Columbia Undergraduate Math Society

Summer 2019 <<  Fall 2019 Lectures >> Spring 2020

Wednesdays, 7:30pm; Room TBD, Mathematics
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Date Speaker Title Abstract
September 11
Raymond Cheng
Polytopes, Mixed Volumes, and Inequalities
Polytopes are higher-dimensional generalizations of polygons in the plane. Not only are they intrinsically beautiful objects, they lie at the crossroads of diverse areas of mathematics. In this talk, I will discuss a quantity associated with polytopes, called the mixed volume, the fundamental Alexandrov—Fenchel Inequality for mixed volumes, and the many connections that these quantities have with other parts of mathematics.
September 18
Stanislav Atanasov
“Number fields contain the
elements of their own transcending”
A gentle yet rapid introduction to class field theory, its conjectured generalization – the Langlands program, and the role of L-functions therein. Our starting point is a density result due to Frobenius, which implies that the Galois (hence all) extensions of a number field are uniquely determined by the set of primes that split completely. The rest of the talk traces historically the quest of understanding this set culminating in its close connection with Artin L-functions, certain meromorphic functions on the complex plane reminiscent of the Riemann zeta function.
September 25
Quang Dao
Plane Partitions of the
Rectangle and Trapezoid Posets
Given a partially ordered set (poset) P, a plane partition is a way of stacking boxes on each element that respect the partial order of P. In 1983, Proctor proved that the number of plane partitions of a rectangle poset is the same as that of its corresponding trapezoid poset. Proctor’s proof, however, is non-bijective. In this talk, I will describe a recent combinatorial bijection that comes from a surprising connection with K-theoretic Schubert calculus, in particular the operation of K-jeu-de-taquin on increasing Young tableaux.
October 2
Micah Gay
Geometric Representation Theory,
or Why Everything is a Sheaf
Geometric representation theory uses geometric techniques such as sheaves and D-modules in order to study representations of symmetry objects such as Hecke algebras, quantum groups, quivers, or algebraic groups. In this talk, we will develop some geometric and representation-theoretic tools, and give an overview of several major results in geometric representation theory, including the Beilinson-Bernstein correspondence, Riemann-Hilbert correspondence, and Kazhdan-Lusztig conjectures.
October 9
Oleg Lazarev
Legendrian knots
Legendrian knots appear in many different guises in mathematics and physics: as wavefronts of light propagation, trajectories of a rolling coin that does not slip, generalizations of the derivative of smooth functions, and as graphs of critical points of a family of functions. I will discuss these classical examples and some recent developments in the field related to symplectic flexibility and microlocal sheaf theory.
October 16
Ben Church
The Logic of Provability
The field of provability logic grows largely out of Godel’s second incompleteness result: the impossibility of a sufficiently powerful consistent formal system proving its own consistency. The main interest of this theorem, besides some philosophical musings, is the general method of proof which goes about formalizing within number theory itself the Godelian argument of constructing a sentence which is equivalent to its own unprovability. That is, number theory is sufficiently powerful to derive Godel’s first incompleteness theorem and show that the truth value of any Godel sentence is equivalent to the consistency of the theory (which, informally, is clear since if G is false then there must exist a proof of G, which is false). Godel’s second incompleteness result is intimately related to Lob’s theorem which expresses a sort of “epistemic modesty” that number theory has about its its own soundness. Lob’s theorem is the vital step in constructing GL or provability logic which succeeds in stripping number theory of all the extra junk about numbers while entirely capturing its self-referential behavior regarding provability captured within the language.
October 23
Will Sawin
The Cap Set Problem
The cap set problem in combinatorics involves finding configurations with no valid moves in a generalization of the classic card game Set. In 2016, breakthrough work by Croot, Lev, and Pach and Ellenberg and Gijswijt made massive progress on the problem by an elementary proof that was totally different from previous methods. I will explain the problem and discuss some of the ideas in the proof.
October 30
No Meeting
November 6
Matthew Lerner-Brecher
The Hardy-Ramanujan Theorem
and Probabilistic Number Theory
In this talk, we will give some intuition for how probabilistic techniques can be useful in number theory by studying the function $\omega(n)$, which counts the number of distinct prime factors of $n$. After giving some motivation and basic results from analytic number theory, we will prove the Hardy-Ramanujan theorem, which gives a simple bound satisfied for $o(N)$ integers $n \le N$. From there, we will use basic probability theory to reinterpret Hardy-Ramanujan. Lastly, we will introduce a more general result about the distribution of $\omega(n)$: the Erdos-Kac theorem. Time permitting, we will give a heuristical proof of Erdos-Kac.
November 13
Francesco Lin
Thurston's geometrization
In this talk we'll discuss the circle of ideas introduced by Thurston to study 3-dimensional spaces: namely, how tools from geometry can be extremely powerful when studying topology. By the end of this talk, we'll state Thurston famous geometrization conjecture (now a theorem thanks to the work of Perelman).
November 20
Alexandra Florea
Moments and Zeros of L-functions
The Riemann zeta function, a classical example of an L-function, encodes important information about the prime numbers. For example, knowing the location of its zeros implies strong results for counting how many primes there are up to X, when X goes to infinity. I will describe the classical conjectures in the field, such as the Riemann hypothesis and the Lindelof hypothesis, which provides a strong upper bound on the size of the Riemann zeta function. I will then explain what moments are and how they can be used to deduce some weaker results towards these conjectures.
November 27
No Meeting
December 4
Ben Church
Spin, Clifford Algebras, and the
Representation Theory of the Lorentz Group
We will explore the quantum mechanical property of spin through the framework of representation theory. We will use the real form SU(2) of the lie algebra of SL(2, C) to classify representations of SL(2, C) then discuss how these are related to representations of the Lorentz group via the theory of projective representations. In doing so we will venture through Lie algebra cohomology. We will then consider how parity enters the picture and, in doing so, define Clifford algebras, the Spin and Pin groups, and spinor spaces.
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