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SPRING 2018 KOLCHIN LECTURE

Abstract

“Just over 50 years ago the modern era of Set Theory began with Cohen’s discovery of the method of forcing and his proof of the independence of the Continuum Hypothesis from the ZFC axioms of Set Theory. 25 years before Cohen’s discovery of forcing, Gödel discovered the Constructible Universe of Sets and defined the  axiom “V = L” which is the axiom that asserts that every set is constructible.  This axiom implies the Continuum Hypothesis and more importantly,  Cohen’s method of forcing cannot be used in the context of the axiom “V = L”.

However the axiom “V = L” must be rejected since it limits the fundamental nature of infinity. In particular the axiom refutes (most) strong axioms of infinity.

A key question emerges. Is there an “ultimate” version of Gödel’s constructible universe yielding an axiom “V = Ultimate L” which retains the power of the axiom “V = L” for resolving questions like that of the Continuum Hypothesis, which is also immune against Cohen’s method of forcing, and yet which does not refute strong axioms of infinity?

Until recently there seemed to be a number of convincing arguments as to why no such ultimate L can possibly exist. But the situation is now changed.”

*Kolchin Lecture Flyer*

Wednesday, April 18, 2018 at 4:30 p.m.

Mathematics Hall, Room 520

2990 Broadway at 117th Street

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CONGRATULATIONS to Professor Weinstein!

2018 SIAM’s Martin Kruskal Prize Lecture

Congratulations to Michael Weinstein who was selected as the 2018 Martin Kruskal Prize Lecturer. The prize will be awarded by the SIAM Activity Group on Nonlinear Waves and Coherent Structures (SIAG/NWCS) at their meeting in June 2018.

For more information please visit the link below;

http://www.siam.org/prizes/sponsored/siagnwcs.php

 

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Spring 2018 JOSEPH FELS RITT LECTURES

**NOTICE: Due to inclement weather, the talk for 3/21 has been postponed to 3/22. **

The spring 2018 Ritt Lectures, by Professor Michael Eichmair, will take place on Tuesday, March 20, 2018 between 2:45 – 3:45pm in Rm 417 and Thursday, March 22, 2018 between 5:30 – 6:30pm in Rm 417. Professor Eichmair (University of Vienna), will deliver a two talk series titled:

“Scalar Curvature & Isoperimetry in the Large”

Abstract

According to the initial value formulation of general relativity, all that is future and all that is past is contained in a glimpse of a space-time. This correspondence between the physics of the evolving space-time and the geometry of initial data for the Einstein equations is highly non-linear. The works of H. Bray, D. Christodoulou, G. Huisken, R. Schoen, S.-T. Yau, and others suggests isoperimetry (How much area is needed to enclose a given amount of volume in initial data for the space-time?) as a tool for extracting physical information about the space-time from the initial data. I will discuss recent proofs of a number of their conjectures in my two lectures.

This is joint work with S. Brendle, with O. Chodosh, with O. Chodosh, Y. Shi, and H. Yu, and with O. Chodosh, Y. Shi, and J. Zhu.

Tea will be served at 4 pm in 508 Mathematics.

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Berkeley–Columbia Meeting in Engineering & Statistics

The Berkeley-Columbia Meeting provides a biannual, interdisciplinary forum for research in Engineering, Finance, Mathematics and Statistics. The first meeting was held at UC Berkeley in 2016. The second meeting is hosted by Columbia on Friday, April 6 and Saturday, April 7, 2018. Registration is not necessary. However, seating is limited—first come, first served.

For more information please visit the link below;

https://sites.google.com/site/berkeleycolumbia18/

 

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“Elliptic Curves, Heegner Points, and Beyond”

Special Seminar

Come join us Wednesday, February 14, 2018 at 12 pm in RM 507, Professor Chao Li (Columbia University) will be giving a special lecture about “Elliptic Curves, Heegner Points, and Beyond”.

Abstract

An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which led to the celebrated conjecture of Birch and Swinnerton-Dyer. We will start with a history of this problem and then discuss our recent work (with D. Kriz) on certain families of elliptic curves (such as y^2=x^3-d), which in particular proves a conjecture of Goldfeld. Our approach uses Heegner points and their deep connection with L-functions. We will explain these key ingredients and ideas. If time permits, we will also illustrate a framework to go beyond the case of Heegner points and to build a connection with L-functions of more general symplectic motives.

Mathematics Hall, Room 507

Wednesday, February 14, 2018 at 12 pm

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“The Threshold Theorem for the Hyperbolic Yang-Mills Equation”

Special Seminar

Come join us Monday February 12, 2018 at 12 pm in RM 507, Professor Sung-Jin Oh (KIAS) will be giving a special lecture about “The Threshold Theorem for the Hyperbolic Yang-Mills Equation”.

Abstract

In this lecture, I will present the recent proof (joint with D. Tataru) of the threshold theorem for the energy critical hyperbolic Yang-Mills equation in (4+1) dimensions. This theorem provides a sharp criterion for global existence and scattering in terms of the energy of the initial data. Moreover, we prove that failure of global existence/scattering is characterized by “bubbling” of a solution to the harmonic Yang-Mills equation.

Our proof lies at the intersection of many recent developments, such as null form estimates and function spaces; parametrix construction via pseudodifferential gauge renormalization; induction on energy; monotonicity formulae arising from the normalized scaling vector field, etc. Also of note is the use of the associated parabolic flow, namely the Yang-Mills heat flow, to construct a high quality global gauge (called the caloric gauge), extending the idea of Tao for the harmonic map heat flow.

Mathematics Hall, Room 507

Monday February 12, 2018 at 12 pm

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“Equivariant Commutative Ring Spectra”

Special Topology Seminar

Come join us Monday February 12, 2018 at 4:30 pm in RM 520, Professor Andrew Blumberg (UT Austin) will be giving a special lecture about “Equivariant Commutative Ring Spectra”.

Abstract

One of the central developments in algebraic topology in the last 30 years has been the study of the chromatic filtration on the stable homotopy category.  The chromatic perspective relates the fine structure of the stable category to the algebraic geometry of formal groups.  In particular, an important role in the story is played by an elliptic cohomology theory called topological modular forms (tmf).

Calculations with forms of tmf that incorporate information about level structures suggest that the framework of equivariant derived algebraic geometry would be extremely useful.  To develop this framework, it is necessary to understand equivariant commutative ring spectra.  This turns out to be a subtle problem; in this talk I will explain the situation and what the answers look like.

Mathematics Hall, Room 520

Monday February 12, 2018 at 4:30 pm

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“Estimates for Modular Forms Using Exponential Sums”

Special Seminar

Come join us Friday February 9, 2018 at 12 pm in RM 507, Professor Will Sawin (ETH Zürich) will be giving a special lecture about “Estimates for Modular Forms Using Exponential Sums”.

Abstract

Together with Emmanuel Kowalski and Philippe Michel, I proved two estimates on sums of Fourier coefficients of modular forms — one on the second moment of critical value of the L-function twisted by Dirichlet characters, and the other on the level of distribution. We did this using bounds for exponential sums proved by geometric and topological methods. I will explain these techniques and how they can be used for these two problems.

Mathematics Hall, Room 507

Friday February 9, 2018 at 12 pm

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“Rare Behavior in Models of Random Geometry”

Special Seminar

Come join us Wednesday February 7, 2018 at 12 pm in RM 507, Professor Shirshendu Ganguly (Berkeley) will be giving a special lecture about “Rare Behavior in Models of Random Geometry”.

Abstract

Models of random geometry have long been investigated in contexts such as real life networks, fluid flow in porous media, and interface dynamics in statistical physics. To develop a refined understanding of such models, one often needs to study not only typical fluctuation theory but also the realm of atypical events. In this talk we describe such a program for two classical models of random geometry: percolation on the complete graph and random distortions of the Euclidean lattice. In particular, we will consider the rare events that a sparse random network has an atypical number of some local structure and that a geodesic in a random metric space has atypical length; both of the corresponding random variables being non-linear functions of the underlying randomness. The geometry associated to typical instances of these rare events is an important topic of inquiry: it can involve merely local objects, or more global ones. We will discuss recent resolutions of certain long standing questions concerning such phenomena, and connections to other areas of mathematics including random matrix theory, information theory, and algebraic and extremal combinatorics.

Mathematics Hall, Room 507

Wednesday February 7, 2018 at 12 pm

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“Counting in Algebraic Geometry and Modular Forms”

Special Seminar

Come join us Monday February 5, 2018 at 12 pm in RM 507, Professor Georg Oberdeck (MIT) will be giving a special lecture about “Counting in Algebraic Geometry and Modular Forms”.

Abstract

It is generally expected that the enumerative geometry of Calabi-Yau geometries
should be closely related to the theory of modular forms. A basic example
(which I will recall) is the count of uniruled divisors on hyper-Kaehler
manifolds by Fourier coefficients of Jacobi forms. Another interesting but less
understood case is the count of algebraic curves and sheaves on Calabi-Yau
threefolds. I will discuss recent work with A. Pixton and J. Shen which, for a
particularly Calabi-Yau threefold, proves the modularity of its curve counting
partition function using geometric arguments.

Mathematics Hall, Room 507

Monday February 5, 2018 at 12 pm

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