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Seminar Calendar
for events the day of Tuesday, April 18, 2017.

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Tuesday, April 18, 2017

11:00 am in 345 Altgeld Hall,Tuesday, April 18, 2017

Representable Cartesian Fibrations

Nima Rasekh (UIUC)

Abstract: The goal of this talk is to introduce a notion of a representable Cartesian fibration. Getting there will take us on a journey to many different places in higher category land. We will start by discussing right fibrations, which model presheaves, and then move on to generalize it to Cartesian fibrations. Finally we will have our last stop in the realm of complete Segal objects, which will enable us to define and discuss representable Cartesian fibrations.

12:00 pm in 243 Altgeld Hall,Tuesday, April 18, 2017

Compatibility of length functions and the geometry of their deformation spaces

Edgar Bering (UIC Math)

Abstract: Given two length functions l, m of minimal irreducible G actions on R-trees A, B, when is l + m again the length function of a minimal irreducible G action on an R-tree? We will show that additivity is characterized by the geometry of the Guirardel core of A and B, and also by a combinatorial compatibility condition generalizing the condition given by Behrstock, Bestvina, and Clay for Fn actions on simplicial trees. This compatibility condition allows us to characterize the PL-geometry of common deformation spaces of R-trees, such as the closure of Culler-Vogtmann Outer Space or the space of small actions of a hyperbolic group G.

1:00 pm in 347 Altgeld Hall,Tuesday, April 18, 2017

Ground states for nonlinear Schr\"odinger equation on a dumbbell graph

Jeremy Marzuola (University of North Carolina at Chapel Hill)

Abstract: With Dmitry Pelinovsky, we describe families of standing waves on a closed quantum graph in the shape of a dumbbell, namely having two loops connected by a link with Kirchhoff boundary conditions. We describe symmetry breaking bifurcations and prove a remarkable asymptotic property that is a bit surprising in terms of the energy minimizing solutions at a given mass.

2:00 pm in 347 Altgeld Hall,Tuesday, April 18, 2017

Path transformations of excursion landscape

Ju-Yi Yen (University of Cincinnati)

Abstract: In this talk, we study the process obtained from a Brownian bridge after excising all the excursions below the waterline level which reach zero. Three variables of interest are the maximum of this process, the value where this maximum is attained, and the total length of the excursions which are excised. Our analysis relies on some interesting transformations connecting Brownian path fragments and the 3-dimensional Bessel process.

3:00 pm in 241 Altgeld Hall,Tuesday, April 18, 2017

Permutations fixing k-sets and applications to group theory

Kevin Ford (Illinois Math)

Abstract: A $k$-set of a permutation $\pi\in S_n$ is a subset $I\in [n]$ of size $k$ which is itself permuted by $\pi$. Equivalently, $I$ is a product of a subset of the cycles of $\pi$. In this talk, we discuss two problems: (1) If one chooses $r>1$ permutations at random, what is the likelihood that for some large $k$ each contains a $k$-set? This has application to the problem of invariable generation of $S_n$, and is connected with a famous old problem of Erdős: to show that almost all integers have two divisors in some dyadic interval $(y,2y]$. (2) Given $k_1, k_2, \ldots, k_m$ what is the likelihood that a random $\pi$ has a $k_1$-set, $k_2$-set, ..., $k_m$-set (all disjoint)? Such bounds are applied to the problem of estimating how many permutations $\pi \in S_n$ lie in transitive subgroups of $S_n$ other than $S_n$ or $A_n$. This is joint work with Sean Eberhard, Ben Green and Dimitrios Koukoulopoulos.

3:00 pm in 243 Altgeld Hall,Tuesday, April 18, 2017

Stable quotients and the B-model

Rahul Pandharipande (ETH Zurich)

Abstract: I will give an account of recent progress on stable quotient invariants, especially from the point of view of the B-model and present a geometrical derivation of the holomorphic anomaly equation for local CY cases (joint work with Hyenho Lho).

4:00 pm in 245 Altgeld Hall,Tuesday, April 18, 2017

Faculty Discussion Session: Grants

4:00 pm in 131 English Building,Tuesday, April 18, 2017

Free rods under tension and compression: cascading and phantom spectral lines

Jooyeon Chung (UIUC Math)

Abstract: In this talk, I will consider the spectrum of the one-dimensional vibrating free rod equation $u'''' − \tau u'' = \mu u$ under tension ($\tau > 0$) or compression ($\tau < 0$). The eigenvalues $\mu$ as functions of the tension/compression parameter $\tau$ are shown to exhibit three distinct types of behavior. In particular, eigenvalue branches in the lower half-plane exhibit a cascading pattern of barely-avoided crossings. I will graphically illustrate properties of the eigenvalue curves such as monotonicity, crossings, asymptotic growth, cascading and phantom spectral lines.