Columbia Undergraduate Math Society

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Wednesdays, 7:30pm; Room 507, Mathematics
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Date Speaker Title Abstract
January 30
Stanislav Atanasov
Class Field Theory
We will talk about class field theory -- non-adelically, for the masses
February 6
Ben Church
Self-Reference in Number Theory
Godel famously proved his incompleteness theorems by constructing a version of the liar’s paradox inside number theory: a statement about the natural numbers which expresses its own unprovability. We will show how number theory can be formalized in first-order logic by Robinson arithmetic and Peano arithmetic. Neither of these theories seems to include the “machinery” for sentences to reference themselves or other sentences (i.e. express the metalanguage) or to encode notions such as provability which belong to the metatheory. However, we will discuss how Godel was able to cleverly encode this metatheory in number-theoretical statements and thus make number theory “talk about itself.” We will then prove the self-reference lemma which implies the existence of such “self-denying” sentences (formally showing the existence of modal fixed-points in provability logic). Finally, we will prove Lob’s theorem which forms the foundation of GL-provability logic and deduce from it both of Godel’s celebrated incompleteness theorems.
February 13
Nick Salter
Math magic tricks get a bad rap, and indeed most of them are pretty lame. The main result of the talk will be the construction of a math magic trick originally due to Conway which provides a substantially improved lower bound on the known coolness of a math magic trick. I will even risk the peculiar wrath of of the magic community by explaining how the trick works. The mathematics hiding behind the scenes is astonishingly beautiful and deep and will bring us to the intersection of ideas from topology, number theory, and algebraic geometry. I can assure you that you will be perfectly safe, and that no one will get kicked in the face.
February 20
Hugo Falconet
Conformal invariance and
2D statistical physics
I will discuss several breakthroughs from the last two decades establishing rigorously the conformal invariance of several statistical physics models taken at their critical parameter.
February 27
Alex Pieloch
An Introduction to Spectra
We give an elementary introduction to spectra. We begin by introducing (stable) homotopy groups of spaces. Once motivated, we will define spectra and show that they recover stable homotopy groups. Finally, we will discuss the relationship between spectra and cohomology theories, discussing the Eilenberg-MacLane spectrum and Brown representability.
March 6
Micah Gay
Dr Drinfeld and Jimbo, or
How I Learned to Stop Worrying
and Love Quantum Groups
Quantum groups arise as Hopf Algebras which can be thought of as deformations of certain universal enveloping algebras. We will discuss Lie theory and notions of duality, then see what happens when we stick the word “quantum” in front of everything to learn what goes into constructing a quantum group.
March 13
Kyler Siegel
Counting Rational Curves in the Plane
An ancient observation is that there is a unique line passing through any two distinct points in the plane. A more modern incarnation replaces the "plane" with the complex projective plane and "lines" with degree one complex algebraic curves. From this perspective, a very natural question asks for the number of genus zero curves of a given degree (after imposing appropriate constraints to make the number finite). Some progress was made in the 19th century, but it took an influx of ideas from string theory, symplectic geometry, and algebraic geometry to compute these numbers in full. I will describe Kontsevich's recursive formula, discovered around 1994. I will also try to convey some sense of the key ingredients, such as quantum cohomology and Gromov-Witten theory, and how they relate to subsequent developments.
March 20
No Meeting
Spring Break
March 27
Thibault Langlais
Some Geometric aspects
of Quantization
Hamiltonian systems arise as classical models for physical evolutions. Given such a Hamiltonian system, one wants to build a quantum model representing the same evolution; this is called quantization. In this talk, we will develop the theory of Hamiltonian systems from scratch, justify the axioms of quantization, and depending on time, explain a first geometric approach to quantization.
April 3
Oleg Lazarev
Morse Theory
Morse functions are smooth functions whose critical points have non-degenerate matrix of second-derivatives. I will explain how these functions serve as a powerful link between analysis and topology and discuss their role in the topology of complex varieties, the Poincare conjecture about high-dimensional spheres, and the existence of geodesics on manifolds.
April 10
Romain Panis
Introduction to Percolation
Percolation models first appeared in probability theory to model physical situations. The simplest model, Bond percolation on the lattice Z^d, was introduced by Broadbent and Hammersley in 1957. We will give a formal construction of this model and study the transition phenomena arising from its definition. This will lead to several questions (Many of which remain unsolved to this day!). If we have enough time, we will prove a beautiful result which states the almost sure unicity of infinite clusters in the case where percolation happens with non zero probability.
April 17
Evan Warner
A moduli space is just a parameter space for some reasonable class of mathematical (usually geometric) objects. We'll meet a few basic examples, see a couple of important facts that can be proved with them, and observe how things can go wrong.
April 24
Jacob Shapiro
Anderson Localization
We give a rudimentary introduction to the mathematics of Anderson localization from the point of view of single-particle electrons moving in a solid in the tight-binding regime. We begin with a short introduction to random ergodic operators and define the relevant notions, go on to analyze a model with sufficiently high disorder and show via the fractional-moment method that its entire spectrum is localized. This then leads to: (a) vanishing of the DC conductivity for time-reversal invariant systems via the Kubo formula (b) exp. off-diagonal decay of the Fermi projection and consequently the vanishing of the direct-conductivity also in systems with broken-time reversal and a well-defined Hall conductivity. This last fact is the starting point for the field of topological insulators, as it insures that all invariants are well-defined.
May 1
Akram Alishahi
Khovanov Homology and
the Unknotting Number
Knot theory is the study of knots i.e. embeddings of circle in the 3-dimensional space. We will sketch some problems in knot theory, and discuss some knot invariants including Jones polynomial and Khovanov homology. We will explain how these invariants or their refined versions can (sometimes) be used to answer these questions.
May 8
No meeting
Reading Week

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