# 세미나 및 콜로퀴엄

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Let f be a nonzero polynomial. A pair of matrices (A, B) of polynomials is called a matrix factorization of f if both AB and BA are f times identity matrices. Eisenbud introduced this notion to study free resolutions over hypersurface rings and complete intersection rings. Among several applications of matrix factorization, we focus on the correspondence between aCM/Ulrich sheaves in this talk. Then we provide a matrix factorization of a general cubic hypersurface of dimension at most 7 using Shamash's construction. Note that this is an alternative proof of the existence of rank 9 Ulrich bundle on a general cubic sevenfold, which is first known by Iliev and Manivel. If time permits, we discuss on classification of cubic forms whose Hessian matrices induce matrix factorization of themselves. This is a joint work with F.-O. Schreyer.

Let
M
=
(
M
i
:
i
∈
K
)
be a finite or infinite family consisting of finitary and cofinitary matroids on a common ground set
E
.
We prove the following Cantor-Bernstein-type result: if
E
can be covered by sets
(
B
i
:
i
∈
K
)
which are bases in the corresponding matroids and there are also pairwise disjoint bases of the matroids
M
i
then
E
can be partitioned into bases with respect to
M
.

In this talk, I propose a computer model that generalizes most if not all previous computer models including Turing machine, Boolean circuit, continuous time system, quantum computer, Blum-Shub-Smale model, and deep learning. I will describe a computational complexity theory for this new model, and construct infinite dimensional algebraic varieties that parametrize deterministic/non-deterministic polynomial time languages.

We will see several applications of model theory to other mathematics, mainly to number theory, arithmetic geometry, and algebraic geometry.
Model theory is a branch of mathematical logic. There are mainly four branches: computability theory, model theory, proof theory, and set theory. W. Hodges described model theory as "algebraic geometry minus fields" in his book "A Shorter Model Theory". In model theory, we study definable sets in mathematical structures, for example, algebraic sets in the field of complex numbers or semialgebraic sets in the field of real numbers, and we classify their first order theories of the structures.
We recall several basic concepts of model theory: elementary equivalence, quantifier elimination, and elimination of imaginaries. And we will see how such concepts can be applied to number theory, arithmetic geometry, and algebraic geometry.

To an abelian category A satisfying certain finiteness conditions, one can associate an algebra H_A (the Hall algebra of A) which encodes the structures of the space of extensions between objects in A. For a non-additive setting, Dyckerhoff and Kapranov introduced the notion of proto-exact categories, as a non-additive generalization of an exact category, which is shown to suffice for the construction of an associative Hall algebra. In this talk, I will discuss the category of matroids in this perspective.

We introduce a notion of PAC structures, which generalizes perfect PAC fields, and we study their elementary theories via their Galois groups.
It is well known that elementary equivalences of PAC fields are controlled by their Galois groups by Embedding Lemma of M. Jarden and U. Kiehne. Also, G. Cherlin, L. van den Dries, and A. Macintyre introduced a notion of complete systems of profinite groups, axiomatized by first order logic, and they showed that the category of complete systems whose morphisms are embedding is equivalent to the category of profinite groups whose morphisms are epimorphisms. And they showed that the complete systems of Galois groups of PAC fields determine elementary equivalences and elementary extensions between PAC fields. Recently, Z. Chatzidakis and N. Ramsey showed that complete systems of Galois groups are very crucial role in the classification program of first order theories of PAC fields: If the complete system of Galois group of a given PAC field is SNOPn, then so is the PAC field.
We generalize these previous results for PAC fields into PAC structures: We generalize Embedding Lemma for PAC fields into PAC structures. We also introduce a notion of sorted complete systems, which generalizes the complete systems and we show that first order theories of PAC structures are determined by their sorted complete systems. And we generalize NSOPn criteria for PAC fields into PAC structures in terms of sorted complete systems.
This talk is based on joint works of J. Dobrowolski, D. M. Hoffmann, and of D. M. Hoffmann.

Given any integers $s,t\geq 2$, we show there exists some $c=c(s,t)>0$ such that any $K_{s,t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $cd^{\frac{1}{2}\frac{s}{s-1}}$ vertices. In particular, when $s=2$ this resolves in a strong sense the conjecture of Mader in 1999 that every $C_4$-free graph has a subdivision of a clique with order linear in the average degree of the original graph. In general, the widely conjectured asymptotic behaviour of the extremal density of $K_{s,t}$-free graphs suggests our result is tight up to the constant $c(s,t)$. This is joint work with Richard Montgomery.

In this presentation, we consider the random-cluster model which is a generalization of the standard edge percolation model. For the random-cluster model on lattice, we prove that the Glauber dynamics exhibits a phenomenon known as the cut-off, especially for the very subcritical regime for all dimensions. This is a joint work with Shirshendu Ganguly.

The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion, defined as first-order interpretations of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth decompositions, replacing treedepth by its dense analogue called shrubdepth.

We use circles on a sphere to illustrate important concepts in symplectic topology. We explain the difficulties encountered in higher dimensions and to what extent it can be overcome. Subsequently, we introduce the Fukaya category and connect the story to Khovanov homology.

The subject of p-adic differential equations was pioneered by Dwork in 1950’s, who investigated p-adic properties of solutions of a certain hypergeometric differential equation. This study of Dwork’s study led to extremely fascinating applications in number theory; especially, on elliptic curves and modular forms. The main goal of this colloquium talk is to explain the motivating example of the p-adic hypergeometric differential equation studied by Dwork and its link to the Legendre family of elliptic curves. If time permits, I’d like to discuss some generalization of Dwork’s study to families of abelian varieties and its potential applications.

The Swan conductor is a local invariant measuring wild ramification (for Artin representations or suitable variants thereof). It is one of the central objects to study in the ramification theory.
In this expository talk, let me start with a review of ramification subgroups and the theory of Swan conductors and Swan characters. At the end of the talk, I would like to pose some questions on ‘p-adic analogue of Swan characters’ in the theory of p-adic differential equations, motivated by equivariant BSD conjecture over global function fields.

Seminar Talk

Seminar Talk