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Columbia Summer Math Research Experience for Undergraduates

The Mathematics Department runs a 10 week summer research program, aimed at rising junior and senior undergraduate math majors. Students participating in the program work closely with faculty members and graduate students in a small group setting.

Accepting applications through Friday, March 8, 2019

May 28 – August 1, 2019

For more information please visit;

www.math.columbia.edu/reu

 

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Liouville Quantum Gravity As a Metric Space and a Scaling Limit

Special Seminar

Come join us Monday, February 11, 2019 at 4:30 pm in RM 507, Professor Jason Miller (University of Cambridge) will be giving a special lecture about “Liouville Quantum Gravity As a Metric Space and a Scaling Limit”.

Abstract

Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has its roots in string theory and conformal field theory from the 1980s and 1990s. The second is the Brownian map, which has its roots in planar map combinatorics from the 1960s together with recent scaling limit results.  We will describe work with Sheffield in which it is shown that Liouville quantum gravity (LQG) with parameter $\gamma=\sqrt{8/3}$ is equivalent to the Brownian map and work with Gwynne which use the $\sqrt{8/3}$-LQG metric to prove the convergence of self-avoiding walks and percolation on random planar maps towards SLE(8/3) and SLE(6), respectively, on a Brownian surface.

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Cardy Embedding of Random Planar Maps

Special Seminar

Come join us Wednesday, February 6, 2019 at 4:30 pm in RM 507, Professor Nina Holden (ETH, Zürich) will be giving a special lecture about “Cardy Embedding of Random Planar Maps”.

Abstract

A random planar map is a canonical model for a discrete random surface which is studied in probability, combinatorics, mathematical physics, and geometry. Liouville quantum gravity is a canonical model for a random 2d Riemannian manifold with roots in the physics literature. In joint work with Xin Sun, we prove a strong relationship between these two natural models for random surfaces. Namely, we prove that the random planar map converges in the scaling limit to Liouville quantum gravity under a discrete conformal embedding which we call the Cardy embedding.

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On the Black Hole Stability Problem

Special Seminar

Come join us Monday, February 4, 2019 at 4:30 pm in RM 507, Professor Jérémie Szeftel (Université Pierre et Marie Curie) will be giving a special lecture about “On the Black Hole Stability Problem”.

Abstract

I will introduce the celebrated Kerr black hole stability problem, which asks whether metrics of the Kerr family are stable as solutions to the Einstein vacuum equations of general relativity. I will then focus on recent progress obtained in my joint work with Sergiu Klainerman.

Mathematics Hall, Room 507

Monday, February 4, 2019 at 4:30 pm

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Memorial Service for Michael Zhao

 

A memorial service for Michael Zhao will be held in Garden Room 2 of Faculty House on Saturday,  February 2 at 12 pm.  A reception will follow the service.  All are welcome.

Michael Zhao was a first-year doctoral student whose sudden and unexpected loss on December 8 was shattering for everyone in the Department.  He touched many of our lives during the short time he spent here and will be missed by all.

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Super-Teichmüller Spaces, Spin Structures, Penner Coordinates, and Applications

Special Seminar

Come join us Friday, February 1, 2019 at 4:30 pm in RM 507, Professor Anton Zeitlin (Louisiana State University) will be giving a special lecture about “Super-Teichmüller Spaces, Spin Structures, Penner Coordinates, and Applications”.

Abstract

The Teichmüller space, which parametrizes Riemann surfaces, is a

fundamental space that is important in many areas of mathematics and physics.

Recently, generalizations of this space have been intensely studied.  Examples of 

such higher Teichmüller spaces are the so-called super-Teichmüller

spaces. They appear in the application of the combinatorial approach to spin

structures on Riemann surfaces and generalizations to supermanifolds. The

super-Teichmüller spaces naturally  arise as higher Teichmüller spaces,

corresponding to supergroups that play an important role in geometric

topology, algebraic geometry, and mathematical physics.

 

In this talk, I will give a solution of the long-standing problem

of describing an analogue of Penner coordinates on super-Teichmüller

spaces and their generalizations. The importance of these coordinates is

justified by two remarkable properties: the action of the mapping class

group is rational and the Weil-Petersson form is given by a simple explicit

formula.

 

I will end my presentation with a description of some of the emerging

applications of this theory.

Mathematics Hall, Room 507

Friday, February 1, 2019 at 4:30 pm

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Principal Bundles and Diophantine Geometry

Special Seminar

Come join us Monday, January 28, 2019 at 4:30 pm in RM 507, Professor Minhyong Kim (University of Oxford) will be giving a special lecture about “Principal Bundles and Diophantine Geometry”.

Abstract

Principal bundles and their moduli spaces have been important objects of study and essential tools in the geometry and topology of manifolds for at least the last fifty years. This talk will describe their applications to number theory, especially the theory of polynomial equations and their rational or integral solutions.

Mathematics Hall, Room 507

Monday, January 28, 2019 at 4:30 pm

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Graduate Topics Courses

The Graduate topics courses for Spring 2019 are now available through the following link:

Graduate Topics Courses


 

 

 

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Fall 2018 JOSEPH FELS RITT LECTURES

Come join us on Wednesday, October 31, 2018 between 4:30 – 5:30pm in Rm 520 and Thursday, November 1, 2018 between 5:30 – 6:30pm in Rm 417, Professor Eugenia Malinnikova (Norwegian University of Science and Technology) will be giving a special lecture titled An improvement of Liouville’s theorem for discrete harmonic functions.

ABSTRACT
“The classical Liouville theorem says that if a harmonic function on the plane is bounded then it is a constant. At the same time for any angle on the plane, there exist non-constant harmonic functions that are bounded everywhere outside the angle. The situation is different for discrete harmonic functions on the standard square lattices. The following strong version of the Liouville theorem holds on the two-dimensional lattice. If a discrete harmonic function is bounded on 99% of the lattice then it is constant. Simple counter-example shows that in higher dimensions such improvement is no longer true.

We will present some discrete methods, compare the behavior of continuous and discrete harmonic functions and discuss some related questions and motivation.
The lectures are based on a joint work with L. Buhovsky, A. Logunov and M. Sodin.”

Tea will be served at 4 pm in 508 Mathematics.

*Joseph Fels Ritt Lecture Flyer*

Time & Location

Wednesday, October 31, 2018 between 4:30 – 5:30 pm in Rm 520

&

Thursday, November 1, 2018 between 5:30 – 6:30 pm in Rm 417

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Graduate Topics Courses

The Graduate topics courses for Fall 2018 are now available through the following link;

Graduate Topics Courses

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