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January 19: Junliang Shen

Title: The P=W conjecture and hyper-Kähler geometry.

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January 14: Elena Giorgi

Title: The stability of black holes with matter.

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January 12: Daniel Álvarez-Gavela

Title: K-theoretic Torsion Invariants in Symplectic and Contact Topology.

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Incoming class of 2020

The incoming graduate class for 2020 has been announced. For a list of the new students, please visit the Incoming class page.

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On February 12, Samuel Eilenberg Professor of Mathematics Andrei Okounkov was elected to Foreign Membership in the Royal Swedish Academy of Sciences.  He joins a distinguished company of Foreign Members including Pierre Deligne and Jean-Pierre Serre.  Many congratulations to Andrei!

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Heat Equations in Analysis, Geometry, and Probability

“In its most basic form, the heat equation is a partial differential equation that describes the evolution of temperature in space over time. This and similar equations play a central role in diverse fields. The goal is to provide an introduction to the heat equation and its generalizations, including connections to geometry and geometric flows, probability, and, of course, analysis.”

*Samuel Eilenberg Lecture Flyer*

Tuesdays at 2:40 pm

Room 520, Mathematics Hall

2990 Broadway (117th Street)

First lecture: January 28, 2020

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Graduate Topics Courses in Spring 2020

The Department is offering 7 graduate topics courses in Spring 2020.  Titles and abstracts may be viewed here.

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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 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 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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