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.

 

 

(This is a reading seminar for graduate students.)  Algebraic $K$-theory of schemes has a sequence of $K$-groups for a closed immersion which is exact except only one place, namely $K_0$ of the open immersion of the complement, as the proto-localization theorem shows. Following the idea of Hyman Bass, we define a non-connective spectrum of a scheme whose non-negative part coincides with the usual algebraic $K$-theory defined by perfect complexes. This non-connective $K$-theory spectrum in particular gives a long exact sequence for a closed immersion. Our aim is to construct it, which is done by descending induction and requires the projective bundle formula that is also important regardless of our situation.

We introduce the mechanical model designed  by W. Malkus and L. Howard for the Lorenz system. Some mechanical equation explaining this will be derived. Based on this mechanical equations, the Fourier modes and velocity of the system will be presented as a complete system following Lorenz's model. Some key factors, including Rayleigh number, will be introduced to compare Malkus's model and the fluid convection, key interest of Lorenz that gave birth to his model.

Abstract: Linkage is a classical topic in algebraic geometry and commutative algebra. Fix an affine space A. We say two subschemes X, Y of A are linked if their union is a complete intersection in A and X and Y do not have a common component. Two linked subschemes share several properties in common. Linkage has been studied by various people, Artin-Nagata, Peskine-Szprio, Huneke-Ulrich, to name a few.

 

In 2014, Niu showed that if Y is a generic link of a variety X, then LCT (A, X) <= LCT (A, Y), where LCT stands for log canonical threshold. In this talk, we show that if Y is a generic link of a determinantal variety X, then X and Y have the same log canonical threshold. This is joint work with Lance E. Miller and Wenbo Niu.

The models of directed polymers are based on Gibbs measures on paths with the reference measure usually describing a process with independent increments where the energy of the interaction between the path and the environment is given by a space-time random potential accumulated along the path. One of the intriguing phenomena that these models exhibit is the localization/delocalization transition between high/low temperature regimes. In order to study directed polymers in Euclidean space, we will first review a new metrization of the Mukherjee-Varadhan topology, introduced as a translation-invariant compactification of the space of probability measures on Euclidean spaces. This new metrization allows us to prove that the asymptotic clusterization (a natural continuous analogue of the asymptotic pure atomicity property) holds in the low temperature regime and that the endpoint distribution is geometrically localized with positive density if and only if the system is in the low temperature regime.

In the first part of the talk, I will present my recent working paper on continuous time game between a suspect and a defender. Economically, this is a sequential game model with asymmetric information, imperfect observation, and optimal stopping. Mathematically, search of a Markov equilibrium boils down to finding a solution of the system of equations (a HJB equation from the suspect’s optimal control, variational inequality from the defender’s optimal stopping, and a SDE from the filtering equation).

In the second part of the talk, I will roughly overview subjects in mathematical finance, and place where my past, current, and future research topics lie.