2023 IMS Fellow

Marcel F. Nutz, Professor, Columbia University, has been named Fellow of the Institute of Mathematical Statistics (IMS).  Dr. Nutz received the award for outstanding contributions to probability, in particular to optimal transport, stochastic analysis, and mathematical finance; and for dedicated service to the profession.

The designation of IMS Fellow has been a significant honor for over 85 years. Each Fellow is assessed by a committee of their peers and has demonstrated distinction in research or leadership that has profoundly influenced the field. Established in 1935, the Institute of Mathematical Statistics is a member organization that fosters the development and dissemination of the theory and applications of statistics and probability. The IMS has over 4,700 active members throughout the world. Approximately 10% of the current IMS membership has earned the status of fellowship. The announcement of the 2023 class of IMS Fellows can be viewed here.

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The incoming graduate class for 2023 has been announced. For a list of the new students, please visit the incoming class page.

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Will Sawin, an Associate Professor in the Mathematics Department, was recently selected as a recipient for the prestigious Sloan Fellowship award. From the Sloan Research Fellowship website:

“The Sloan Research Fellowships seek to stimulate fundamental research by early-career scientists and scholars of outstanding promise. These two-year fellowships are awarded yearly to 126 researchers in recognition of distinguished performance and a unique potential to make substantial contributions to their field.”

For more information please visit: http://www.sloan.org/sloan-research-fellowships/

To view all 2023 Sloan research fellows, please visit: https://sloan.org/fellowships/2023-Fellows

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Speaker: Tomer Schlank (Hebrew University in Jerusalem)
Title:  Knots Invariants and Arithmetic Statistics.

Abstract: The Grothendieck school introduced \’etale topology to attach algebraic-topological invariants such as cohomology to varieties and schemes. Although the original motivations came from studying varieties over fields, interesting phenomena such as Artin–Verdier duality also arise when considering the spectra of integer rings in number fields and related schemes. A deep insight, due to B. Mazur, is that through the lens of \’etale topology, spectra of integer rings behave as 3-dimensional manifolds while prime ideals correspond to knots in these manifolds. This knots and primes analogy provides a dictionary between knot theory and number theory, giving some surprising analogies. For example, this theory relates the linking number to the Legendre symbol and the Alexander polynomial to Iwasawa theory.  In this talk, we shall start by describing some of the classical ideas in this theory. I shall then proceed by describing how via this theory, giving a random model on knots and links can be used to predict the statistical behavior of arithmetic functions. This is joint work with Ariel Davis.

Where: Mathematics Hall, room 520
When: Wednesday, April 19, 2023 at 04:30pm

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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.

Flyer

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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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