In this talk, we present the notion of Stark units in function field arithmetic. This notion was first introduced for investigations on the construction of certain units from the L-function of Drinfeld modules, i.e. log-algebraicity identities. More generally, to a Drinfeld module of higher dimension defined over a function field, we can associate its module of Stark units. We give basic properties of this object and state its connection with Taelman's class formula. Then we will describe the module of Stark units attached to the Carlitz module and its power tensors defined over certain abelian extensions of function fields.

Second day abstract Metamaterials are manmade composite media structured on a scale much smaller than a wavelength. The Minnaert resonance phenomenon makes air bubbles good candidates for the basic building blocks for acoustic metamaterials. Firstly we show the existence of a subwavelength phononic bandgap in bubble phononic crystals, which is proved by an original formula for the quasi-periodic Minnaert resonance frequencies of an arbitrarily shaped bubble. This phenomena can be explained by the periodic inference of cell resonance which is due to the high contrast in both the density and bulk modulus between the bubbles and the surrounding medium. Secondly we show that the bubbly fluid functions like an acoustic metamaterial. Near the Minnaert resonant frequency, an effective medium theory can be derived in the dilute regime. Furthermore, above the Minnaert resonant frequency, the real part of the effective bulk modulus is negative, and consequently the bubbly fluid behaves as a diffusive medium for the acoustic waves. Meanwhile, below the Minnaert resonant frequency, with an appropriate bubble volume fraction, a high contrast effective medium can be obtained, making the subwavelength focusing or superfocusing of waves achievable.

In this talk, we present three related topics on the collective modeling of many-body systems. In the first story, we discuss universal triality relation between bacteria aggregation, Cucker-Smale flocking and Kuramoto synchronization. These three seemingly different phenomena can be integrated into a common nonlinear consensus framework. In our second story, we present a second-order Cucker-Smale modeling on Riemannian manifolds such as the unit circle, the unit sphere in R^3 and Poincare upper half plane model for hyperbolic geometry. Finally, in our third story, we explain how aforementioned collective modeling can be used in the first-order consensus-based optimization algorithm.

First day abstract In acoustics, it is known that air bubbles are subwavelength resonators. Due to the high contrast between the air density inside and outside an air bubble in a fluid, a quasi-static acoustic resonance known as the Minnaert resonance occurs and the bubble behaves as a strong monopole scatterer of sound. Through the application of layer potential techniques and Gohberg–Sigal theory we derive an original formula for the Minnaert resonance frequencies of arbitrarily shaped bubbles. We also provide a mathematical justification for the monopole approximation of scattering of acoustic waves by bubbles at their Minnaert resonant frequency. An acoustic meta-screen is a thin sheet with patterned subwavelength structures, which nevertheless has a macroscopic effect on acoustic wave propagation. When periodic subwavelength bubbles mounted on a reflective surface (with Dirichlet boundary condition) are considered, it is shown that the structure behaves as an equivalent surface with Neumann boundary condition at the Minnaert resonant frequency which corresponds to a wavelength much greater than the size of the bubbles. An analytical formula for this resonance is derived and some numerical simulations confirm its accuracy.

We would give a characterization of those motivic spectra for which the associated slice spectral sequence converges strongly. The characterization is given in terms of the birational covers introduced by the author in order to study the Bloch-Beilinson filtration.

For a graph H , its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in L p , p ≥ e ( H ) , denoted by t H ( W ) . One may then define corresponding functionals ∥ W ∥ H := | t H ( W ) | 1 / e ( H ) and ∥ W ∥ r ( H ) := t H ( | W | ) 1 / e ( H ) and say that H is (semi-)norming if ∥ . ∥ H is a (semi-)norm and that H is weakly norming if ∥ . ∥ r ( H ) is a norm. We obtain some results that contribute to the theory of (weakly) norming graphs. Firstly, we show that ‘twisted’ blow-ups of cycles, which include K 5 , 5 ∖ C 10 and C 6 □ K 2 , are not weakly norming. This answers two questions of Hatami, who asked whether the two graphs are weakly norming. Secondly, we prove that ∥ . ∥ r ( H ) is not uniformly convex nor uniformly smooth, provided that H is weakly norming. This answers another question of Hatami, who estimated the modulus of convexity and smoothness of ∥ . ∥ H . We also prove that every graph H without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of H when studying graph norms. Based on joint work with Frederik Garbe, Jan Hladký, and Bjarne Schülke.

According to a 2018 preprint by Nobuaki Yagita, the conjecture on a relationship between K- and Chow theories for a generically twisted flag variety of a split semisimple algebraic group G, due to the speaker, fails for G the spinor group Spin(17). Yagita's tools include a Brown-Peterson version of algebraic cobordism, ordinary and connective Morava K-theories, as well as Grothendieck motives related to various cohomology theories over fields of characteristic 0. The talk presents a simpler proof using only the K- and Chow theories themselves and, in particular, extending the (slightlymodified) example to arbitrary characteristic.

The Boussinesq abcd system was originally derived by Bona, Chen and Saut [J. Nonlinear. Sci. (2002)] as a rst order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime. Among many particular regimes, the Hamiltonian generic regime is characterized by the setting b = d > 0 and a; c < 0. It is known that the system in this regime is globally well-posed for small data in the energy space H1 H1 by Bona, Chen and Saut [Nonlinearity (2004)]. In this talk, we are going to discuss about the decay of small solutions to abcd system in three directions: First, for a weakly dispersive abcd systems (characterized only in terms of parameters a; b and c), all small solutions must decay to zero, locally strongly in the energy space, in proper subset of the light cone jxj jtj. Second, for every ray x = vt, jvj < 1 inside the light cone, small solutions to suciently dispersive system (smallness and dispersion are characterized by v) decay to zero, in proper subset along the ray. Last, small solutions decay to zero in exterior regions jxj jtj under suitable conditions of parameters (a; b; c). All results rule out, among other things, the existence of zero or nonzero speed or super-luminical small solitary waves in each regime where decay is present.

This is joint work with Claudio Munoz.

 

 

We introduce the algebraic connective K-theory and discuss its relations with some other oriented cohomology theories. Then we present recent results on connective K-theory of varieties of parabolic subgroups in semisimple algebraic groups.

For a graph H and an integer k ≥ 1 , the k -color Ramsey number R k ( H ) is the least integer N such that every k -coloring of the edges of the complete graph K N contains a monochromatic copy of H . Let C m denote the cycle on m ≥ 4 vertices. For odd cycles, Bondy and Erd\H{o}s in 1973 conjectured that for all k ≥ 1 and n ≥ 2 , R k ( C 2 n + 1 ) = n ⋅ 2 k + 1 . Recently, this conjecture has been verified to be true for all fixed k and all n sufficiently large by Jenssen and Skokan; and false for all fixed n and all k sufficiently large by Day and Johnson. Even cycles behave rather differently in this context. Little is known about the behavior of R k ( C 2 n ) in general. In this talk we will present our recent results on Ramsey numbers of cycles under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles. We prove that the aforementioned conjecture holds for all k and all n under Gallai colorings. We also completely determine the Ramsey number of even cycles under Gallai colorings. Joint work with Dylan Bruce, Christian Bosse, Yaojun Chen and Fangfang Zhang.

In this talk, we study the asymptotic stability of a non-autonomous linear system of ordinary differential equations

Let $G$ be a simply-connected reductive algebraic group over $\mathbb{C}$. For a dominant integral weight $\lambda$, and a reduced decomposition $\mathbf i$ of the longest element in the Weyl group of $G$, the string polytope $\Delta_{\mathbf i}(\lambda)$ is a combinatorial object which encodes weights of $G$-irreducible. It has been observed that the combinatorics of string polytopes depend on a choice of $\mathbf{i}$. Note that string polytopes are non-simple polytopes. Hence they define singular toric varieties. In this talk, we introduce string polytopes when $G = \textrm{SL}_{n+1}(\mathbb{C})$, and we present small resolutions of toric varieties $X_{\Delta_{\mathbf i}(\lambda)}$ for some special $\mathbf i$ using Bott manifolds. This talk is based on joint work with Yunhyung Cho, Yoosik Kim, and Kyeong-Dong Park.

A graph or graph property is ℓ ℓ -reconstructible if it is determined by the multiset of all subgraphs obtained by deleting ℓ ℓ vertices. Apart from the famous Graph Reconstruction Conjecture, Kelly conjectured in 1957 that for each ℓ∈N ℓ∈N , there is an integer n=n(ℓ) n=n(ℓ) such that every graph with at least n n vertices is ℓ ℓ -reconstructible. We show that for each n≥7 n≥7 and every n n -vertex graph G G , the degree list and connectedness of G G are 3 3 -reconstructible, and the threshold n≥7 n≥7 is sharp for both properties.‌ We also show that all 3 3 -regular graphs are 2 2 -reconstructible.

(This is a reading seminar for graduate students.) Recall that there is a spectral sequence strongly converging to the connective $K$-groups whose second page is given by the Zariski cohomology of connective $K$-theory sheaf. In the proof of this result by Quillen, the localization theorem is the most important ingredient. We prove an analogous statement for non-connective $K$-theory with both Zariski and Nisnevich cohomology for noetherian schemes of finite Krull dimension. This theorem is usually phrased as "non-connective algebraic $K$-theory satisfies Zariski and Nisnevich descent". It is known that non-connective algebraic $K$-theory does not satisfy étale descent.

Regularization methods for modeling and prediction are popular in statistics and machine learning. They may be viewed as the procedures that modify the maximum likelihood principle or empirical risk minimization. In particular, methods of regularization defined in reproducing kernel Hilbert spaces (known as kernel methods) provide versatile tools for statistical learning. Primary examples include smoothing splines and support vector machines. I will describe kernel methods focusing on these two examples and discuss some relevant statistical and computational issues. Further I will provide a general description of kernel methods covering mathematical elements and results underlying the methods. Part I: Smoothing Splines (September 30, Monday) Part II: Support Vector Machines (October 2, Wednesday and October 4, Friday) Part III: Kernel Methods (October 8, Tuesday)

A well-known Ramsey-type puzzle for children is to prove that among any 6 people either there are 3 who know each other or there are 3 who do not know each other. More generally, a graph G arrows a graph H if for any coloring of the edges of G with two colors, there is a monochromatic copy of H. In these terms, the above puzzle claims that the complete 6-vertex graph K_6 arrows the complete 3-vertex graph K_3. We consider sufficient conditions on the dense host graphs G to arrow long paths and even cycles. In particular, for large n we describe all multipartite graphs that arrow paths and cycles with 2n edges. This implies a conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi from 2007 for such n. Also for large n we find which minimum degree in a (3n-1)-vertex graph G guarantees that G arrows the 2n-vertex path. This yields a more recent conjecture of Schelp. This is joint work with Jozsef Balogh, Mikhail Lavrov and Xujun Liu. (*Joint Colloquium between KAIST Mathematical Sciences and IBS Discrete Mathematics Group)

영상 복원(Image restoration, IR)은 low-level vision에서 매우 중요하게 다루는 근본적인 문제 중 하나로서 denoising, deblur, super-resolution 등의 다양한 영상 처리 문제를 포괄한다. 이 발표에서는 영상 복원 분야 중에서도 super-resolution 문제에 대해 집중적으로 다루겠다. 전통적인 model-based optimization 방식과 deep learning을 적용하여 문제를 푸는 방식에 대해, 각각의 장단점과 최신 연구 발전 흐름을 소개한다. 마지막으로는 이 둘을 하나로 잇는 통일된 관점을 제시하고 관련 연구들 살펴본 후, super-resolution 분야에서 아직 남아있는 문제점들을 정리하겠다

We consider an optimal consumption/investment problem to maximize expected utility from consumption. In this market model, the investor is allowed to choose a portfolio that consists of one bond, one liquid risky asset (no transaction costs), and one illiquid risky asset (proportional transaction costs). I fully characterize the optimal consumption and trading strategies in terms of the solution of the free boundary ordinary differential equation (ODE) with an integral constraint. I find an explicit characterization of model parameters for the well-posedness of the problem, and show that the problem is well posed if and only if there exists a shadow price process. Finally, I describe how the investor’s optimal strategy is affected by the additional opportunity of trading the liquid risky asset, compared to the simpler model with one bond and one illiquid risky asset.

Using a simple binomial model, I will present topics in Mathematical Finance (derivative pricing, optimal investment, equilibrium asset pricing). Then we will consider market models with Brownian motion on these topics. Lastly, I will present some result of mine in equilibrium asset pricing in incomplete market setup.

Regularization methods for modeling and prediction are popular in statistics and machine learning. They may be viewed as the procedures that modify the maximum likelihood principle or empirical risk minimization. In particular, methods of regularization defined in reproducing kernel Hilbert spaces (known as kernel methods) provide versatile tools for statistical learning. Primary examples include smoothing splines and support vector machines. I will describe kernel methods focusing on these two examples and discuss some relevant statistical and computational issues. Further I will provide a general description of kernel methods covering mathematical elements and results underlying the methods. Part I: Smoothing Splines (September 30, Monday) Part II: Support Vector Machines (October 2, Wednesday and October 4, Friday) Part III: Kernel Methods (October 8, Tuesday)

This is joint work with Sławomir Kołodziej. We show that the complex m-Hessian
operator of a Holder continuous m-subharmonic function is well dominated by the corresponding capacity. As consequence we obtain the Holder continuous subsolution theorem for the complex m-Hessian equation.

We show finite-time blow up for strong solutions to the 3D Euler equations in two types of corner domains. The first result is for axi-symmetric domains with corners and the data is allowed to be $C^\infty$-smooth if the corner angle is small. In the second case, we utilize the fundamental domain for the octahedral symmetry group. Inside the domain, the data is smooth and can be extended to entire $\mathbb{R}^3$ by a sequence of reflections. In both cases, the solutions have Lipschitz continuous velocity with compact support and have finite energy in particular. This talk is based on joint works with T. Elgindi.

Regularization methods for modeling and prediction are popular in statistics and machine learning. They may be viewed as the procedures that modify the maximum likelihood principle or empirical risk minimization. In particular, methods of regularization defined in reproducing kernel Hilbert spaces (known as kernel methods) provide versatile tools for statistical learning. Primary examples include smoothing splines and support vector machines. I will describe kernel methods focusing on these two examples and discuss some relevant statistical and computational issues. Further I will provide a general description of kernel methods covering mathematical elements and results underlying the methods. Part I: Smoothing Splines (September 30, Monday) Part II: Support Vector Machines (October 2, Wednesday and October 4, Friday) Part III: Kernel Methods (October 8, Tuesday)

A diffusion equation is one of most famous partial differential equations. Lots of generalized diffusion equations have appeared on the basis of scientific meaning. Equations describing degenerate or unbounded diffusion including stochastic noises are some of them. In this talk, we are going to discuss change of regularity of solutions depending on degeneracy and unboundedness of diffusion and stochastic noise.

We introduce the incompressible Euler equations, which describe the dynamics of volume-preserving inviscid fluids, and describe a few open problems in relation to turbulence. Then we discuss a priori estimates, which give upper bounds on the solution in function spaces. In particular, these estimates guarantee that if the initial fluid velocity is smooth, then there is a unique smooth solution for some time interval. After that, we shall review some recent results towards the opposite direction: attempts in showing lower bounds on the solution instead, with the goal of establishing finite-time singularity formation.

Given a graph G , there are several natural hypergraph families one can define. Among the least restrictive is the family B G of so-called Berge copies of the graph G . In this talk, we discuss Turán problems for families B G in r -uniform hypergraphs for various graphs G . In particular, we are interested in general results in two settings: the case when r is large and G is any graph where this Turán number is shown to be eventually subquadratic, as well as the case when G is a tree where several exact results can be obtained. The results in the first part are joint with Grósz and Methuku, and the second part with Győri, Salia and Zamora.

Regularization methods for modeling and prediction are popular in statistics and machine learning. They may be viewed as the procedures that modify the maximum likelihood principle or empirical risk minimization. In particular, methods of regularization defined in reproducing kernel Hilbert spaces (known as kernel methods) provide versatile tools for statistical learning. Primary examples include smoothing splines and support vector machines. I will describe kernel methods focusing on these two examples and discuss some relevant statistical and computational issues. Further I will provide a general description of kernel methods covering mathematical elements and results underlying the methods. Part I: Smoothing Splines (September 30, Monday) Part II: Support Vector Machines (October 2, Wednesday and October 4, Friday) Part III: Kernel Methods (October 8, Tuesday)

Synchronization phenomenon is ubiquitous in an ensemble of coupled oscillators, e.g., hand clapping in opera and musical halls, flashing of fireflies and heart beating of pacemaker cells, etc. In the last forty years, the Kuramoto model served as a prototype model for describing such synchronization phenomena. In particular, we will consider the Kuramoto model under the stochastic noise. As the number of oscillators tends to infinity, we can derive the kinetic

equation for the Kuramoto model by using the standard BBGKY hierarchy. In this talk, we will consider the asymptotic behavior for the kinetic Kuramoto models under the stochastic noise, and talk about their large time behaviors.

Support vector machine (SVM) is a very popular technique for classification. A key property of SVM is that its discriminant function depends only on a subset of data points called support vectors. This comes from the representation of the discriminant function as a linear combination of kernel functions associated with individual cases. Despite the direct relation between each case and the corresponding coefficient in the representation, the influence of cases and outliers on the classification rule has not been examined formally. Borrowing ideas from regression diagnostics, we define case influence measures for SVM and study how the classification rule changes as each case is perturbed. To measure case sensitivity, we introduce a weight parameter for each case and reduce the weight from one to zero to link the full data solution to the leave-one-out solution. We develop an efficient algorithm to generate case-weight adjusted solution paths for SVM. The solution paths and the resulting case influence graphs facilitate evaluation of the influence measures and allow us to examine the relation between the coefficients of individual cases in SVM and their influences comprehensively. We present numerical results to illustrate the benefit of this approach.

 It is a common theme in algebraic geometry that many constructions have only been done for schemes and morphisms of finite type. However, in arithmetic geometry one would also like to work with infinite objects, as for example infinite level modular curve. In my talk I motivate and define schemes and morphism satisfying a weaker finiteness property, which contain many examples from arithmetic geometry. The aim of this talk is to extend the definition of cohomology with compact support to them; in fact, we even obtain Grothendieck's six operations for this class of morphism.

Stochastic heat equations usually refer to heat equations perturbed by noise. Depending on noise, stochastic heat equations have similar properties as heat equations such as strict positivity or properties which cannot be seen from heat equations such as intermittency. We consider various properties of stochastic heat equations in this talk. (This talk will be a survey talk and should be accessible to all graduate students.)

Let F and H be graphs. The subgraph counting function ex(n,H,F) is defined as the maximum possible number of subgraphs H in an n-vertex F-free graph. This function is a direct generalization of the Turán function as ex(n,F)=ex(n,K2,F). The systematic study of ex(n,H,F) was initiated by Alon and Shikhelman in 2016 who generalized several classical results in extremal graph theory to the subgraph counting setting. Prior to their paper, a number of individual cases were investigated; a well-known example is the question to determine the maximum number of pentagons in a triangle-free graph. In this talk we will survey results on the function ex(n,H,F) including a number of recent papers. We will also discuss this function’s connection to hypergraph Turán problems.

Given a graph $G$, we define ${\text{ex}}_c\left(G\right)$ to be the minimum value of $t$ for which there exists a constant $N(t,G)$ such that every $t$-connected graph with at least $N(t,G)$ vertices contains $G$ as a minor. The value of ${\text{ex}}_c\left(G\right)$ is known to be tied to the vertex cover number $\tau \left(G\right)$, and in fact $\tau \left(G\right)\le {\text{ex}}_c\left(G\right)\le \frac{31}{2}\left(\tau \right(G)+1)$. We give the precise value of ${\text{ex}}_c\left(G\right)$ when $G$ is a forest. In particular we find that ${\text{ex}}_c\left(G\right)\le \tau \left(G\right)+2$ in this setting, which is tight for infinitely many forests.

In this talk we consider a reaction-diffusion model for the spreading of farmers in Europe, which was occupied by hunter-gatherers; this process is known as the Neolithic agricultural revolution. The spreading of farmers is modelled by a nonlinear porous medium type diffusion equation which coincides with the singular limit of another model for the dispersal of farmers as a small parameter tends to zero. From the ecological viewpoint, the nonlinear diffusion takes into account the population density pressure of the farmers on their dispersal. The interaction between farmers and hunter-gatherers is of the Lotka-Volterra prey-predator type. We show the existence and uniqueness of a global in time solution and study its asymptotic behaviour as time tends to infinity.

After the discovery of algebraic $K$-theory of schemes by Quillen, it was anticipated that algebraic cycles would play a fundamental role in computing the $K$-groups of schemes. These groups are usually very hard to compute even as they contain a lot of information about the underlying scheme. For smooth schemes, the problem of describing $K$-theory by algebraic cycles was satisfactorily settled by works of several mathematicians, spanned over many years.

However, solving this problem under the presence of singularities is still a very challenging task.

The theory of additive Chow groups and the Chow groups with modulus were invented to address this problem.

Even if it is not yet clear if these Chow groups will finally solve the problem, I shall present some positive results in this direction in my talk. In particular, I shall show that the additive Chow groups do completely describe the $K$-theory of infinitesimal neighborhoods of the origin in the affine line. This was the initial motivation of the discovery of additive Chow groups by Bloch and Esnault. This talk is based on a joint work with Rahul Gupta.

 

After the discovery of algebraic $K$-theory of schemes by Quillen, it was anticipated that algebraic cycles would play a fundamental role in computing the $K$-groups of schemes. These groups are usually very hard to compute even as they contain a lot of information about the underlying scheme. For smooth schemes, the problem of describing $K$-theory by algebraic cycles was satisfactorily settled by works of several mathematicians, spanned over many years.

However, solving this problem under the presence of singularities is still a very challenging task.

 

The theory of additive Chow groups and the Chow groups with modulus were invented to address this problem.

 

Even if it is not yet clear if these Chow groups will finally solve the problem, I shall present some positive

 

results in this direction in my talk. In particular, I shall show that the additive Chow groups do completely describe the $K$-theory of infinitesimal neighborhoods of the origin in the affine line. This was the initial motivation of the discovery of additive Chow groups by Bloch and Esnault. This talk is based on a joint work with Rahul Gupta.