Title: Renormalization, fractal geometry and the Newhouse phenomenon

Speaker: Artur Avila (IMPA)

Date, Time, Location: Wednesday, April 23rd @4:30PM – 5:30PM in Math Hall 520

Abstract:

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The Lars Tyge Nielsen Memorial Conference is on April 25th from 1-7PM. This is a day to honor our late director of the MAFN program Lars Tyge Nielsen. The Memorial Conference will be held at the Innovation Hub of Columbia Engineering (2276 12th Ave, New York, NY 10027) on April 25th, 2025. Register Here.

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Title: Homotopy groups of spheres and algebraic K-theory

Speaker: Ishan Levy (Copenhagen)

Date, Time, Location: Wednesday, March 12th @4:30PM – 5:30PM in Math Hall 520

Abstract:

The homotopy groups of spheres classify continuous functions from an n-dimensional sphere to an m-dimensional sphere up to continuous deformation. They are of great interest as they control many classification problems in geometric topology, but are enormously complicated. I will explain some of what we know about them, and describe the telescopic approach to understanding them systematically. I will then explain how algebraic K-theory has refined our understanding of the telescopic approach, and can be used to obtain asymptotic lower bounds on the complexity of these homotopy groups that were previously out of reach.

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The Spring 2025 Joseph Fels Ritt Lectures will be delivered by Professor Gunther Uhlmann on Wed. April 16th @ 4:30 p.m. in room 520 and on Thurs. April 17th @ 2:45 p.m in room 520.

• Flyer
• Click here for details.

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Starting Tues. Jan 21st, the Spring 2025 Minerva Foundation Lectures
will take place on Tuesdays at 4:10 p.m. in room 507.
[Jan. 21 – March 14, 2025]

Professor Rama Cont (Oxford), will deliver a series of lectures titled Rough Calculus.

Lecture Series Info Sheet
Flyer

The Ito calculus may be viewed as an extension of the Newton-Leibniz calculus to
smooth functions of paths with non-zero quadratic variation. This analytical
viewpoint is exploited to develop a calculus for smooth function(al)s of irregular
paths with non-zero p-th variation for arbitrary p>1. Although this “rough
calculus” is strictly pathwise in nature and does not involve any probabilistic
ingredient, it is applicable to stochastic processes with irregular paths.

We illustrate the concepts and results of this theory in the setting of the Ito-
Föllmer calculus for smooth function(al)s of paths with finite quadratic variation.
We will then show how these results may be extended to the more general setting
of smooth functionals of paths with non-zero p-th variation for arbitrary p>1,
leading to a higher order Ito-type calculus. Finally, we will sketch some examples of
applications to transport equations, optimal control and rough dynamics on
manifolds.

I. Ito calculus without probability
II. Ito-Föllmer calculus for functionals of paths with finite quadratic variation.
Pathwise isometry and rough-smooth decompositions.
III. Rough calculus for function(al)s of path with finite p-th variation.
IV. The case of paths with fractional regularity (*)
V. Transport of measures along rough trajectories.
VI. Rough dynamics on manifolds

*: if time permits

Meeting on Tuesdays at 4:10 p.m.

Room 507, Mathematics Hall

2990 Broadway (117th Street)

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Special Colloquium

Speaker: Lue Pan (Princeton University)
Title: The Fontaine-Mazur conjecture in dimension two
Abstract: The Fontaine-Mazur conjecture predicts which two-dimensional p-adic representations of the absolute Galois group of Q arise from modular forms. In this talk, I will explain this conjecture by some concrete examples and report some recent progress which relies on some recent developments of p-adic geometry.

Date and Time: Tuesday, January 28 @ 2:40PM
Location: 520 Mathematics

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Special Colloquium

Speaker: Alex Smith (UCLA)
Title: The distribution of conjugates of an algebraic integer
Abstract: For every odd prime p, the number 2 + 2cos(2 pi/p) is an algebraic integer whose conjugates are all positive numbers; such a number is known as a totally positive algebraic integer. For large p, the average of the conjugates of this number is close to 2, which is small for a totally positive algebraic integer. The Schur-Siegel-Smyth trace problem, as posed by Borwein in 2002, is to show that no sequence of totally positive algebraic integers could best this bound. In this talk, we will resolve this problem in an unexpected way by constructing infinitely many totally positive algebraic integers whose conjugates have an average of at most 1.899. To do this, we will apply a new method for constructing algebraic integers to an example first considered by Serre. We also will explain how our method can be used to find simple abelian varieties with extreme point counts.

Date and Time: Thursday, January 23 @ 2:40PM
Location: 520 Mathematics

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Special Colloquium

Speaker: Alex Petrov (MIT)
Title: Topology of algebraic varieties in positive characteristic
Abstract: One fruitful way to analyze the topology of an algebraic variety over complex numbers is to consider its cohomology groups; at least if one is content with disregarding torsion in cohomology, these groups can be expressed in terms of algebraic differential forms on the variety, via an algebraic analog of de Rham cohomology. When applied to algebraic varieties in positive characteristic, de Rham cohomology exhibits behavior that in several ways is qualitatively different from the situation in characteristic zero. I will discuss some recent progress in understanding these differences which have to do with the failure of Hodge decomposition in positive characteristic. The methods used turn out to also be helpful for studying torsion in cohomology of families of varieties in characteristic zero.

Date and Time: Wednesday, January 22 @ 4:30PM
Location: 520 Mathematics

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