The spring 2023 Ritt Lectures will take place on Monday, April 17, 2023 (4:30pm start time on Monday) and Friday, April 21, 2023 from 4:45 – 5:45pm in room 520. Professor Camillo De Lellis (Institute for Advanced Study), will deliver a two talk series titled:

Title: Area-minimizing integral currents: singularities and structure
Abstract: Area-minimizing integral currents are a natural generalization of area-minimizing oriented surfaces. The concept was pioneered by De Giorgi for hypersurfaces of the Euclidean space, and extended by Federer and Fleming to any codimension and general Riemannian ambients. These classical works of the fifties and sixties establish a general existence theory for the oriented Plateau problem of finding surfaces of least area spanning a given contour.

Celebrated examples of singular 7-dimensional minimizers in R^8 and of singular 2-dimensional minimizers in R^4 are known since long and in fact in these cases there is no smooth oriented minimizer and any smooth minimizing sequence converges to the singular ones in an appropriate sense. A first theorem which summarizes the work of several mathematicians in the 60es and 70es (De Giorgi, Fleming, Almgren, Simons, and Federer) and a second theorem of Almgren from 1980 give dimension bounds for the singular set which match the one of the examples, in codimension 1 and in general codimension respectively.

In these lectures I will focus on the case of general codimension and address the question of which structural results can be further proved for the singular set. A recent theorem by Liu proves that the latter can in fact be a fractal of any Hausdorff dimension \alpha \leq m-2. On the other hand it seems likely that it is an(m-2)-rectifiable set, i.e. that it can be covered by countably many C^1submanifolds leaving aside a set of zero (m-2)-dimensional Hausdorff measure. This conjecture is the counterpart, in general codimension, of a celebrated work of Leon Simon in the nineties for the codimension 1 case. In these lectures I will explain why the problem is very challenging, how it can be broken down into easier pieces, and present a line of attack based on recent joint works with Anna Skorobogatova and Paul Minter.

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Speaker: Robert McCann (Toronto)
Title: A nonsmooth approach to Einstein‘s theory of gravity

Abstract: While Einstein’s theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under dimension and curvature bounds analogous to the energy conditions in relativity:

Here sectional curvature is reformulated through triangle comparison, while and Ricci curvature is reformulated using entropic convexity along geodesics of probability measures. This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity.

We begin with a simplified approach to Kunzinger and Saemann’s theory of (globally hyperbolid, regularly localizable) Lorentzian length spaces in which the time-separation function takes center stage. We show compatibility of two different notions of time like geodesic used in the literature. We then propose a synthetic (i.e. nonsmooth) reformulation of the null energy condition by relating to the time like curvature-dimension conditions of Cavalletti \& Mondino (and Braun), and discuss its consistency and stability properties.

Where: Mathematics Hall, room 520
When: Wednesday, April 05, 2023 at 04:30pm

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Speaker: Chao Li (NYU)

Title: The stable Bernstein theorem in $R^4$

Abstract: I will discuss the stable Bernstein theorem for minimal hypersurfaces in $R^4$: a complete, two-sided, stable minimal hypersurface in $R^4$ is flat. The proof relies on an intriguing relation between the stability inequality and the geometry of 3-manifolds with uniformly positive scalar curvature. If time permits, I will also talk about an extension to anisotropic minimal hypersurfaces. This is based on joint work with Otis Chodosh.

Where: Mathematics Hall, room 520
When: Wednesday, March 08, 2023 at 04:30pm

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Special Seminar

Come join us on Monday, January 30, 2023 at 04:30pm in room 520, Professor Alisa Knizel (The University of Chicago) will be giving a special lecture titled “The Universality Phenomenon for Log-Gas Ensembles”.

Abstract: Though exactly solvable systems are very special, their asymptotic properties
are believed to be representative for larger families of models. In this way,
besides being interesting in their own right, exactly solvable systems are
exemplars of their conjectured universality classes and can be used to build
intuition and tools, as well as to make predictions. I will illustrate the phenomenon of
universality with the examples from my work on log-gas ensembles.

Location: Mathematics Hall, room 520

Date: Monday, January 30, 2023 at 04:30pm

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The Graduate topics courses for the Spring term are now available.

For more information, please visit the following link: Graduate Topics Courses

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Special Seminar

Come join us on Wednesday, January 25, 2023 at 04:30pm in room 520, Professor Eric Ramos (Bowdoin College) will be giving a special lecture titled “The many kinds of uniformity in graph configuration spaces”.

Abstract: For a given topological space X, the (unordered) configuration space of n points on X, F_n(X), is the space of n-element subsets of X. Much of the work on these spaces has considered cases where the underlying space X is a manifold of dimension higher than two. For instance, one famous result of McDuff states that if X is the interior of a compact manifold of dimension at least two with boundary, then for any i the isomorphism class of the homology group H_i(C_n(X)) is independent of n whenever n is big enough. Put more succinctly, if X is a “sufficiently nice” manifold of dimension at least 2, then the configuration spaces C_n(X) exhibit homological stability.

In this talk, we will consider configuration spaces in the cases where X is a graph. That is, when X is 1-dimensional. In this setting we will find that the homology groups H_i(C_n(X)) exhibit extremely regular behaviors in two orthogonal ways. The first, similar to the classical setting, is when X is fixed and n is allowed to grow. In this case we will see that rather than stabilizing, the Betti numbers grow as polynomials in n. The second kind of regular behavior is observed when one fixes n and allows X to vary. In this case we will use extremely powerful structural theorems in graph theory to discover features of the homology groups H_i(C_n(X)) that must be common across all graphs X.

Location: Mathematics Hall, room 520

Date: Wednesday, January 25, 2023 at 04:30pm

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Special Seminar

Come join us on Friday, January 20, 2023 at 10:30am in room 520, Professor Kyle Hayden (Rutgers University-Newark) will be giving a special lecture titled “Surfaces and 4-manifolds“.

Abstract: The topology of smooth manifolds is governed largely by geometry in low dimensions and by algebraic topology in high dimensions. The phase transition occurs in dimension four, where continuous and differential topology split apart and “exotic” phenomena emerges. I will begin by describing how this phase transition can be studied via embedded surfaces in4-manifolds, then I will survey recent developments in this area. In particular, I will explain how quantum invariants (such as Khovanov homology)have recently been used to address existence and uniqueness questions about surfaces in 4-manifolds — and what this might imply for foundational questions about 4-manifolds themselves.

Location: Mathematics Hall, room 520

Date: Friday, January 20, 2023 at 10:30am

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It is with profound sadness that we write to share the news that Igor Krichever passed away on Thursday December 1, 2022. A funeral service was held on Friday, December 2, at the Plaza Jewish Community Chapel.

Our condolences goes out to all. Professor Krichever was beloved by many and we will always remember him for his great work and kindness.

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Speaker: Søren Galatius

Title: On topological Pontryagin classes

Abstract: The Pontryagin classes of a real vector bundle can be defined via Chern classes of its complexification, and appear in Hirzebruch’s formula for the signature of a smooth 4n-dimensional manifold for example.  It was realized long ago that Pontryagin classes can be defined more generally for topological bundles, that is, bundles with fibers homeomorphic to euclidean spaces, even in the absence of linear structures.  I will recall a bit of the classical theory of Pontryagin classes for topological bundles, and discuss some new developments including joint work with Randal-Williams on algebraic independence.

Where: Mathematics Hall, room 520
When: Monday, January 23, 2023 at 04:30pm

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Mathematics Professors Panagiota Daskalopoulos (Columbia University) and Nataša Šešum (Rutgers University) have been selected to receive the 2023 Ruth Lyttle Satter Prize in Mathematics.

The prize, awarded by the American Mathematical Society every two years, recognizes an outstanding contribution to mathematics research by a woman in the previous six years.

The Satter Prize will be awarded in January 2023 at the Joint Mathematical Meetings, which will be held in Boston.

AMS news website

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