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Tuesday, September 15, 2015

11:00 am in 243 Altgeld Hall,Tuesday, September 15, 2015

Computations in 2-local stable homotopy theory

Philip Egger (Northwestern)

Abstract: Computing the stable homotopy groups of spheres is a major focus of modern algebraic topology. One way of finding infinite families of nontrivial elements involves finite CW-complexes with non-nilpotent self-maps. I will show computationally (joint with Bhattacharya and Mahowald) that the spectra $A_1$ whose cohomology is $A(1)$ admits a self-map detected in Morava K-theory by $v_2^{32}$. Time permitting, I will describe Mahowald's proof of the $K(1)$-local telescope conjecture and introduce a finite complex $Z$ that I hope will shed light on the $K(2)$-local telescope conjecture.

12:00 pm in 345 Altgeld Hall,Tuesday, September 15, 2015

Random walks and random group extensions

Giulio Tiozzo (Yale University)

Abstract: Let us consider a group G of isometries of a delta-hyperbolic metric space X, which is not necessarily proper (e.g. it could be a locally infinite graph). We can define a random walk by picking random products of elements of G, and projecting this sample path to X. We show that such a random walk converges almost surely to the Gromov boundary of X, and with positive speed. As an application, we prove that a random k-generated subgroup of the mapping class group is convex cocompact, and a similar statement holds for Out(F_n). This is joint work, partially with J. Maher and partially with S. Taylor.

1:00 pm in 241 Altgeld Hall,Tuesday, September 15, 2015

Subelliptic heat kernels for some CR model spaces

Jing Wang (UIUC)

Abstract: I will present some results on an explicit expression of the heat kernel on CR Sasakian model spaces. In particular, we study heat kernels on the sphere $S^{2n+1}$ and on the anti deSitter spaces $H^{2n+1}$ which are known to be constantly positively and negatively curved model spaces. On each space there is a canonical diffusion operator $L$. The sub-Laplacian $L$ is not elliptic but only subelliptic. The symmetries of these model spaces enable us to obtain an explicit and geometrically meaningful formula for each associated heat kernel. As an application of such results, we can deduce the small-time behavior of the heat kernels on the diagonal, on the vertical cut-locus, and outside of the cut-locus. The key point is to work in cylindrical coordinates that reflect the symmetries coming from the Hopf fibration of these model spaces. This work is joint with F. Baudoin.

1:00 pm in 345 Altgeld Hall,Tuesday, September 15, 2015

On "Surreal numbers, derivations and transseries" by A. Berarducci and V. Mantova

Lou van den Dries   [email] (UIUC Math)

Abstract: This is to introduce the paper mentioned in the title. We plan to continue the study of this paper in Friday talks by other speakers. Here is its arXiv link:

2:00 pm in Altgeld Hall,Tuesday, September 15, 2015

Perturbations of delay differential equations at the verge of instability.

Nishanth Lingala (UIUC Aerospace Eng)

Abstract: The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. This talk deals with linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation lie on the imaginary axis of the complex plane (critical eigenvalues), and all other roots have negative real parts (stable eigenvalues). We show that, when the system is perturbed by small noise, under an appropriate change of time scale, the distribution of the amplitude of projection onto the critical eigenspace is close to the distribution of a certain one-dimensional stochastic differential equation (SDE) without delay. Further, we show that the projection onto the stable eigenspace is small. These results allow us to give an approximate description of the delay-system using an SDE (without delay) of just one dimension.

3:00 pm in 243 Altgeld Hall,Tuesday, September 15, 2015

Stable envelopes for Nakajima quiver varieties

Josh Wen (UIUC )

Abstract: Introduced by Maulik and Okounkov, stable envelopes provide an alternate form of the localization isomorphism for the equivariant cohomology of a space. This new isomorphism plays a central role in the their theory linking the quantum cohomology of Nakajima quiver varieties and a geometrically constructed Yangian. I’ll present bits of the general construction and then focus on its specialization to the cases of cotangent bundles to flag varieties and Hilbert schemes of points on the plane, where it is related to characteristic varieties of Verma modules and symmetric polynomials, respectively.

3:00 pm in 241 Altgeld Hall,Tuesday, September 15, 2015

On some problems of Cameron and Erdős

József Balogh   [email] (UIUC Math)

Abstract: Cameron and Erdős proposed several of enumeration problems in additive combinatorics, for example what is the number of sum-free sets in [n], or what is the number of sets in [n] which do not contain k-term arithmetic progression. I plan to survey the recent progress on these type of questions, focusing on the applications of a recent powerful tool, the hypergraph container lemma of Balogh-Morris-Samotij, and of Saxton-Thomason.