know that proofs and programs are, in some sense the same thing? Come learn about category theory, a beautiful and abstract mathematical language which is useful for unifying various areas of math.

M7074: Counting Past Infinity in Spark! 2013 (Mar. 16, 2013)
Did you ever have arguments about whether or not $$\infty + 1 = \infty$$? Come learn how to count past infinity, graphically!

M7075: Counting Past Infinity with Ordinal Arithmetic in Spark! 2013 (Mar. 16, 2013)
Did you ever have arguments about whether or not $$\infty + 1 = \infty$$? Have you ever wondered how to define numbers, rigorously? Come learn how to count past infinity, graphically! We'll then go over the formal definition of numbers as sets, and talk a little bit about ordinal arithmetic.

M6052: Your classical intuition is wrong! in ESPrinkler Summer 2012 (Jul. 08 - Aug. 19, 2012)
Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe!

S5724: Your classical intuition is wrong! in Spark! 2012 (Mar. 10, 2012)
Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe!

M5725: Counting Past Infinity in Spark! 2012 (Mar. 10, 2012)
Did you ever have arguments about whether or not $$\infty + 1 = \infty$$? Come learn how to count past infinity, graphically! Time permitting, we'll then go over the formal definition of numbers as sets, and talk a little bit about cardinal and ordinal arithmetic.

C5726: Learn how to [learn] LaTeX in Spark! 2012 (Mar. 10, 2012)
Want to learn how to use LaTeX to format your mathematical formulae like this: $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$? Want to make your English teachers think you’re crazy for having your papers formatted nicely in scientific form? Come learn the basics of LaTeX, the standard mathematical typesetting language. Works on any platform. We provide the computers. But I’ll tell you how to install it on your own computers. If you want to use your laptop instead, you should install MiKTeX and TeXnicCenter (either together from http://www.tug.org/protext/ or separately from http://miktex.org/2.8/setup and http://www.texniccenter.org/resources/downloads/29), or another LaTeX editor (if you don’t use windows) before you arrive; the installation of MiKTeX can take about half an hour to an hour.

S4903: Your classical intuition is wrong! in Splash! 2011 (Nov. 19 - 20, 2011)
Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe!

C4904: Learn how to [learn] LaTeX in Splash! 2011 (Nov. 19 - 20, 2011)
Want to learn how to use LaTeX to format your mathematical formulae like this: $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$? Want to make your English teachers think you’re crazy for having your papers formatted nicely in scientific form? Come learn the basics of LaTeX, the standard mathematical typesetting language. Works on any platform. We provide the computers. But I’ll tell you how to install it on your own computers. If you want to use your laptop instead, you should install MiKTeX and TeXnicCenter (either together from http://www.tug.org/protext/, or separately from http://miktex.org/2.8/setup and http://www.texniccenter.org/resources/downloads/29), or another LaTeX editor (if you don’t use windows) before you arrive; the installation of MiKTeX can take about half an hour to an hour.

S5276: Your classical intuition is wrong! in SPICY Delve 2011 (Oct. 23, 2011)
Do you think you have a good grasp of physics? A good intuition for the physical world? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe!

S4747: How Quantum Mechanics Breaks Your Intuition About the World in ESPrinkler Summer 2011 (Jul. 10 - Aug. 21, 2011)
Do you think you have good physical intuition and a good grasp of physics? Come learn about the predictions of quantum mechanics (and the experiments that validate these predictions) which violate your fundamental beliefs about the universe!

C4588: Become a LaTeXer! in Spark! 2011 (Mar. 12, 2011)
Want to learn how to use LaTeX to format your mathematical formulae like this: $$\sum_{i=1}^\infty\frac{1}{n^2} = \frac{\pi^2}{6}$$? Want to make your English teachers think you’re crazy for having your papers formatted nicely in scientific form? Come learn the basics of LaTeX, the standard mathematical typesetting language. Works on any platform. We provide the computers. But we’ll tell you how to install it on your own computers. Although we’ll provide example mathematics to typeset, you’ll probably get more out of the class if you bring your own mathematics to typeset (e.g. notes or homework from your math class). If you want to use your laptop instead, you should install MiKTeX and TeXnicCenter (either together from http://www.tug.org/protext/, or separately from http://miktex.org/2.8/setup and http://www.texniccenter.org/resources/downloads/29), or another LaTeX editor (if you don’t use windows) before you arrive; the installation of MiKTeX can take about half an hour to an hour.

H4589: Experimental Philosophy in Spark! 2011 (Mar. 12, 2011)
Do you think philosophy is all in your head? Experimental philosophy is a growing field that uses experimental data—usually, surveys of ordinary non-philosophers—to answer philosophical questions. We'll look at some of the questions experimental philosophers are trying to answer. How do people's intuitions about ethical problems vary across cultures? What is consciousness? When are people responsible for their actions? And finally—Can we learn anything about philosophy by doing experiments?

M3849: Become a LaTeXer! in Splash! 2010 (Nov. 20 - 21, 2010)
Want to learn how to use LaTeX to format your mathematical formulae like this: $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$? Want to make your English teachers think you’re crazy for having your papers formatted nicely in scientific form? Come learn the basics of LaTeX, the standard mathematical typesetting language. Works on any platform. We provide the computers. But I'll tell you how to install it on your own computers. Although we’ll provide example mathematics to typeset, you’ll probably get more out of the class if you bring your own mathematics to typeset (e.g. notes or homework from your math class). If you want to use your laptop instead, you should install MiKTeX and TeXnicCenter (either together from http://www.tug.org/protext/, or separately from http://miktex.org/2.8/setup and http://www.texniccenter.org/resources/downloads/29), or another LaTeX editor (if you don’t use windows) before you arrive; the installation of MiKTeX can take about half an hour to an hour.

H4023: Free-Form Philosophy in Splash! 2010 (Nov. 20 - 21, 2010)
This isn't a class, really. In the tradition of philosophers since Socrates, we don't know anything: so we're not going to try to teach you anything. We will discuss questions, however. We'll talk about some classic philosophical dilemmas, but feel free to ask about your own ideas! No previous philosophy experience is required, but willingness to participate is a must.

M3579: The Magic of Matrices in HSSP Summer 2010 (Jul. 11, 2010 - Jul. 11, 2011)
In arithmetic, if you multiply two numbers, you get the same result no matter which order you put them in. Learn what a matrix is and why the product of two matrices depends on which order you multiply them in. The class will cover determinants, matrix multiplication, row operations, and inverses.

M3175: Circuits over Sets of Natural Numbers in Spark! 2010 (Mar. 13, 2010)
This class will be a discussion about the math research I’ve been doing for the past three years. I will talk about the various problems I’ve been investigating (see below for a brief overview), and then we will discuss the progress that I’ve made, the open problems I’m currently working on, or whatever aspects of the topic you find interesting. Consider the operations union, intersection, complementation, addition, and multiplication on sets of natural numbers (where the sum or product of two sets is the set of sums or products). What are all the sets that can be constructed, starting from one element sets of natural numbers? For example, the set of composite numbers (along with 0) is definable as $$\overline{\{1\}} \times \overline{\{1\}}$$ (where ¯ is complementation). The set of squares (1, 4, 9, 16, …), however, is believed to not be definable. Another question we can ask is, is it possible (and if so, how long will it take) to decide if a given number is in the set defined by a given formula? What if you disallow some of the operations?

M3198: Become a LaTeXer! in Spark! 2010 (Mar. 13, 2010)
Want to learn how to use LaTeX to format your mathematical formulae like this: $$\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$$? Want to make your English teachers think you’re crazy for having your papers formatted nicely in scientific form? Come learn the basics of LaTeX, the standard mathematical typesetting language. Works on any platform. We provide the computers. But we’ll tell you how to install it on your own computers. Although we’ll provide example mathematics to typeset, you’ll probably get more out of the class if you bring your own mathematics to typeset (e.g. notes or homework from your math class). If you want to use your laptop instead, you should install MiKTeX and TeXnicCenter (either together from <a href="http://www.tug.org/protext/">http://www.tug.org/protext/</a>, or separately from <a href="http://miktex.org/2.8/setup">http://miktex.org/2.8/setup</a> and <a href="http://www.texniccenter.org/resources/downloads/29">http://www.texniccenter.org/resources/downloads/29</a>), or another LaTeX editor (if you don’t use windows) before you arrive; the installation of MiKTeX can take about half an hour to an hour.

H3286: Free-form Philosophy in Spark! 2010 (Mar. 13, 2010)
Since the days of Socrates, and his dialogues with random passers-by in the town square, wandering philosophical discussion has been an important part of the noble history of philosophy. So come and philosophize!! The discussion will be mostly driven by the students; we'll provide some questions to spark the conversation and some background on the Socratic method and useful philosophical debate.

M3343: Become a LaTeX Guru in HSSP Spring 2010 (Apr. 17, 2010)
LaTeX is typesetting language that can be used to make fancy formulas look beautiful. LaTeX is also a <a href="http://en.wikipedia.org/wiki/Turing_completeness">Turing complete</a> programming language that can be used to do just about anything you want it to! I intend for this class to be mostly exercise driven; I will give out one or more exercises each lecture, teach you a few of the basic concepts or point you to useful resources, and spend the rest of the time helping you do the exercises. In the first class, I will teach you the basics of LaTeX. The exercise sheet will have exercises of all levels, from easy to diabolical (so all experience levels are welcome). The topics of the remaining four weeks will vary, and requests are welcome. Do you want to know how to make pretty bibliographies? Draw pretty pictures in LaTeX? Make LaTeX do your math homework for you? Have LaTeX typeset as many lines of Pascal's Triangle as you tell it to? Write your own package (so you can \usepackage{my-package})? Write a replacement for the article document class, so that LaTeX makes your pages look however you want them to? Get LaTeX to do things that it was never meant to do (like search the Internet for you, or become your favorite programming language (albeit slower))? I can help you figure out how to do all of these things, and more, if you attend my class! Come learn just how powerful LaTeX is! Computers will be provided.

M2673: Circuits over Sets of Natural Numbers in Splash! 2009 (Nov. 21 - 22, 2009)
This class will be a discussion about the math research I've been doing for the past three years. I will talk about the various problems I've been investigating (see below for a brief overview), and then we will discuss the progress that I've made, the open problems I'm currently working on, or whatever aspects of the topic you find interesting. Consider the operations union, intersection, complementation, addition, and multiplication on sets of natural numbers (where the sum or product of two sets is the set of sums or products). What are all the sets that can be constructed, starting from one element sets of natural numbers? For example, the set of composite numbers (along with 0) is definable as $$\overline{\{1\}}\times \overline{\{1\}}$$ (where ¯ is complementation). The set of squares (1, 4, 9, 16, ...), however, is believed to not be definable. Another question we can ask is, is it possible (and if so, how long will it take) to decide if a given number is in the set defined by a given formula? What if you disallow some of the operations?

M2852: Become a LaTeXer! in Splash! 2009 (Nov. 21 - 22, 2009)
Want to learn how to use LaTeX to format your mathematical formulae like this: $$\sum_{n = 1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$$? Want to make your English teachers think you're crazy for having your papers formatted nicely in scientific form? Come learn the basics of LaTeX, the standard mathematical typesetting language. Works on any platform. We provide the computers. But we'll tell you how to install it on your own computers. Although we'll provide example mathematics to typeset, you'll probably get more out of the class if you bring your own mathematics to typeset (e.g. notes or homework from your math class). If you want to use your laptop instead, you should install MiKTeX and TeXnicCenter (either together from http://www.tug.org/protext/, or separately from http://miktex.org/2.8/setup and http://www.texniccenter.org/resources/downloads/29), or another LaTeX editor (if you don't use windows) before you arrive; the installation of MiKTeX can take about half an hour to an hour.