Department of

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for events the day of Tuesday, November 12, 2019.

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Tuesday, November 12, 2019

11:00 am in 347 Altgeld Hall,Tuesday, November 12, 2019

Equivariant symmetric monoidal categories and K-theory

Peter Bonventre (University of Kentucky)

Abstract: Symmetric monoidal categories have (at least) two allures to homotopy theorists: they describe the algebraic structure appearing in many categories of interest, and they effectively model all connective spectra. Equivariantly, both of these tasks become more interesting, as the category of (connective) genuine equivariant spectra is significantly more subtle. In this talk, I introduce a new model for equivariant symmetric monoidal categories which both describes the algebraic structure in equivariant categories and has a K-theory functor to genuine G-spectra. I will give several examples and applications, including comparisons to many previously proposed models of genuine symmetric monoidal categories and equivariant algebra.

11:00 am in 347 Altgeld Hall,Tuesday, November 12, 2019

Equivariant symmetric monoidal categories and K-theory

Peter Bonventre (University of Kentucky)

Abstract: Symmetric monoidal categories have (at least) two allures to homotopy theorists: they describe the algebraic structure appearing in many categories of interest, and they effectively model all connective spectra. Equivariantly, both of these tasks become more interesting, as the category of (connective) genuine equivariant spectra is significantly more subtle. In this talk, I introduce a new model for equivariant symmetric monoidal categories which both describes the algebraic structure in equivariant categories and has a K-theory functor to genuine G-spectra. I will give several examples and applications, including comparisons to many previously proposed models of genuine symmetric monoidal categories and equivariant algebra.

1:00 pm in 345 Altgeld Hall,Tuesday, November 12, 2019

Model theory of $\mathbb{R}$-trees and of ultrametric spaces

Ward Henson (UIUC Math)

Abstract: First, we consider the class of metric spaces $(M,d)$ that are $\mathbb{R}$-trees with a convex metric. To treat this class using continuous first order logic, we fix a base point $p$ in $M$ and require that $M$ have radius at most $r$ with respect to $p (r>0)$. The class of these structures $(M,d,p)$ is axiomatizable. Moreover, the theory of this class has a model companion $T$, whose models we describe precisely. This theory is a well behaved continuous theory. For example, $T$ has QE and is complete; it is stable (but not superstable) and has the maximum possible number of models in each infinite cardinal.
  Second, given a model $M = (M,d,p)$ of $T$, we consider the closed subset $E_r(M) := \{x \in M | d(p,x)=r\}$. This is a definable set for $ $T, and the entire structure $M$ can be reconstructed from $(E_r(M),d)$. The metric $d$ on $E_r(M)$ is an ultrametric; further, at every $x \in E_r(M)$, the set of distances $\{d(x,y) | y \in E_r(M)\}$ is dense in the interval $[0,2r]$. These properties are easily seen to be axiomatizable in continuous logic, and we let $T^*$ denote the resulting theory. We show that $T^*$ has QE, so it is complete; further, $T$ and $T^*$ are bi-interpretable.
  This is joint work with Sylvia Carlisle.

1:00 pm in 347 Altgeld Hall,Tuesday, November 12, 2019

Turing patterns in the Schnakenberg equations: From normal to anomalous diffusion

Yanzhi Zhang (Department of Mathematics and Statistics, Missouri University of Science and Technology)

Abstract: In recent years, anomalous diffusion has been observed in many biological experiments. Instead of classical Laplace operator, the anomalous diffusion is described by the fractional Laplacian. In this talk, we will discuss the Turing patterns of the Schnakenberg equation and compare the pattern selection under normal and anomalous diffusion. Our analysis shows that the wave number of the Turing instability increases with the exponent of the fractional Laplacian. The interplay of the nonlinearity and long-range diffusion are studied numerically, and especially comparisons are provided to understand the nonlocal effects of the fractional Laplacian.

2:00 pm in 243 Altgeld Hall,Tuesday, November 12, 2019

Extremal graphs without 4-cycles

Zoltán Füredi (Alfréd Rényi Institute of Mathematics and UIUC)

Abstract: Let $\text{ex}(n,C_4)$ denote the maximum number of edges of a graph on $n$ vertices which does not contain a cycle of length $4$. It is known since the 1930's (Erdos and E Klein) that $\text{ex}(n,C_4)=\Theta(n^{3/2})$, and Brown (1966) and independently Erdos, Renyi, and Sos (1966) showed that it is asymptotic to $(1/2) n^{3/2}$. The determination of the exact values seems to be hopeless, but the speaker established their conjecture and showed (1983, 1996) that $\text{ex}(q^2+q+1, C_4)= (1/2)q^2(q+1)$ for primepower $q$.

Firke, Kosek, Nash, and Williford (2013) found another infinite series of exact values ($n=q^2+q$, $q$ is a power of $2$).

In this talk we sketch the proofs and also show (in the case of $n=q^2+q+1 > 100$) that all the extremal graphs can be obtained from a polarity graph. If time permits, other related algebraic constructions are discussed.