Mathematics Department

# Upcoming Talks

This is a list of all upcoming talks for the next two weeks. Talks are from 3:45-4:45 p.m. in the Colloquium or Seminar Room, unless otherwise specified.

• Mar
25
• Ergodicity on Fractal Spaces via Hyperbolic Geometry
Anton Lukyanenko
George Mason University
Time: 03:45 PM
Operator Algebras and Dynamics Seminars

#### View Abstract

Continued fractions on the real numbers have far-reaching applications, including connections to dynamics, Diophantine approximation, and hyperbolic geometry. Their generalizations, both in R and in higher dimensions have been a topic of extensive study over the last few decades. A central question has been the extent to which all points have the same CF-based properties, i.e. whether the associated Gauss map is ergodic. I will discuss an approach used by Hensley, based on the use of transfer operators, to argue that the Gauss map for complex continued fractions is ergodic; with connections to the work of Mauldin, Urbanski, and others. Then, I will describe the recent generalization of the theory to a more general class of spaces, where instead one can use hyperbolic geometry to prove ergodicity, extending the classical approach of Artin and Series.
• Mar
26
• Binomial Proofs and the Area Principle
Will Traves
USNA Math
Time: 12:00 PM
Basic Notions Seminars

#### View Abstract

I'll discuss some elementary geometric ideas which were new to me when I first heard about them several years ago. In particular, I'll explain the role that determinants play in plane geometry and how these ideas lead to an elegant proof technique. Along the way, we'll see why determinants deserve their name, encounter the Fundamental Theorem of Invariant Theory, and learn an important application of Cramer's rule.
• Mar
27
• Discussion of Grading Policies for Core Courses
Time: 12:00 PM
Teaching Seminar

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Discussion of how we determine grades in core courses. How do we assign extra credit? How should 6- and 12-week grades compare to exam grades and final grades? How do our expectations compare to the learning goals of the course? How can we balance academic freedom of instructors with fairness to students over different sections of the course?
• Mar
27
• Counting in flag manifolds
Leonardo Mihalcea
Virginia Tech
Time: 03:45 PM
Colloquium Series

#### View Abstract

Consider the following enumerative questions: how many points are there in a projective space over a finite field; how many lines pass through 4 given lines in 3-space; how many lines are on a non-singular cubic surface; how many (rational) plane curves pass through a number of points ? All of these can be answered by analyzing classical and quantum intersection rings for appropriate parameter (or moduli) spaces. The intersection rings have a particularly rich structure in the case flag manifolds, with connections to many areas in mathematics. I will give a brief overview of these rings, and sketch some techniques useful for calculations such as those above.
• Mar
29
• Data-informed stochastic model reduction for complex dynamics
Fei Lu
Johns Hopkins University
Applied Math Seminar

#### View Abstract

The need to develop reduced nonlinear statistical-dynamical models for complex dynamical systems arises in many applications such as geophysics, biology and engineering. The challenges come from memory effects due to the nonlinear interactions between resolved and unresolved scales, and from the difficulty in inference from discrete partial data. We address these challenges by learning stochastic reduced models, in forms of nonlinear time series, that can account for the memory effects due to truncation/coarse-graining and the numerical errors due to large time-stepping. We show by examples that the stochastic reduced models can capture the key statistical and dynamical properties and can improve the performance of ensemble prediction in data assimilation. The examples include dissipative chaotic/stochastic ODEs and PDEs. We will discuss related open questions in inference and in theoretical understanding of the model reduction.
• Apr
01
• Group actions on product systems and K-theory
Valentin Deaconu
Product systems $Y$ over various semigroups were introduced by N. Fowler, inspired by work of W. Arveson. We will recall the definition of $Y$ and introduce group actions and crossed products $Y\rtimes G$. One motivation is group actions on higher rank graphs. We generalize a result of C. Schafhauser for a row-finite and faithful product system $Y$ indexed by ${\mathbb N}^k$ concerning the $K$-theory of the crossed product by the gauge action $\gamma$. The main result is $K_*({\mathcal O}_A(Y)\rtimes_\gamma{\mathbb T}^k)\cong \varinjlim_{n \in {\mathbb N}^k} (K_*(A),[Y_n]), where$[Y_n]$denotes the homomorphism induced by$Y_n\$ via Fredholm operators. We apply this result to a product system constructed from group representations.