Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, February 2, 2016.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, February 2, 2016

1:00 pm in 347 Altgeld Hall,Tuesday, February 2, 2016

Modulational Instability in KdV, BBM, and Boussinesq type equations

Ashish Pandey (UIUC Math)

Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM and regularized Boussinesq type equations.

1:00 pm in 347 Altgeld Hall,Tuesday, February 2, 2016

Symplectic non-squeezing for the discrete nonlinear Schr\"odinger equation(Part II)

Alexander Tumanov (UIUC Math)

Abstract: The speaker will continue to talk more details on the topic of last seminar. In particular, the proof of the main result will be discussed.

3:00 pm in 241 Altgeld Hall,Tuesday, February 2, 2016

Minimum Degree and Dominating Paths

Douglas B. West   [email] (Zhejiang Normal University and University of Illinois)

Abstract: A dominating path in a graph is a path $P$ such that every vertex outside $P$ has a neighbor on $P$. A result of Broersma from 1988 implies that if $G$ is an $n$-vertex $k$-connected graph and $\delta(G)>(n-k)/(k+2)-1$, then $G$ contains a dominating path. We seek short dominating paths. For $\delta(G)>an$, where $a$ is a constant greater than $1/3$, the minimum length of a dominating path is at most logarithmic in $n$ when $n$ is sufficiently large (the base of the logarithm depends on $a$). For constant $s$ and $c'<1$, an $s$-vertex dominating path is guaranteed by $\delta(G)\ge n-1-c'n^{1-1/s}$ when $n$ is sufficiently large, but $\delta(G)\ge n-c(s\ln n)^{1/s}n^{1-1/s}$ (where $c>1$) does not even guarantee a dominating set of size $s$. This is joint work with Ralph Faudree, Ronald Gould, and Michael Jacobson.

3:00 pm in 243 Altgeld Hall,Tuesday, February 2, 2016

Limiting Mixed Hodge Structures and the Boundary of the Moduli Space of IIA String Theory

Sheldon Katz (Illinois Math)

Abstract: The moduli space of Type IIA string theory is a quantum-corrected version of the Calabi-Yau integrable system of Donagi and Markman. In this talk, I explain how the limiting mixed Hodge structure associated to degenerations of a Calabi-Yau threefold to a threefold containing a curve of singularities of ADE type describes the IIA boundary and contains all the information of the Hitchin system and various gauge-theoretic generalizations arising in F-theory. This talk is based on joint work in progress with L. Anderson, J. Heckman, and L. Schaposnik.

4:00 pm in 243 Altgeld Hall,Tuesday, February 2, 2016

The Krichever correspondence

Matej Penciak (UIUC Math)

Abstract: In this talk I will describe an unexpected equivalence between two seemingly unrelated pieces of data. On one side of the correspondence are commutative algebras of differential operators, and on the other are curves with additional spectral data. In the remaining time I will try to describe how generalizations of this correspondence fit into the modern theory of integrable hierarchies associated to Lie algebras.

4:00 pm in 314 Altgeld Hall,Tuesday, February 2, 2016

Classification of II$_1$ factors arising from free groups acting on spaces

Sorin Popa (UCLA)

Abstract: An old conjecture of Higman from the 1930s states that non-isomoprih torsion free groups $\Gamma$ have non-isomorphic (complex) algebras $\Bbb C\Gamma$. The group algebra $\Bbb C\Gamma$ acts by left convolution on the Hilbert space $\ell^2 \Gamma$ and its closure in the week operator topology gives rise to the group von Neumann algebra $L(\Gamma) \subset \mathcal B(\ell^2\Gamma)$. But the isomorphism class of these much bigger algebras may dramatically "forget" the initial data $\Gamma$. For instance, all wreath product groups $\Gamma = H \wr \Lambda$, with $H, \Lambda$ infinite amenable (e.g. $\Gamma_n=\Bbb Z \wr \Bbb Z^n$, $n=1, 2,$) give rise to isomorphic $L(\Gamma)$'s (Connes 1976). A famous problem of Murray-von Neumann (1943) asks whether the von Neumann algebras $L(\Bbb F_n)$ (II$_1$ factors), arising from the free groups $\Bbb F_n$, $2\leq n \leq \infty$, are non-isomorphic for distinct $n$'s. While this so called free group factor problem is still open, its "group measure space" version, showing that the von Neumann algebras arising from free ergodic actions of $\Bbb F_n$ on probability measure spaces are non-isomoprphic for different $n$, independently of the actions, has recently been settled by Stefaan Vaes and myself. This shows in particular that $L(\Bbb Z \wr \Bbb F_n)$, $n=1, 2, ,$ are non-isomorphic. I will comment on this result and on the related free group factor problem.

A reception will follow the Feb 2 lecture in 239 Altgeld Hall from 5-6 p.m.