# 세미나 및 콜로퀴엄

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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.

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 consider generalized MHD equations with fractional diffusions. Existence of local solutionshas been well-known for initial datain Sobolev spaces $H^s$, if $s$ is sufficiently large. The goal of the talk is to present local existence results with initial datain $H^{s_1} \times H^{s_2}$ of lower orders $s_1$ and/or $s_2$, which extend recent existence results by Fefferman, McCormick, Robinson, and Rodrigo(2014, 2017). This is a joint work with Yong Zhou at the Sun Yat-Sen University (Zhuhai), China.

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.

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
.

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.

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

In this talk we consider the ring of equivariant cohomology of moment-angle complex. We discuss how to compute it and talk about properties of this ring, in particular, we take a look at conditions under which the equivariant cohomology ring is a free module over equivariant cohomology ring of a point.

Torus actions on symplectic toric manifolds and contact toric manifolds share certain similar properties, so-called “local standards”. This allows us to recover the original manifold from the orbit space together with torus action data. We discuss certain wide class of manifolds which includes both of above manifolds and study their equivariant classification.

The debate about the correct diffusion model is related to the way to handle the randomness. In this talk, we will see an example which shows that the Stratonovitch interal is the correct way to handle it.
The classical kinetic equation is related to the Ito integral. We will construct a new kinetic equation of Stratonovitch type.

In this talk, we will study a triply periodic polyhedral surface whose vertices correspond to the Weierstrass points on the underlying Riemann surface. The symmetries of the surface allow us to construct hyperbolic structures and various translation structures that are compatible with its conformal type. With this explicit data, one can find its algebraic description, automorphism group, Veech group, etc.

We discuss a triangle of viewpoints for circle diffeomorphism groups: analysis, dynamics and group theory. In particular, we illustrate how the regularities (from the analytic side) of diffeomorphisms govern the dynamics and the group theoretical properties of diffeomorphisms. This line of study can be traced back to the works of Hölder, Denjoy, Tsuboi, Thurston and many more.

Haviv (European Journal of Combinatorics, 2019) has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over R. We show that this holds actually for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over R – an important graph invariant from coding theory – and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed.
This is joint work with Meysam Alishahi.

The Bayesian approach to inverse problems, in which the posterior probabil-
ity distribution on an unknown eld is sampled for the purposes of computing
posterior expectations of quantities of interest, is starting to become computa-
tionally feasible for partial dierential equation (PDE) inverse problems. Bal-
ancing the sources of error arising from nite-dimensional approximation of the
unknown eld, the PDE forward solution map and the sampling of the prob-
ability space under the posterior distribution are essential for the design of
ecient computational Bayesian methods for PDE inverse problems. We study
Bayesian inversion for a model elliptic PDE with an unknown diusion coef-
cient. We consider both the case where the PDE is uniformly elliptic with
respect to all the realizations, and the case where uniform ellipticity does not
hold, i.e. the coecient can get arbitrarily close to 0 and arbitrarily large as in
the log-normal model. We provide complexity analysis of Markov chain Monte
Carlo (MCMC) methods for numerical evaluation of expectations under the
Bayesian posterior distribution given data, in particular bounds on the overall
work required to achieve a prescribed error level. We rst bound the computa-
tional complexity of plain MCMC, based on combining MCMC sampling with
linear complexity multi-level solvers for elliptic PDEs. The work versus accu-
racy bounds show that the complexity of this approach can be quite prohibitive.
We then present a novel multi-level Markov chain Monte Carlo strategy which
utilizes sampling from a multi-level discretization of the posterior and the for-
ward PDE. The strategy achieves an essentially optimal complexity level that is
essentially equal to that for performing only one step on the plain MCMC. The
essentially optimal accuracy and complexity of the method are mathematically
rigorously proven. Numerical results conrm our analysis. This is a joint work
with Jia Hao Quek (NTU, Singapore), Christoph Schwab (ETH, Switzerland)
and Andrew Stuart (Warwick, England).

Let $F$ be a graph. We say that a hypergraph $\mathcal H$ is an induced Berge $F$ if there exists a bijective mapping $f$ from the edges of $F$ to the hyperedges of $\mathcal H$ such that for all $xy \in E(F)$, $f(xy) \cap V(F) = \{x,y\}$. In this talk, we show asymptotics for the maximum number of edges in $r$-uniform hypergraphs with no induced Berge $F$. In particular, this function is strongly related to the generalized Turán function $ex(n,K_r, F)$, i.e., the maximum number of cliques of size $r$ in $n$-vertex, $F$-free graphs. Joint work with Zoltan Füredi.

Encoder-decoder networks using convolutional neural network (CNN) architecture have been extensively used in deep learning approaches for inverse problems thanks to its excellent performance. However, it is still difficult to obtain coherent geometric view why such an architecture gives the desired performance. Inspired by recent theoretical understanding on generalizability, expressivity and optimization landscape of neural networks, as well as the theory of deep convolutional framelets, here we provide a unified theoretical framework that leads to a better understanding of geometry of encoder-decoder CNNs. Our unified framework shows that encoder-decoder CNN architecture is closely related to nonlinear frame basis representation using combinatorial convolution frames, whose expressivity increases exponentially with the network depth and channels. We also demonstrate the importance of skipped connection in terms of expressivity and optimization landscape. We provide extensive experimental results from various biomedical imaging reconstruction problems to verify the performance encoder-decoder CNNs.

It is a classic result that the maximum weight stable set problem is efficiently solvable for bipartite graphs. The recent bimodular algorithm of Artmann, Weismantel and Zenklusen shows that it is also efficiently solvable for graphs without two disjoint odd cycles. The complexity of the stable set problem for graphs without disjoint odd cycles is a long-standing open problem for all other values of . We prove that under the additional assumption that the input graph is embedded in a surface of bounded genus, there is a polynomial-time algorithm for each fixed . Moreover, we obtain polynomial-size extended formulations for the respective stable set polytopes.
To this end, we show that 2-sided odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed surface. This result may be of independent interest and extends a theorem of Kawarabayashi and Nakamoto asserting that odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed orientable surface.
Eventually, our findings allow us to reduce the original problem to the problem of finding a minimum-cost non-negative integer circulation of a certain homology class, which we prove to be efficiently solvable in our case.
This is joint work with Michele Conforti, Samuel Fiorini, Gwenaël Joret, and Stefan Weltge.

In this talk, I will discuss some recent developments on the
study of singular stochastic wave equations. I also describe some
similarities and differences between stochastic wave and heat equations,
indicating particular difficulty of the dispersive/hyperbolic problem.

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. We proved each of these identities under three different interpretations for the double series, and showed that they are intimately connected with the classical circle and divisor problems in number theory. Furthermore, we established many analogues and generalizations of them. This is joint work with Bruce C. Berndt and Alexandru Zaharescu.