seminars:topsem
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| - | * **March 28th** \\ Speaker: **Mark Pentigore** (University of Virginia) \\ Title: **Residual finiteness growth functions of the mapping class group and the question of linearity** <WRAP box>// Abstract: // Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. One potential application of these functions is towards linearity of the mapping class group, and we will present some partial progress towards understanding these functions for the mapping class group.\\ | + | * **March 28th** \\ Speaker: **Mark Pengitore** (University of Virginia) \\ Title: **Residual finiteness growth functions of the mapping class group and the question of linearity** <WRAP box>// Abstract: // Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. One potential application of these functions is towards linearity of the mapping class group, and we will present some partial progress towards understanding these functions for the mapping class group.\\ |
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| - | * **April 4th** \\ Speaker: **Giuseppe Martone** (Sam Houston State University) \\ Title: **TBA** <WRAP box>// Abstract: // TBA\\ | + | * **April 4th** \\ Speaker: **Giuseppe Martone** (Sam Houston State University) \\ Title: **Correlation theorem and (cusped) Hitchin representations** <WRAP box>// Abstract: // Given distinct hyperbolic structures m and m' on a closed orientable surface, how many closed curves have m- and m'-length roughly equal to x, as x gets large? Schwartz and Sharp's correlation theorem answers this question. Their explicit asymptotic formula involves a term exp(Mx) and 0<M<1 is the correlation number of the hyperbolic structures m and m'. |
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| + | In this talk, we will show that the correlation number can decay to zero as we vary m and m', answering a question of Schwartz and Sharp. Then, we discuss extensions of this correlation theorem to the context of higher rank Teichmüller theory and find diverging sequences of SL(3,R)-Hitchin representations along which the correlation number stays uniformly bounded away from zero. | ||
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| + | This talk is based on joint work with Xian Dai and joint work in progress with Nyima Kao.\\ | ||
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| - | * **April 18th** \\ Speaker: **Alex Wright** (University of Michigan) \\ Title: **TBA** <WRAP box>// Abstract: // TBA\\ | + | * **April 18th** \\ Speaker: **Alex Wright** (University of Michigan) \\ Title: **Spheres in the curve graph and linear connectivity of the Gromov boundary** <WRAP box>// Abstract: //For a vertex $c$ and an integer radius $r$, the sphere $S_r(c)$ is the induced graph on the set of vertices of distance $r$ from $c$. We will show that spheres in the curve graph are typically connected, and discuss connectivity properties of the Gromov boundary. We will also explain the motivation and context for this work, touching tangentially on Cannon's conjecture and convex cocompactness. \\ |
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| - | * **April 25th** \\ No seminar this week (Tuesday classes meet) | + | * **April 25th** \\ No seminar this week (Monday classes meet) |
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| + | * **May 2nd** \\ Speaker: **Cary Malkiewich** (Binghamton University) \\ Title: **A Solomon-Tits theorem for arbitrary hyperplane collections** <WRAP box>// Abstract: // Suppose we take an arbitrary collection of hyperplanes in n-dimensional Euclidean, hyperbolic, or spherical geometry, along with all of their nonempty intersections. These form a partially ordered set, so we can take the realization and get a topological space, called the Tits complex. One version of the Solomon-Tits theorem says that, if we were to take *all* hyperplanes, the space we get is homotopy equivalent to a wedge of spheres of dimension (n-1). | ||
| - | * **May 2nd** \\ Speaker: **Zhouli Xu** (University of California San Diego) \\ Title: **TBA** <WRAP box>// Abstract: // TBA\\ | + | In this talk I'll describe how to prove a variant of this theorem where we can take just about any reasonable subset of the hyperplanes, and the result still holds. We can furthermore give a presentation of the homology of the resulting space: it has a generator for each polytope cut out by the hyperplanes, and the relations encode subdivision of the polytopes. The proof is quite fun, it's an inductive proof where we add the hyperplanes one at a time and count how many new polytopes, and spheres in the Tits complex, are created. Our main application is to the groups of cut-and-paste operations between these polytopes. |
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seminars/topsem.1710984737.txt · Last modified: 2024/03/20 21:32 by sapir