Large course meetings have been inefficient in the past, especially when meetings focus on delivering updates about course administration and policies that are effectively communicated via email. Instead, TAs need an opportunity to discuss their teaching in a smaller group where they can get feedback and guidance about their specific issues. We are working to design a structure that will require less time from course supervisors and lecturers than weekly course meetings by having our Lead TAs run smaller meetings that are more focused on preparing lessons and developing teaching strategies.

Speaker: Jared Speck, Massachusetts Institute of Technology

Title: The Formation of Shock Singularities in Solutions to Wave Equation Systems with Multiple Speeds

Date: 04/10/2017

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

In this talk, I will describe my recent work on the formation of shock singularities in solutions to quasilinear wave equation systems in 2D with more than one speed, that is, systems with at least two distinct wave operators. In the systems under study, the fast wave forms a shock singularity while the slow wave remains regular, even though the two waves interact all the way up to the singularity. This work represents an extension of the remarkable proofs of shock formation for scalar quasilinear wave equations provided by S. Alinhac, as well as the breakthrough sharpening of Alinhac's results by D. Christodoulou for the scalar wave equations of irrotational fluid mechanics. Both results crucially relied on the construction of geometric vectorfields that are adapted to the wave characteristics, whose intersection drives the singularity formation. THe key new difficulty for systems with multiple speeds is that the geometric vectorfields are, by necessity, precisely adapted to the characteristics of the shock-forming (fast) wave. Thus, there is no freedom left to adapt the vectorfields to the characteristics of the slow wave, and for this reason, they exhibit very poor commutation properties with the slow wave operator. To overcome this difficulty, we rely in part on some ideas from our recent joint work with J. Luk, in which we proved a shock formation result for the compressible Euler equations with vorticity, which we formulated as a wave-transport system featuring precisely one wave operator.

As part of or program on noncommutative laurent phenomenon, we
introduce and study noncommutative Catalan 'numbers' as Laurent
polynomials in infinitely many free variables and related theory of
noncommutative binomial coefficients. We also study their (commutative
and noncommutative) specializations, relate them with Garsia-Haiman
(q,t)-versions, and establish total positivity of the corresponding
Hankel matrices. Joint work with Arkady Berenstein (Univ. of Oregon).

In a partially ordered set P, let a pair of elements (x,y) be called alpha-balanced if the proportion of linear extensions that has x before y is between alpha and 1-alpha. The 1/3-2/3 Conjecture states that every finite poset which is not a chain has some 1/3-balanced pair. While the conjecture remains unsolved, we extend the list of posets that satisfy the conjecture by adding certain lattices, including products of two chains, as well as posets that correspond to Young diagrams.

Khovanov homology is a combinatorially-defined knot invariant which refines the Jones polynomial. After recalling the definition of Khovanov homology we will sketch a construction of a stable homotopy refinement of Khovanov homology. We will conclude with some modest applications and some work in progress. This is joint work with Tyler Lawson and Sucharit Sarkar. Another construction of the Khovanov stable homotopy type was given by Hu-Kriz-Kriz.

Persistent homology is a method for computing topological features of a space at different spatial resolutions. In 2004, Zomorodian and Carlsson figured out an algorithm to compute persistent homology when the coefficient ring is a field F. I will mostly be focusing on this. It is supposed to be a very basic talk.

Speaker: Antoine Ayache, University of Lille 1, France

Title: Uniformly and Strongly Consistent Estimation for the Hurst Function of a Linear Multifrational Stable Motion

Date: 04/13/2017

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

Multifractional processes have been introduced in the 90's in order to overcome some limitations of the well-known Fractional Brownian Motion (FBM) due to the constancy in time of its Hurst parameter; in their context, this parameter becomes a Hölder continuous function. Global and local path roughness of a multifractional process are determined by values of this function; therefore, several authors have been interested in their statistical estimation, starting from discrete variations of the process. Because of the complex dependence structure of variations, showing consistency of estimators is a tricky problem which often requires hard computations.
The main goal of our talk, is to introduce in the setting of the non-anticipative moving average Linear Multifractional Stable Motion (LMSM) with a stability parameter 'alpha' strictly larger than 1, a new strategy for dealing with the latter problem. In contrast with the previous strategies, this new one, does not require to look for sharp estimates of covariances related to variations; roughly speaking, it consists in expressing them in such a way that they become independent up to negligible remainders.
This is a joint work with Julien Hamonier at University of Lille 2.

Title: The algebra of box splines, hyperplane arrangements, and zonotopes

Date: 04/13/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Zonotopal algebra is a framework for studying various algebraic, combinatorial, and analytic objects associated to a linear map $\phi: \R^N \rightarrow \R^n$, where $n\le N$. This unified perspective gives formulas for volumes and lattice point enumerators of certain zonotopes, among other things.
This framework was inspired by the theory of box splines, which are piecewise-polynomial functions supported on zonotopes, whose chambers are determined by the matrix $X$ of the map $\phi$. Box splines can be thought of as fiber volume functions, as they measure the volume of the $(N-n)$-dimensional preimage of their $n$-dimensional argument, where the preimage is restricted
to the ``box' $[0,1]^N$.
I'll explain how the theory of zonotopal algebra connects these analytic phenomena to the algebraic properties of the linear map $X$. In particular, how the matroidal structure of $X$ is related to:
1. a family of polynomial ideals associated to $X$,
2. the kernels of those ideals, i.e., the spaces of polynomials annihilated by those ideals,
3. the discrete geometry of the associated hyperplane arrangement, and
4. the tilings of the associated zonotope.
This new line of research allows to study combinatorial and algebraic objects using techniques of analysis. Examples include recent results of de Concini, Procesi, Vergne, Moci, Lenz, and others.

Speaker: Olga V. Holtz, University of California, Berkeley

Title: The blessings and curses of complexity

Date: 04/15/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Ever since the invention of the first algorithms, mathematicians wondered how 'complex' such computational procedures are. This talk will offer an excursion into the world of complexity. How fast can we determine if a given number is prime, find the greatest common divisor of two
polynomials, or multiply two matrices? What problems are solvable in polynomial time? What are randomized algorithms and how complex are they? What is communication complexity? And why should we care whether or not P equals NP?