# 세미나 및 콜로퀴엄

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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 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.

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.

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.

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.

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.

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.