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Fall 2019 JOSEPH FELS RITT LECTURES

The Fall 2019 Joseph Fels Ritt Lectures, by Professor Neshan Wickramasekera, will take place on Wednesday, October 23 and Thursday, October 24, 2019Professor Neshan Wickramasekera (Cambridge University) , will deliver a two talk series titled:

Lecture 1: Variational theory of minimal hypersurfaces of Riemannian manifolds

Lecture 2: Regularity of CMC and prescribed-mean-curvature hypersurfaces

A fundamental idea in the study of partial differential equations is to introduce generalized solutions and establish, possibly subject to further conditions, smoothness of the generalized solutions. Having in place such a “regularity theory” is useful in many ways: besides reducing the question of existence of classical (i.e. smooth) solutions to that of existence of generalized solutions,
it enables, in some instances, finding functions that are classical solutions away from a small set of non-smooth points—the best hope when everywhere smooth solutions do not exist. It also often facilitates study of weak limits of classical solutions. For much the same reasons as in PDE, it is of interest to study regularity of appropriately defined generalized submanifolds of a Riemannian manifold satisfying natural geomertic constraints related to their variationally defined mean curvature. Unlike in the PDE context however, a serious difficulty in this goemetric setting stems from a priori variable multiplicity of the generalized submanifolds; in fact in arbitrary codimesion, many regularity issues related to multiplicity remainpoorly understood.

These lectures will describe progress made in the past several years for a large class of hypersurfaces where it is shown that this multiplicity issue has a satisfactory answer. The work culminates in a sharp regularity and compactness theory subject to certain structural conditions on the hypersurfaces and appropriate control on their mean curvature, mass and the Morse index (with respect to the relevant functional). The work includes minimal (i.e. zero mean curvature) and constant mean curvature (CMC) hypersurfaces as important special cases. We will also discuss applications of the theory including a streamlined PDE theoretic alternative to the classical Almgren–Pitts min-max existence theory for minimal hypersurfaces.

The first lecture, intended for a general mathematical audience, will focus on the minimal hypersurface theory in fairly broad terms. The second lecture will discuss key differences for general mean curvature constraints (focusing on CMC) and some aspects of the proofs of the main theorems in all cases. The lectures will in part be based on a series of speaker’s works some of which are separate joint projects with C. Bellettini, O. Chodosh and Y. Tonegawa.

Lecture 1

Wednesday, October 23, 2019 in room 520 from 4:30 – 5:30pm

Lecture 2

Thursday, October 24, 2019 in room 417 from 5:30 – 6:30pm

Tea will be served at 4 pm in 508 Mathematics.
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Fall 2019 Minerva Lectures

Come join us on Friday, September 27th, October 4th and 11th from 3:15 – 5pm in room 407. Starting Friday, September 27, 2019 Professor Makiko Sasada (University of  Tokyo), will be giving a special lecture titled;

The Box-ball System, Discrete Integrable Systems and Generalized Pitman’s Transform

The Korteweg-de Vries equation (KdV equation) and the Toda lattice are fundamental examples of classical integrable systems. These integrable systems with random initial measures have been intensively studied in recent years.

In this series of lectures, I will present our recent results on discrete versions of the KdV equation and the Toda lattice starting from random initial conditions. As a fundamental example, in the first lecture, I will focus on the box-ball system (BBS). The model, introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour, and can be understood as a special case of the ultra-discrete KdV equation and the ultra-discrete Toda lattice. The BBS is also known to be obtained from a vertex model by crystallization. I will show a new description of the BBS dynamics using Pitman’s transform of a simple random walk path, and give several results related to the BBS with random initial measures.

In the second and third lectures, I will introduce some generalizations of Pitman’s transform and show that the dynamics of several discrete integrable systems, such as the discrete KdV equation, the ultra-discrete KdV equation, the discrete Toda lattice and the ultra-discrete Toda lattice are given by them. This observation is applied to define the dynamics uniquely on the infinite configuration space and study the invariant measures. Some analogy between the discrete Toda lattice (resp. ultra-discrete Toda lattice) and the random polymers (resp. the last passage percolation) and the role of Burke’s theorem will be also discussed.

The lectures are based on the joint work with David Croydon, Tsuyoshi Kato and Satoshi Tsujimoto.”

Friday, September 27th, October 4th and October 11th from 3:15pm – 5pm

Mathematics Hall, room 407

Tea will be served at 3:00 p.m.

*Minerva Lecture Flyer*

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Memorial Conference for Patrick Ximenes Gallagher

A memorial conference for Patrick Ximenes Gallagher http://www.math.columbia.edu/~goldfeld/JointNTS.html will be held on Thursday, October 10. It will be followed by a banquet in the Auditorium of Earl Hall at 7pm. We welcome all to gather and commemorate his life and work. To register for the banquet please email Alenia Reynoso at reynoso@math.columbia.edu by Wednesday, September 25, 2019.

Directions to Conference/Banquet

Patrick Gallagher (1935 – 2019) taught at Columbia University until the end of 2017. He was beloved by his former students and many others.

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Workshop on Free Boundary Problems

This workshop will present recent developments on free Boundary Problems. For more information please visit the link below.

http://www.ki-net.umd.edu/content/conf?event_id=843

Organizers: Daniela De Silva, Nestor Guillen, Ovidiu Savin, and Hui Yu.

May 23  and May 24, 2019

Mathematics Hall, Room 312

*Registration is required

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Spring 2019 JOSEPH FELS RITT LECTURES

The spring 2019 Ritt Lectures, by Professor Laure Saint Raymond, will take place on Wednesday, May 8 and Thursday, May 9, 2019 from 4:30 – 5:30pm in Rm 417. Professor Laure Saint Raymond (École Normale Supérieure de Lyon), will deliver a two talk series titled:

1. Disorder Increases Almost Surely

Consider a system of small hard spheres, which are initially (almost) independent and identically  distributed. Then, in the low density limit, their empirical measure$\frac1N \sum_{i=1}^N \delta_{x_i(t), v_i(t)}$converges almost surely to a non reversible dynamics. Where is the missing information to go backwards?

2. The Structure of Correlations

Although the distribution of hard spheres remains essentially chaotic in this regime, collisions give birth to small correlations. The structure of these dynamical correlations is amazing, going through all scales. How can combinatorial techniques help analyze this departure from chaos?
Tea will be served at 4 pm in 508 Mathematics.
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Michael Harris elected to the American Academy of Arts & Sciences

Please congratulate Professor Michael Harris, who was recently elected as a member of the American Academy of Arts & Sciences!

Professor Michael Harris joins a distinguished list of scholars and thinkers from a variety of fields and disciplines. As posted on the Academy’s website:

“The Academy membership encompasses over 4,600 Fellows and 600 Foreign Honorary Members and reflects the full range of disciplines and professions: mathematics, the physical and biological sciences, medicine, the social sciences and humanities, business, government, public affairs, and the arts. Among the Academy’s Fellows are more than 250 Nobel laureates and 60 Pulitzer Prize winners.”

For a complete list of the 2019 members, including Professor Michael Harris, please visit:
https://www.amacad.org/content/members/newFellows.aspx?s=c

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Anna Wienhard (Heidelberg University)

Title: (Higher) Teichmüller Spaces and Beyond

Abstract:

The Teichmüller space of a surface parametrizes (marked) conformal structures. It covers the moduli space of Riemann surfaces and carries many interesting structures in its own right. Higher Teichmüller spaces are generalizations of Teichmüller space in the context of Lie groups of higher rank such as SL(n,R).

They are related to Higgs bundles, bounded cohomology, dynamics, as well as cluster algebras or total positivity. The talk will provide an introduction to higher Teichmüller spaces and showcase some of the connections.

Wednesday, April 24, 4:30 – 5:30 p.m.

Mathematics 520
Tea will be served at 4:00 p.m.

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Spring 2019 Kolchin Lecture

Spring 2019 Kolchin Lecture

Come join us on Wednesday, April 3, 2019 between 4:30 – 5:30pm in Rm 520, Professor Lars Hesselholt (Nagoya University & University of Copenhagen) will be giving a special lecture titled Higher Algebra and Arithmetic.

ABSTRACT

The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry.

Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to a cohomological interpretation of the Hasse-Weil zeta function by regularized determinants, as envisioned by Deninger.

Tea will be served at 4 pm in 508 Mathematics.

*Kolchin Lecture Flyer*

Time & Location

Wednesday, April 3, 2019 between 4:30 – 5:30 pm in Rm 520

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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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