Wednesdays, 7:30pm; Room 507, Mathematics

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*ums* [at] *math.columbia.edu*

Date | Speaker | Title | Abstract |

Jan 27 | Carl Lian | Lines in 3-space | Given four (sufficiently general) lines in 3-space, how many lines intersect all four? To answer this question, we will introduce some fundamental notions in algebraic geometry and intersection theory. We will attempt to keep the discussion as elementary as possible. |

Feb 3 | Rob Castellano |
Boundaries of manifolds and classical |
One of the first questions that one can ask is "what geometric objects are boundaries of other geometric objects?" This question turns out to be very important and, depending on the context, also interesting. This talk will go over some basic topology and introduce the field of symplectic geometry. |

Feb 10 | Anand Deopurkar | Fermat's last theorem for polynomials |
Fermat's last theorem says that X^{n }+ Y^{n }= Z^{n} has no non-trivial solutions for integers X, Y, Z, if n > 2. We will prove Fermat's last theorem where "integers" is replaced by "polynomials" (in one variable). We will see many different proofs, some elementary and some more sophisticated, using ideas from algebra, topology, and algebraic geometry. |

Feb 17 | Vivek Pal | Puzzles and Paradoxes | In this talk I will show you many puzzles and paradoxes. They will highlight some cool math concepts and are generally fun to work on. |

Feb 24 | James Cornish | Knot Theory and Knot Genus | In this talk we'll explore knot theory through looking at a particular invariant, the genus of a knot. What is knot genus good for? How can you find the genus of a knot? For the later question, we'll learn about the Alexander polynomial and, if there's time, talk a little about knot Floer homology. |

March 2 | Alex Zhang | Morse theory | I plan to do some intro in Morse theory, namely the homotopy type in terms of critical values of Morse functions and some examples, if time allows, I will talk about the Morse inequality also. |

March 9 | Feiqi Jiang | Modular forms | Modular forms are holomorphic functions on the upper-half-plane that are nearly invariant under actions by a subgroup of the group of fractional linear transformations. Roughly speaking they provide the tools required to parametrize complex elliptic curves (and some additional data) with quotients of the upper-half-plane. Using these ideas we will sketch the proof of a surprising result in complex analysis about the image of holomorphic functions. |

March 16 | Spring break | ||

March 23 | Daniel Halpern-Leistner | The representation theory of SU(2) | The group of 2x2 special unitary matrices is one of the most fundamental objects in pure mathematics. It has deep connections with quantum physics, and it pops up unexpectedly in many other places as well, such as in Hodge theory. I will discuss the representation theory of SU(2) as an introduction to the theory of Lie groups and representation theory. In fact, understanding the representation theory of SU(2) is more than a toy example -- it plays a key role in both of these subjects. I will also discuss connections with quantum mechanics and the theory of spherical harmonics. |

March 30 | Daniel Litt | Zeros in integer linear progressions | I'll discuss the locations of zeros in sequences of integers defined by linear recurrences, and use these questions to motivate the use of p-adic analysis. Time permitting, I'll prove a beautiful theorem of Mahler, Skolem, and Lech describing the possible zero sets of integer linear recurrences. |

April 6 | Pak-Hin Lee | Rational elliptic curves have no 11-torsion | One principle in modern algebraic geometry is that questions about geometric objects can be fruitfully answered by studying the family of all objects of the same type, i.e., by looking at the moduli space. As an illustration of this principle, we will sketch a proof that elliptic curves over the rational numbers have no rational points of order 11 -- a very special case of Mazur's celebrated torsion theorem. The essential ingredients, such as modular curves and modular forms, will be introduced along the way. |

April 13 | Ina Petkova | TBD | |

April 20 | Simon Brendle | TBD | |

April 27 | Karsten Gimre | TBD |

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