Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, January 29, 2019.

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Tuesday, January 29, 2019

12:00 pm in 243 Altgeld Hall,Tuesday, January 29, 2019

The Farey Sequence Next-Term Algorithm, and the Boca-Cobeli-Zaharescu Map Analogue for Hecke Triangle Groups G_q

Diaaeldin Taha (University of Washington)

Abstract: The Farey sequence is a famous enumeration of the rationals that permeates number theory. In the early 2000s, F. Boca, C. Cobeli, and A. Zaharescu encoded a surprisingly simple algorithm for generating--in increasing order--the elements of each level of the Farey sequence as what grew to be known as the BCZ map, and demonstrated how that map can be used to study the statistics of subsets of the Farey fractions. In this talk, we present a generalization of the BCZ map to all Hekce triangle groups G_q, q \geq 3, with the G_3 = SL(2, \mathbb{Z}) case being the "classical" BCZ map. If time permits, we will present some applications of the G_q-BCZ maps to the statistics of the discrete G_q linear orbits in the plane \mathbb{R}^2 (i.e. the discrete sets \Lambda_q = G_q (1, 0)^T).

1:00 pm in 347 Altgeld Hall,Tuesday, January 29, 2019

Traveling waves in an inclined channel and their stability

Zhao Yang (Indiana University Bloomington)

Abstract: The inviscid Saint-Venant equations are commonly used to model fluid flow in a dam or spillway. To classify known traveling wave solutions to the St. Venant equations, the condition of hydrodynamic stability introduces a dichotomy on the parameter F (Froude number): Namely, the constant flow solution is stable for F < 2 where one expect persistent asymptotically-constant traveling wave solutions and unstable for F > 2 where one expect rather complex pattern formation. We will discuss for F>2 Dressler's construction of the inviscid roll wave solution and for F<2 Yang-Zumbrun's construction of the smooth/discontinuous hydraulic shock profiles. We will then present recent stability results of these traveling waves. That is a complete spectral stability diagram for F>2 roll wave case obtained in [JNRYZ18] and spectral, linear orbital, and nonlinear orbital stability of all the hydraulic shock profiles obtained in [YZ18] and [SYZ18].

1:00 pm in 345 Altgeld Hall,Tuesday, January 29, 2019

Self-similar structures

Garret Ervin (Carnegie Mellon)

Abstract: An iterated function system is a finite collection $f_1, …, f_n$ of contraction mappings on a complete metric space. Every such system determines a unique compact subspace $X$, called the attractor of the system, such that $X = \bigcup f_i[X]$. Many well-known fractals, like the Cantor set and Sierpinski triangle, are realized as attractors of iterated function systems.
 A surprisingly rich analysis can be carried out even when the functions $f_i$ are only assumed to be non-surjective injections from a set to itself. Moreover, in many cases this analysis can be used to characterize when a structure $X$, like a group or linear order, is isomorphic to a product of itself, or to its own square. Such structures behave much like attractors of iterated function systems. We present the technique, and cite solutions to two old problems of Sierpinski as an application.

2:00 pm in 243 Altgeld Hall,Tuesday, January 29, 2019

Eigenvalues and graph factors

Suil O (Stony Brook University)

Abstract: An (even or odd) $[a,b]$-factor is a spanning subgraph $H$ such that ($d_H(v)$ is even or odd respectively, and) $a \le d_H(v) \le b$ for all $v \in V(G)$. When $a=b=k$, it is called a $k$-factor.

In this talk, we give sharp conditions for a graph to have an even $[a,b]$-factor. For a positive integer $k$, we also prove a sharp lower bound for the spectral radius in an $n$-vertex graph to have a $k$-factor. Furthermore, we give a sharp lower bound for the third largest eigenvalue in an $n$-vertex $r$-regular graph to have odd $[1,b]$-factor.

This is joint work partly with Eun-Kyung Cho, Jongyoon Hyun, Jeongrae Park, and Douglas B. West.

3:00 pm in 243 Altgeld Hall,Tuesday, January 29, 2019

Non-reduced Parabolic Group Schemes

William Haboush (UIUC Math)

Abstract: In the 90’s I and my student N. Lauritzen described all possible non reduced parabolic subgroup schemes of a semisimple algebraic group. These lead to complete homogeneous spaces with very interesting properties. Among other things they provide counterexamples which were crucial to the Mori program. Now that the Lusztig conjecture has been shown to be completely false I am revisiting this material hoping to make some interesting contribution to the decomposition problem for Weyl modules.

4:00 pm in 243 Altgeld Hall,Tuesday, January 29, 2019

Backprop in Neural Nets and Automatic Differentiation

George Francis   [email] (University of Illinois at Urbana–Champaign)

Abstract: In 1988 Rumelhart et al brought backpropagation into prominence throughout the Connectionist School of AI (neural nets, hidden layers, deep learning, etc). The technique was used earlier, but had remained obscure til then. Now, 3 decades later, backprop is a well established component of ML theory and practice. But it often comes wrapped in dense mathematical obscurity. In my latter day efforts to understand backprop I finally found some comprehensible answers in Baydin, Pearlmutter, Radul, and Siskind's survey paper "Automatic Differentiation in Machine Learning", J. Machine Learning Res 18 (2018) pp 1-43. I hope to pass along what I learned by working through a very illuminating example, leaving the context and (informal) definitions to the ample Q/A part of the seminar. For more information about our seminar, please see its webpage at http://new.math.uiuc.edu/MathMLseminar/