Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, March 8, 2016.

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Tuesday, March 8, 2016

11:00 am in 345 Altgeld Hall,Tuesday, March 8, 2016

Topological Hochschild homology of the connective cover of the K(1)-local sphere

Gabriel Angelini-Knoll (Wayne State)

Abstract: In the early 1980’s Waldhausen proposed a program for studying the algebraic K-theory of the E(n) Bousfield localizations of the sphere using his localization theorem. The E(0) localization of the sphere, which is equivalent to the rational Eiilengberg-Maclane spectrum is the only case where the algebraic K-theory is known. Due to the work of McCarthy and Dundas, relative algebraic K-theory and relative topological cyclic homology are homotopy equivalent after p completion for certain maps of ring spectra. Topological cyclic homology is built out of topological Hochschild homology using the S^1-equivariant and cyclotomic structure. We therefore compute mod (p,v_1) homotopy of THH of the connective cover of the K(1)-local sphere, which is the p-completion of E(1)-local sphere. We construct a filtration of the connective cover of the K(1)-local sphere and then use a May-type spectral sequence to compute THH of the connective K(1)-local sphere. The connective K(1)-local sphere is a chromatic height 1 spectrum, so we expect height 2 phenomena in its algebraic K-theory according to the chromatic red-shift conjecture of Ausoni and Rognes. Though this is not visible in THH, it can already be seen by examining the homotopy fixed points with respect to the circle action. Specifically, I will show how the element v_2 appears in the homotopy fixed points with respect to the circle action.

12:00 pm in Altgeld Hall 243,Tuesday, March 8, 2016

Mapping class group and right-angled Artin group actions on the circle

Thomas Koberda (University of Virginia)

Abstract: I will discuss virtual mapping class group actions on compact one-manifolds. The main result will be that there exists no faithful C^2 action of a finite index subgroup of the mapping class group on the circle, which generalizes results of Farb-Franks and establishes a higher rank phenomenon for the mapping class group, mirroring a result of Ghys and Burger-Monod. This talk will represent joint work with H. Baik and S. Kim.

1:00 pm in 345 Altgeld Hall,Tuesday, March 8, 2016

Chief factors in Polish groups

Phillip Wesolek (Université catholique de Louvain, visiting Northwestern University)

Abstract: For a Polish group $G$, closed normal subgroups $L < K$ of $G$ form a chief factor $K/L$ if there is no closed normal subgroup of $G$ strictly between $L$ and $K$. Chief factors play an important role in the structure theory of finite groups, and recently, they have appeared in the structure theory of compactly generated locally compact groups. Surprisingly, the theory of chief factors admits a natural and useful extension to the setting of Polish groups. In this talk, we develop this theory. In particular, we define the association relation and the partially ordered space of chief blocks. We then outline a Schreier refinement theorem and a trichotomy theorem for topologically characteristically simple Polish groups. Time permitting, we discuss applications to locally compact Polish groups and finitely generated branch groups. This is joint work with Colin Reid.

2:00 pm in 347 Altgeld Hall,Tuesday, March 8, 2016

Random walks on abelian sandpiles

John Pike   [email] (Cornell University)

Abstract: Given a simple connected graph $G=(V,E)$, the abelian sandpile Markov chain evolves by adding chips to random vertices and then stabilizing according to certain toppling rules. The recurrent states form an abelian group $\Gamma$, the sandpile group of $G$. I will discuss joint work with Dan Jerison and Lionel Levine in which we characterize the eigenvalues and eigenfunctions of the chain restricted to $\Gamma$ in terms of ``multiplicative harmonic functions'' on $V$. We show that the moduli of the eigenvalues are determined up to a constant factor by the lengths of vectors in an appropriate dual Laplacian lattice and use this observation to bound the mixing time of the sandpile chain in terms of the number of vertices and maximum vertex degree of $G$. We also derive a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on $G$.

3:00 pm in 241 Altgeld Hall,Tuesday, March 8, 2016

Rainbow arithmetic progressions

Ryan Martin (Iowa State University)

Abstract: In this talk, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers $n$ and $k$, the expression ${\rm aw}([n],k)$ denotes the smallest number of colors with which the integers $\{1,\ldots,n\}$ can be colored and still guaranteee there is a rainbow arithmetic progression of length $k$. We establish that ${\rm aw}([n],3)=\Theta(\log n)$ and ${\rm aw}([n],k)=n^{1-o(1)}$ for $k\geq 4$. For positive integers $n$ and $k$, the expression ${\rm aw}(\mathbb{Z}_n,k)$ denotes the smallest number of colors with which elements of the cyclic group of order $n$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. In this setting, arithmetic progressions can ``wrap around,'' and ${\rm aw}(\mathbb{Z}_n,k)$ behaves quite differently from ${\rm aw}([n],3)$, depending on the divisibility of $n$. As showin by Jungić, et al., ${\rm aw}(\mathbb{Z}_{2^m},3)=3$ for any positive integer $m$. We establish that ${\rm aw}(\mathbb{Z}_n,3)$ can be computed from knowledge of ${\rm aw}(\mathbb{Z}_p,3)$ for all of the prime factors $p$ of $n$. However, for $k\geq 4$, the behavior is similar to the previous case, that is, ${\rm aw}(\mathbb{Z}_n,k)=n^{1-o(1)}$. This work was part of what we called the "working seminar," done in a workshop-style setting over the course of two semesters. This is joint work with Steve Butler, Craig Erickson, Leslie Hogben, Kirsten Hogenson, Lucas Kramer, Richard L. Kramer, Jephian Chin-Hung Lin, Derrick Stolee, Nathan Warnberg and Michael Young.

4:00 pm in 243 Altgeld Hall,Tuesday, March 8, 2016

Introduction to toric ideals

Eliana Duarte (UIUC Math)

Abstract: In this talk I will give a brief introduction to affine toric varieties and the ideals used to describe their coordinate rings, known as toric ideals. I will present a theorem by Hochster that describes the multigraded Betti numbers of toric ideals in terms of the reduced homology of a certian simplicial complex. If time permits I will explain how Hochster’s theorem relates to the implicitization problem for monomial ideals. No prior knowledge of these topics will be assumed.

4:00 pm in 245 Altgeld Hall,Tuesday, March 8, 2016

On Approximation of Functions by Exponential Sums

Gregory Beylkin (University of Colorado at Boulder)

Abstract: We consider efficient approximations of functions (and number sequences) by linear combinations of exponentials or Gaussians and their applications. These approximations are obtained for a finite but arbitrary accuracy and may have significantly fewer terms than associated Fourier representations. It is well known that exponential functions and Gaussians are ubiquitous in mathematics and often lend themselves as tools in diverse applications. As examples of applications we will consider construction of quadratures for bandlimited exponentials, new algorithm and resolution improvements in X-ray tomography, computation of oscillatory integrals, approximation of operator kernels and their use in quantum chemistry and, briefly, a possible use of approximations of discount functions via exponentials in economic models.