Mathematics

Word and conjugacy problems in finitely generated groups

Arman Darbinyan (Vanderbilt Math)

Abstract: Word and conjugacy problems are two of the main decision problems associated with finitely generated groups. In particular, there are deep results which bridge some of the main concepts of the theories of computability and computational complexity with group theoretical invariants through the word problem in groups. In this talk we will recall some of the well-known facts about word and conjugacy problems in groups as well as discuss new results concerning the relationship between these decision problems.

Spectral heat content for Levy processes

Hyunchul Park (SUNY at New Paltz)

Abstract: In this talk, we study a short time asymptotic behavior of spectral heat content for Levy processes. The spectral heat content of a domain D can be interpreted as the amount of heat if the initial temperature on D is 1 and temperature outside D is identically 0 and the motion of heat particle is governed by underlying Levy processes. We study spectral heat content for arbitrary open sets with finite Lebesgue measure in a real line under some growth condition on the characteristic expo- nents of the L ́evy processes. We observe that the behavior is very different from the classical heat content for Brownian motions. We also study the spectral heat content in general dimensions when the processes are of bounded varia- tion. Finally we prove that asymptotic expansion of spectral heat content is stable under integrable perturbation when heat loss is sufficiently large. This is a joint work with Renming Song and Tomasz Grzywny.

An improved upper bound for the Hales-Jewett number HJ(4,2)

Mikhail Lavrov (Illinois Math)

Abstract: In the $n$-dimensional grid $[t]^n = \{1, 2, \dots, t\}^n$, a combinatorial line is an injective function $\ell : [t] \to [t]^n$ such that for each coordinate $1 \le i \le n$, $\ell_i$ is either constant or the identity function on $[t]$. An example of such a line in $[4]^5$ is the function $\ell(x) = (3, x, 1, x, 4)$ whose image is the set of four points (3,1,1,1,4), (3,2,1,2,4), (3, 3,1,3,4), (3,4,1,4,4). A classic result in Ramsey theory, the Hales-Jewett theorem, asserts that for all values of the parameters $t$ and $r$, there is a sufficiently large $n = HJ(t,r)$ such that any $r$-coloring of $[t]^n$ will contain a combinatorial line whose points are monochromatic. Apart from $HJ(2,r)$ for variable $r$ and $HJ(3,2)$ which have all been determined exactly, good bounds on values of $HJ(t,r)$ are generally not known. Even for the small case $HJ(4,2)$, the best result known is a general result due to Shelah, which gives an upper bound between $2 \uparrow \uparrow 7$ and $2 \uparrow \uparrow 8$: quite far from the best known lower bound of 12. We show that the upper bound can be improved to $10^{11}$.