2018
Algebra Seminars 2018
November 12 (Mon) 2018, 10:30~12:00, 13:30~15:00
Title: Lectures on perfectoid Shimura varieties VII, VIII, 아산이학관 526
Speaker: Wansu Kim (KIAS)
Abstract:
The purpose of this lecture series is to explain the main results of Scholze’s paper “On torsion in the cohomology of locally symmetric varieties” in the case of Siegel modular varieties. More specifically, I’d like to explain perfectoid Shimura varieties with Hodge-Tate period map (in the Siegel case), and how it can be used to construct automorphic Galois representations (in the case of GL(n) over Q).
There are a lot of technical details and rigour one could delve into, but to make this task manageable I will try to focus on how to work with various technical tools (instead of defining and explaining them rigorously).
I will try to plan so that lectures each day can “stand on its own”, and I will not assume that the audience have attended the overview lectures on the first day (15th Oct).
November 5 (Mon) 2018, 10:30~12:00, 13:30~15:00
Title: Lectures on perfectoid Shimura varieties V, VI, 아산이학관 526
Speaker: Wansu Kim (KIAS)
Abstract:
The purpose of this lecture series is to explain the main results of Scholze’s paper “On torsion in the cohomology of locally symmetric varieties” in the case of Siegel modular varieties. More specifically, I’d like to explain perfectoid Shimura varieties with Hodge-Tate period map (in the Siegel case), and how it can be used to construct automorphic Galois representations (in the case of GL(n) over Q).
There are a lot of technical details and rigour one could delve into, but to make this task manageable I will try to focus on how to work with various technical tools (instead of defining and explaining them rigorously).
I will try to plan so that lectures each day can “stand on its own”, and I will not assume that the audience have attended the overview lectures on the first day (15th Oct).
October 29 (Mon) 2018, 10:30~12:00, 13:30~15:00
Title: Lectures on perfectoid Shimura varieties III, IV, 아산이학관 526
Speaker: Wansu Kim (KIAS)
Abstract:
The purpose of this lecture series is to explain the main results of Scholze’s paper “On torsion in the cohomology of locally symmetric varieties” in the case of Siegel modular varieties. More specifically, I’d like to explain perfectoid Shimura varieties with Hodge-Tate period map (in the Siegel case), and how it can be used to construct automorphic Galois representations (in the case of GL(n) over Q).
There are a lot of technical details and rigour one could delve into, but to make this task manageable I will try to focus on how to work with various technical tools (instead of defining and explaining them rigorously).
I will try to plan so that lectures each day can “stand on its own”, and I will not assume that the audience have attended the overview lectures on the first day (15th Oct).
October 15 (Mon) 2018, 10:30~12:00, 13:30~15:00
Title: Lectures on perfectoid Shimura varieties I, II, 아산이학관 526
Speaker: Wansu Kim (KIAS)
Abstract:
The purpose of this lecture series is to explain the main results of Scholze’s paper “On torsion in the cohomology of locally symmetric varieties” in the case of Siegel modular varieties. More specifically, I’d like to explain perfectoid Shimura varieties with Hodge-Tate period map (in the Siegel case), and how it can be used to construct automorphic Galois representations (in the case of GL(n) over Q).
There are a lot of technical details and rigour one could delve into, but to make this task manageable I will try to focus on how to work with various technical tools (instead of defining and explaining them rigorously).
On the first day, I will explain the main results in the case of modular curves. Note that the construction of perfectoid modular curve with Hodge-Tate period map is already new (and is due to Scholze).
August 2018, 21 (Tue) 16:00~17:30, 22~24 (Wen,Thur,Fri) 15:00~16:30
Title: Lectures on Fontaine-Laffaille theory and ramification of crystalline representations, 아산이학관 111
Speaker: Hattori Shin (Tokyo City University)
Abstract:
Let K be a complete discrete valuation field of mixed characteristic (0,p) with perfect residue field and G_K its absolute Galois group. For a variety X over K, etale cohomology groups with coefficients in Q_p yield p-adic G_K-representations, which can be considered as linearizations of geometric information of X. p-adic Hodge theory enables us not only to understand etale cohomology groups via (variants of) de Rham cohomology groups, but also to classify good p-adic G_K-representations by certain semi-linear algebraic data.
On the other hand, we often encounter G_K-representations with Z_p- or p-power torsion coefficients, which are more subtle than those with Q_p-coefficients. Integral p-adic Hodge theory is a tool to study them in a compatible way with p-adic Hodge theory, and typically it is very powerful for investigating deformation of G_K-representations. The Fontaine-Laffaille theory is the most primitive form of integral p-adic Hodge theory established in 1980s. Since then integral p-adic Hodge theory has been highly developed in various ways, while the Fontaine-Laffaille theory is still useful for arithmetic study of Galois representations.
In this series of lectures, I will talk about what the Fontaine-Laffaille theory is like, present a sketch of proofs of its main theorems and explain its classical application to ramification bound for torsion etale cohomology groups due to Abrashkin and Fontaine.
June 14 (Thu), 2018, 16:30~17:30,
Title: On the unit groups of some orders and its application, 아산이학관 433
Speaker: Jun Ho Lee (Mokpo University)
Abstract: In this talk, we are interested in the unit group of some order, particularly, a system of fundamental units. First, we introduce known results for our topic and, then share with you our results in this direction. Finally, as its application, we consider some diophantine equation related to some cubic field and show you how to solve it. It is a joint work with St\'{e}phane R. Louboutin.
June 5 (Tue), 2018, 15:30~16:30, 아산이학관 526
Title: Elliptic curves and Selmer companion curves
Speaker: Myungjun Yu (University of Michigan)
Abstract: Mazur and Rubin introduced the so-called Selmer-companion curves in 2015. Let $E$ be an elliptic curve over a number field $K$. Suppose there is a function that sends a quadratic character of $K$ to the $p$-Selmer rank of $E$ twisted by that character. How much information of $E$ can be read off from the function?
In this talk, we give a sketch of proof of a conjecture on $p$-Selmer near-companion posed by Mazur and Rubin when $p=2$.
May 4 (Fri), 2018, 14:00~15:00, 아산이학관 526
Title: Hook length property of d-complete posets via q-integrals
Speaker: Meesue Yoo (Sungkyunkwan University)
Abstract: In this work, we prove the hook length property of the d-complete posets using the q-integral technique developed by Kim and Stanton. For a non-negative integer n, the generating function for the number of partitions of n with no more than k parts, pk(n), is given by
Considering a partition of n with no more than k parts as an order-reversing map from a k-element chain to the set of non-negative integers such that the sum of images equals to n, Stanley extended this concept of partition and defined P-partitions of n. Then Stanley proved that the P-partition generating function for shapes has the hook length property. Proctor and Peterson figured out that the d-complete posets satisfy the hook length property, and Proctor showed that any connected d-complete poset P can be uniquely decomposed into a slant sum of one element posets and irreducible components. Furthermore, he classified 15 disjoint classes of irreducible components and showed that these 15 disjoint classes exhaust the set of all irreducible components. We show that the P-partition generating function for each irreducible d-complete poset can be written as a q-integral and prove the hook length property of them by computing the q-integrals explicitly. This is a joint work with Jang Soo Kim.
May 4 (Fri), 2018, 15:30~16:30, 아산이학관 526
Title: A sum of squares not divisible by a prime
Speaker: Kyoungmin Kim (Sungkyunkwan University)
Abstract: Let p be a prime. We define S(p) the smallest number k such that every positive integer is a sum of at most k squares of integers that are not divisible by p. In this talk, we prove that S(2)=10, S(3)=6, S(5)=5, and S(p)=4 for any prime p greater than 5. In particular, it is proved that every positive integer is a sum of at most four squares not divisible by 5, except the unique positive integer 79. This is a joint work with Byeong-Kweon Oh.
April 20 (Fri), 2018, 14:00~15:00
Title: Decomposition of Primes in a Quadratic Field [학부생을 위한 강연]
Speaker: Dohoon Choi (Korea University)
Abstract: 유리수와 수체 (유리수의 유한 확장체) 상의 원소들은 곱의 관점에서 소수에 의해 구성된다. 유리수상의 소수들은 수체상에서 다른 소수들에 의해 더 분해되기도 한다 이러한 현상의 연구는 유체론의 근간이 되었으면 더 확장되어 오늘날 랑글랜즈 프로그램에 이르게 되었다. 이 발표에서는 이 차체의 경우 이 질문에 관한 중요한 결과인 이차 상호 법칙에 관해 얘기한다.
April 19 (Thu), 2018, 13:00~14:00, 14:30~15:30, 아산이학관 525
Title: An IMO problem, Coxeter groups and fusion rings
Speaker: Chul-hee Lee (KIAS)
Abstract: Problem 3 of the International Mathematical Olympiad in 1986 is as follows :
To each vertex of a regular pentagon an integer is assigned in such a way that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers x,y,z respectively and y<0 then the following operation is allowed: the numbers x,y,z are replaced by x+y,-y,z+y respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.
I will explain this algorithm using the Coxeter groups and how to use it for computations with the affine fusion rings that arise from conformal field theory.
April 6 (Fri), 2018, 16:30~17:30, 아산이학관 526
Title: Local-Global Compatibility
Speaker: Chol Park (KIAS)
Abstract: Let K be a finite extension of Q_p. It is believed that one can attach a smooth F ̅_p-"representation" of 〖"GL" 〗_n (K) (or a packet of such representations) to a continuous Galois representation Gal(Q ̅_p/K)→ 〖"GL" 〗_n ("F" ̅_p) in a natural way, that is called mod p Langlands program for 〖"GL" 〗_n (K) which is known only for 〖"GL" 〗_2 (Q_p). In this talk, we will introduce mod p Langlands program for 〖"GL" 〗_n (K), and their expected compatibility with the classical local Langlands correspondence for 〖"GL" 〗_n as well as the global Langlands program.
March 30 (Fri), 2018, 16:30~17:30, 아산이학관 526
Title: Characterization of Cycle Obstruction Sets for Improper Coloring Planar Graphs
Speaker: Ilkyoo Choi (Hankuk University of Foreign Studies)
Abstract: For nonnegative integers $k, d_1, \ldots, d_k$, a graph is $(d_1, \ldots, d_k)$-colorable if its vertex set can be partitioned into $k$ parts so that the $i$th part induces a graph with maximum degree at most $d_i$ for all $i\in\{1, \ldots, k\}$. A class $\mathcal C$ of graphs is {\it balanced $k$-partitionable} and {\it unbalanced $k$-partitionable} if there exists a nonnegative integer $D$ such that all graphs in $\mathcal C$ are $(D, \ldots, D)$-colorable and $(0, \ldots, 0, D)$-colorable, respectively, where the tuple has length $k$.
A set $X$ of graphs is an {\it obstruction set} of a {graph class $\mathcal C$} if containing none of the graphs in $X$ as a subgraph guarantees membership of $\mathcal C$. In other words, a graph $G$ is a member of $\mathcal C$ if $G$ does not contain any graph in $X$ as a subgraph. This paper characterizes all cycle obstruction sets of planar graphs to be balanced $k$-partitionable and unbalanced $k$-partitionable for all $k$; namely, we identify all inclusion-wise minimal cycle obstruction sets for all $k$. This is joint work with Chun-Hung Liu and Sang-il Oum.
March 30 (Fri), 2018, 13:00~14:00, 14:30~15:30 아산이학관 526
Title: Simple zeros of L-functions
Speaker: Jaehyun Cho (UNIST)
Abstract: We show that every cuspidal automorphic L-function of $GL_2$ over $\mathbb{Q}$ has infinitely many simple zeros. This is a joint work with A. R. Booker and Myoungil Kim.
January 22 (Mon), 2018, 03:20 ~ 04:20 아산이학관 525
Title: Non-negative polynomials and Sums of squares
Speaker: 문현석 박사 (KAIST 수리과학과)
Abstract: It is known that there are differences between cone of non-negative polynomials and cone of sums of squares. We review the known results about their algebraic boundaries and their degrees. And we discuss about the further obstacles and applications
January 22 (Mon), 2018, 01:30 ~ 03:00 아산이학관 525
Title: A tale on two cones of Non-negative polynomials and of Sums of squares
Speaker: 한강진 교수 (DGIST 기초학부)
Abstract: First, we review basic notions and motivations for NP and SOS and investigate its properties and application to optimization problems.