2週內演講

Algebraic Geometry & Number Theory, 日本東北大學 山崎龍雄教授 Tuesday, March 3, 14:00—15:00 數學系3樓會議室 Title: Mixed Hodge Structures with Modulus and 1-Mmotives Abstract: We shall overview how the notion of modulus enables us to generalize many aspects of motive theory. One of the earliest examples can be seen in Laumon's generalization of the theory of Deligne 1-motives. We then explain our construction of modulus version of the mixed Hodge structures. As an application, we generalize Kato-Russell's construction of Albanese varieties with modulus to 1-motives. This is joint work with Ivorra. No knowledge of Hodge theory is assumed.

Algebraic Geometry & Number Theory, 台灣大學數學系 楊一帆教授 Tuesday, March 3, 15:30—16:30 數學系3樓會議室 Title: Modular Units and Cuspidal Divisor Classes on X_0(n^2M) with n|24 and M squarefree Abstract: For a positive integer N, let C(N) be the subgroup of J_0(N) generated by all cuspidal divisors of degree 0 and C(N)(\mathbb Q) be its rational subgroup. Let also C_{\mathbb Q}(N) be the subgroup of C(N)(\mathbb Q) generated by \mathbb Q-rational cuspidal divisors. We prove that when N=n^2M for some integer n dividing 24 and some squarefree integer M, the two groups C(N)(\mathbb Q) and C_{\mathbb Q}(N) are equal. To achieve this, we determine all the modular units on X_0(N) for such level N. This is a joint work with Liuquan Wang.

Colloquium, Purdue University 何孟哲助理教授 Thursday, March 5, 16:10—17:00 數學系3樓會議室 Title: Groups, Logic, and Languages Abstract: The interplay between group theory and logic had played a crucial role in both areas for many decades. The most famous questions in this intersection is the word problem proposed by Dehn in 1911. The word problem is shown to be unsolvable in general by Novikov in 1955. However, as a logician, the (un)solvability of a decision problem is only the beginning. For an unsolvable problem, computable structure theory gives a framework to study "how unsolvable" the problem is. On the other hand, for a solvable problem, formal language theory provides a way to study its complexity. We will survey various past and current results as well as some work in progress in these directions In particular, we will study the linguistic complexity of word problems and geodesic representatives in finitely-generated groups.