#### ** Upcoming Talk**

**May 20 2020 (Wed.)**** 11:00-12:30**

Ser-Wei Fu (National Center for Theoretical Sciences)

Venue: Room 202, Astro-Math. Building

Title:Simple closed curves, foliations, and flat surfaces

**Abstract:**

**Simple closed curves often serve as the example to connect intuition to theorems. This talk will start from Riemann surfaces and conclude with length rigidity and deformations. The focus will be placed on the intermediate object, the space of measured foliations. The aim of the talk is to present the beautiful intertwined picture involving the mapping class group, Teichmuller theory, holomorphic quadratic differentials, and the dynamics of deformations.
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**May 27 2020 (Wed.)**** 11:00-12:30**

Yen-Liang Kuan (National Center for Theoretical Sciences)

Venue: Room 202, Astro-Math. Building

Title:The Mordell-Weil theorem for t-modules

**May 27 2020 (Wed.)**

**11:00-12:30**

Yen-Liang Kuan (National Center for Theoretical Sciences)

Venue: Room 202, Astro-Math. Building

Title:The Mordell-Weil theorem for t-modules

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**Abstract:**

**For each positive characteristic multiple zeta value (defined by Thakur) $\zeta_A(\mathfrak{s})$, Chang-Papanikolas-Yu constructed the $t$-module $E_{\mathfrak{s}}$ defined over $A$ and integral points $\mathbf{v}_{\mathfrak{s}}$, $\mathbf{u}_{\mathfrak{s}} \in E_{\mathfrak{s}}(A)$. They proved that $\zeta_A(\mathfrak{s})$ is Eulerian (resp. zeta-like) if and only if $\mathbf{v}_{\mathfrak{s}}$ is an $\mathbb{F}_q[t]$-torsion point in $E_{\mathfrak{s}}(A)$ (resp. $\mathbf{v}_{\mathfrak{s}}$, $\mathbf{u}_{\mathfrak{s}}$ are $\mathbb{F}_q[t]$-linearly dependent in $E_{\mathfrak{s}}(A)$).
In this talk, we are interested in the structure theory of the $t$-module $E_{\mathfrak{s}}(A)$. Poonen proved an analogue for Drinfeld modules of the Mordell-Weil theorem. We shall generalize his results to the case of specific families of $t$-modules. In particular, we prove that the $t$-module $E_{\mathfrak{s}}(A)$ is the direct sum of its torsion submodule, which is finite, with a free $\mathbb{F}_q[t]$-module of rank $\aleph_0$. **

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