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.

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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.

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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Francesco Lin, Associate Professor, earned The Richard Hamilton Best Paper Award in Geometric Analysis at the 2024 Annual International Congress of Chinese Mathematicians for his paper “The Seiberg-Witten Equations and the Length Spectrum of Hyperbolic Three-manifolds”; the conference was held in Shanghai from January 3-6, 2025, where Professor Lin also presented a lecture on Floer theory and the geometry of hyperbolic $3$-manifolds.

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Professor Dusa McDuff will receive the 2025 American Mathematical Society Leroy P. Steele Prize for Lifetime Achievement. […]

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Title: Symplectic geometry and inscription problems

Speaker: Josh Greene (Boston College)

Date, Time, Location: Wednesday, December 4th @4:30PM in Math Hall 520

Abstract:

The square peg problem was posed by Otto Toeplitz in 1911. It asks whether every Jordan curve in the plane contains the vertices of a square, and it is still open to this day. I will survey the approaches to this problem and its relatives using symplectic geometry. This talk is based on joint work with Andrew Lobb.

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