# ESP Biography

## BENJAMIN CHURCH, ESP Teacher

Major: Mathematics / Astrophysics

College/Employer: Student at Columbia University

Not Available.

## Past Classes

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

M14072: Elliptic Curves, Complex Tori, and Fermat's Last Theorem in HSSP Summer 2020 (Jul. 11, 2020)
Elliptic curves are a class of cubic curves (defined by a cubic polynomial in two variables) with deeply surprising and beautiful properties. The most surprising fact about elliptic curves is that their points naturally form an additive structure under some geometric operation. Elliptic curves show up everywhere in mathematics from tori defined over the complex numbers to generating abelian extensions of number fields. Elliptic curves also feature prominently in the Birch and Swinnerton-Dyer conjecture, one of the Millenium prize problems, and Andrew Wiles' proof of Fermat's last theorem. In this class, we will develop some fundamental results about elliptic curves such as Weierstrass function theory, the group law, and modular forms before diving head-first into some more advanced territory mentioned above. This class will be a meandering relentless rollercoaster through some of the most beautiful connections in modern mathematics rather than a reasonably-paced, well-structured, and rigorous development of a topic. Expect proofs to be "sketched", definitions to be hand waved, and lots of inane terminology to be used. That said, expect to come away with a broader awareness of open problems in mathematics, a much richer appreciation of the interplay between geometry and algebra, and a healthy respect for cubic polynomials in two variables.

M13180: Elliptic Curves, Complex Tori, and maybe a few l-adic Galois Representations in Splash 2019 (Nov. 23 - 24, 2019)
Elliptic curves are a class of cubic curves (defined by a cubic polynomial in two variables) with deeply surprising and beautiful properties. The most surprising fact about elliptic curves is that their points naturally form a group structure under some geometric operation. Elliptic curves show up everywhere in mathematics from tori defined over the complex numbers to generating abelian extensions of number fields. Elliptic curves also feature prominently in the Birch and Swinnerton-Dyer conjecture, one of the millenium prize problems, and Andrew Wiles' proof of Fermat's last theorem. In this class we will develop the fundamental results about elliptic curves such as the Mordell-Weil theorem and Weierstrass function theory before diving head-first into some more advanced territory mentioned above. This class will be a meandering relentless rollercoaster through some of the most beautiful connections in modern mathematics rather than a reasonably-paced, well-structured, and rigorous development of a topic. Expect proofs to be "sketched", definitions to be hand waved, and lots of inane terminology to be used. That said, expect to come away with an broader awareness of open problems in mathematics, a much richer appreciation of the interplay between geometry and algebra, and a healthy respect for cubic polynomials in two variables.

X13188: A Treatise on the Modern Meme in Splash 2019 (Nov. 23 - 24, 2019)
To meme or not to meme, that is the question: Whether ‘tis nobler in the Sub to repost The gifs and jokes of yester year Or to create OC against the sea of reposts And in downvoting end them. To OC–to post; To post perchance to meme–ay there’s the sub, For in that post of OC what memes may come, When we have shuffled off this bottom text, Must give us pause–But that the dread of spici’r OC, That undiscover’d board, for whose born no memelord returns, puzzles the will, And makes us rather post the memes we have Than create others we not of? Thus Four Chan doth make normies of us all.

M13219: A Grand Tour of Rotations, Quaternions, The Hopf Fibration and Spin with a side dish of Lie Groups in Splash 2019 (Nov. 23 - 24, 2019)
We will begin a fantastical journey into some of the most beautiful and useful geometric objects in modern mathematics, Lie groups, by asking the simple question: how do we represent rotations in 3D space. This question will lead us to define a strange algebraic object, the quaternions, investigate the mysterious topology of spheres living in four (and more) dimensions, marvel at a beautiful images of a sphere in dimension four decomposed into tori by the Hopf fibration, and finally discuss how these higher-dimensional geometric objects are, in fact, physically realized by spin in the strange world of quantum mechanics.

M13220: A Headfirst Dive Into "Mathematical Logic" in Splash 2019 (Nov. 23 - 24, 2019)
The meaning of term "Mathematical Logic" is fairly non-trivial. Mathematical logic is, on the one hand, the study of the logic of mathematics rigorizing the notions of "proof" and "example" in the framework of formal logic. But Mathematical logic is also the application of mathematical methods to logic using tools such as induction and set theory to proof meta-theorems about logic. It is even the application of logic to solving (somewhat) concrete math problems. In this course we will discuss all these flavors of mathematical logic as we introduce the basic concepts of completeness, consistency, satisfiability, and categoricity, discuss foundational results linking model theory (the study of examples) to proof theory (the study of formal proofs), then investigate the limitations of first-order logic, and finally prove Godel's momentous incompleteness theorem of first-order arithmetic. On the way, we will naturally develop foundational ideas about the theory of computation and how decidability and incompleteness are intricately linked. Time permitting, we will discuss applications of mathematical logic to problems in algebraic geometry such as the Ax-Grothendieck theorem and Lefschetz principle.

H13436: The Apotheosis of Captain Cook in Splash 2019 (Nov. 23 - 24, 2019)
On the 17 of January, 1779, the sails of The Resolution peaked above the eastern horizon of Kealakekua bay, Hawaii. At her helm was James Cook, an industrious British explorer searching for the western route to Asia. At the time of Cook’s arrival, the Polynesian natives of Kealakekua were joyously assembled on the beach–in the midst of celebrating their annual Makahiki festival honoring the sea god Lono. When Cook landed, he was celebrated by the natives as the god Lono: he was draped in fine red fabrics (the color of Polynesian divinity); prostrated to and given sacrifices of fruit and roast pigs; and paraded around the island while the Hawaiians chanted “Lono, Lono, Lono!” …Or at least, that’s how the story goes. In his book, The Apotheosis of Captain Cook, Gananath Obeyesekere rejected the position held by “every biographer and historian of Cook” that the Englishman was interpreted as a Polynesian God. Obeyesekere argued that the sources supporting the apotheosis are weak at best, and are entangled with gossip and myth. Obeyesekere’s skepticism of apotheosis sources is not unfounded; in a 1982 lecture at Princeton University, Polynesian Historian Marshall Salhins declared that the Hawaiian king was so distraught at the departure of Cook that: “By all accounts, British as well as Hawaiian, they told him such sad stories as the death of kings as to force him to sit upon the ground” (so that Cook could leave). Obeyesekere notes that these “accounts” are a blatant plagiarism of Shakespeare’s Richard II, and therefore cannot be legitimate: “For god’s sake, let us sit upon the ground/ And tell sad stories of the death of kings (Richard II, act III scene II).” This course will examine the historiographic debate surrounding the Apotheosis of Captain cook. The main historiographic raised by the ‘Apotheosis question’ is how to elucidate the beliefs of native peoples in the absence of a substantial source archive. The Polynesians left no written records from 1779; discerning whether they actually understood Captain Cook as Lono presents a serious problem for Polynesian historians. By analyzing the discourse surrounding the ‘Apotheosis question,’ we can make larger statements about historical methodology, epistemological frameworks, and interpretation of an incomplete source record.

S13437: Cortisol Communities: Stress, the Balkan Wars, and the Psychophysical Origins of Turkish Nationalism in Splash 2019 (Nov. 23 - 24, 2019)
The Balkan Wars of 1912-1913 were a series of conflicts between the Ottoman Empire and its former European territories which triggered brutal violence against Balkan Muslim civilians. Of the 2.3 million Muslims living in the Balkans prior to the wars, approximately 27% died in under six years as a result of attrition, famine, and disease outbreaks, and another 35% fled as refugees to Anatolia and Eastern Thrace. This tragedy represented a defining moment in Turkish national history. Before 1912 most Ottoman Muslims had associated with numerous linguistic, cultural, and ethnic backgrounds; however, the traumatic expulsion of 813,000 Balkan Muslims to Anatolia catalyzed a surge in the popularity of ethnic ‘Turkish’ nationalism among the geographically-consolidated Ottoman-Muslim polity. I argue that this defining moment in Turkish history has continued to impact the modern population, not only through historical memory, but through heritable biochemical memory. This course will question whether the psychophysical stress of Balkan Muslim refugees sustained during the Balkan Wars of 1912-1913 left an epigenetic mark on their living descendants. This course will serve as an introduction to the field of intergenerational psychoepigenetics, a discipline which spans biochemistry, psychology, and history. I will provide an overview of the biochemical research project I am planning to perform in Turkey the year after I graduate college in 2020 (I am working towards a dual degree in Biochemistry and Ottoman History). We will perform a literature review of existing intergenerational psychoepigenetic studies, learn the basics of what epigenetics is as a biochemical phenomenon, and parse out common misperseptions about what this discipline can and cannot tell us; specifically, we will be interested in the intergenerational epigenetic impact of trauma. Next we will situate ourselves within the historical context of the Balkan Wars, and their relevance to the modern country of Turkey. Last we will talk about how my proposed biochemical project would be conducted, which living people in the modern population would be best to include in the present study, and methods of biochemical analysis.

S13438: Nuclear Magnetic Resonance (NMR): Physical Theory and Chemical Applications in Splash 2019 (Nov. 23 - 24, 2019)
Nuclear Magnetic Resonance (NMR) is one of the most imporant tools in organic chemistry for identifying the composition and detailed electronic structure of compounds. NMR is a quantum mechanical effect of the interaction of the nucleus of an atom (or generally each nucleus comprising a complex molecule) with a strong applied magnetic field. Specifically, the effect is due to the strange nature of quantum spin, intrinsic angular momentum carried by the nucleus and how spin couples to external magnetic fieds. Furthermore, this interaction effect is senstive to the nearby nucleli and electronic envioronment so that each nucleus will have a distinct NRM resonance figerprint associated to its position in the molecular structure alowing NRM spectroscopy to ID the molecule. In this course we will discuss the physics of quantum mechanical spins in magnetic fields and the process by which they absorb radiation at specific resonance frequencies. We will then shiow how NRM spectroscopy is used in practice and the process by which compounds may be identified. Finally, if time permits we will discuss other chemical and medical applications of NMR phenomena.

H13443: An 'Abrahamic' Empire: The Philosophical and Theological Foundations of the pre-Modern Ottoman State in Splash 2019 (Nov. 23 - 24, 2019)
I have a chapter of my senior thesis due Monday I am grinding on, so I will submit the course abstract columbus day weekend.

M12376: The Foundation Crisis in Mathematics in Splash 2018 (Nov. 17 - 18, 2018)
For centuries, mathematics was considered to be the most stable and deductive reasoning, which gives results with absolute certainty, it was long believed that mathematical knowledge was beyond doubt. But at the end of the 19th century and the beginning of the 20th, several developments shook our faith in the unshakable nature of mathematical reasoning. The emergence of non-Euclidean geometry undermined absolute acceptance of the theory of space and shape that had reigned since classical Greece. Gregor Cantor’s work on the nature of infinity forced us to rethink our sense of numbers. And Kurt Gödel’s incompleteness theorem cast doubt on the possibility of a completely well-grounded notion of mathematical truth. In this class, we will explore the fundamental philosophical uncertainty in mathematics, and hopefully I will convince you that the math you have known and loved your whole life is built on shaky ground. This class will include discussions on the philosophy of math and whether it is grounded in the real world, or own minds, or somewhere far stranger.

P12416: Neon Genesis Evangelion: A Dialectic in Splash 2018 (Nov. 17 - 18, 2018)
What is Evangelion? A cult phenomenon 90s anime which ran out of money and ideas? An overhyped robot show that was just a vehicle for a lonely man's insecurities? Or a misunderstood masterpiece of modern culture? In this talk, we will discuss how a strange Japanese cartoon two decades old might have startling implications for your life and the nature of the human spirit. Evangelion is a show about middle-school kids getting in big "robots" and saving the world. It is also a deconstructive subversion of anime itself, a terrifying exploration of the mentality of such child soldiers and the fundamental brokenness of human relationships. Expect to leave singing the words "come, sweet death."

M12418: A Headfirst Dive Into "Mathematical Logic" in Splash 2018 (Nov. 17 - 18, 2018)
The meaning of term "Mathematical Logic" is fairly non-trivial. Mathematical logic is, on the one hand, the study of the logic of mathematics rigorizing the notions of "proof" and "example" in the framework of formal logic. But Mathematical logic is also the application of mathematical methods to logic using tools such as induction and set theory to proof meta-theorems about logic. It is even the application of logic to solving (somewhat) concrete math problems. In this course we will discuss all these flavors of mathematical logic as we introduce the basic concepts of completeness, consistency, satisfiability, and categoricity, discuss foundational results linking model theory (the study of examples) to proof theory (the study of formal proofs), then investigate the limitations of first-order logic, and finally prove Godel's momentous incompleteness theorem of first-order arithmetic. On the way, we will naturally develop foundational ideas about the theory of computation and how decidability and incompleteness are intricately linked. Time permitting, we will discuss applications of mathematical logic to problems in algebraic geometry such as the Ax-Grothendieck theorem and Lefschetz principle.

M12419: The Baire Category Theorem in Splash 2018 (Nov. 17 - 18, 2018)
Does there exist a function which is continuous exactly on the irrational numbers? It turns out the answer is yes: https://en.wikipedia.org/wiki/Thomae%27s_function. The natural next question is: "What about he complement? Is there a function exactly continuous on the rationals?" The answer to this is, somewhat surprisingly, no! To answer this question we will need to develop some powerful tools in analysis such as the Baire Category Theorem and Borel Hierarchy which tell us about what happens when we take repeated intersections, unions, and complements of infinitely many open sets.

S12422: A Rigorous Development of Classical Mechanics in Splash 2018 (Nov. 17 - 18, 2018)
This class will discuss classical mechanics from a fairly abstract perspective. We will first introduce the Lagrangian formalism and discuss the principle of least action, its motivation, and its consequences. Next, using the Lagrangian perspective, we will develop the Hamiltonian framework which elegantly formalizes and extends the notions of momentum and energy. From there, we will discuss the theory of canonical transformations leading to Hamilton-Jacobi theory. If time permits, we will investigate advanced topics such as adiabatic invariants and show how Hamilton-Jacobi theory is the "ray-optics approximation" to the Schrodinger equation and how this insight lead Schrodinger to develop quantum theory. Along the way, we will discuss Noether's theorem on the connection between symmetries and conservation laws from each of these perspectives since each one gives their own flavor to this wonderful theorem.

S12480: Medicine of the Future: Discovering the Cure to Cancer in Splash 2018 (Nov. 17 - 18, 2018)
Cancer is an inevitable part of Human life. This phenomenon occurs because the human genome mutates innumerable times a day, and eventually these mutations will lead to uncontrolled cell growth (i.e. cancer). Despite significant pharmacological efforts over the past seventy years to treat various forms of cancer, most treatments are still insufficient, and many types of cancer are still considered “un-drugable”. But today, we are on the brink of a pharmaceutical revolution that will change how we think about treating cancer. In this class, I will be talking about my own lab research on combating mutated genes that lead to cancer, and how this research is into the larger field of experimental pharmacology. For those interested, I work on identifying potential allosteric sites on oncogenic proteins that can be targeted by small organic molecules as a means to inhibit constitutive activation. (All big science words, I know. I don’t expect you to know any of them, just come ready to learn and ask questions!)

X12481: A Treatise on the Modern Meme in Splash 2018 (Nov. 17 - 18, 2018)
To meme or not to meme, that is the question: Whether ‘tis nobler in the Sub to repost The gifs and jokes of yester year Or to create OC against the sea of reposts And in downvoting end them. To OC–to post; To post perchance to meme–ay there’s the sub, For in that post of OC what memes may come, When we have shuffled off this bottom text, Must give us pause–But that the dread of spici’r OC, That undiscover’d board, for whose born no memelord returns, puzzles the will, And makes us rather post the memes we have Than create others we not of? Thus Four Chan doth make normies of us all.

M12581: Elliptic Curves, Complex Torii, and maybe a few l-adic Galois Representations in Splash 2018 (Nov. 17 - 18, 2018)
Elliptic curves are a class of cubic curves (defined by a cubic polynomial in two variables) with deeply surprising and beautiful properties. The most surprising fact about elliptic curves is that their points naturally form a group structure under some geometric operation. Elliptic curves show up everywhere in mathematics from torii defined over the complex numbers to generating abelian extensions of number fields. Elliptic curves also feature prominently in the Birch and Swinnerton-Dyer conjecture, one of the millenium prize problems, and Andrew Wiles' proof of Fermat's last theorem. In this class we will develop the fundamental results about elliptic curves such as the Mordell-Weil theorem and Weierstrass function theory before diving head-first into some more advanced territory mentioned above. This class will be a meandering relentless rollercoaster through some of the most beautiful connections in modern mathematics rather than a reasonably-paced, well-structured, and rigorous development of a topic. Expect proofs to be "sketched", definitions to be hand waved, and lots of inane terminology to be used. That said, expect to come away with an broader awareness of open problems in mathematics, a much richer appreciation of the interplay between geometry and algebra, and a healthy respect for cubic polynomials in two variables.