Thermal diffusion has been studied for over 150 years. Despite of the long history and the increasing importance of the phenomenon, the physics of thermal diffusion remains poorly understood until now. In this paper the author shows that Ludwig’s thermal diffusion can be completely understood using Einstein’s random walk. The only thing that needs to be added to Einstein’s random walk is a location dependency, which is needed to work with the temperature gradient of thermal diffusion. Hence, the walk length and the walk speed are location dependent functions in the random walk system suggested in this paper. Then, a mathematical understanding of such a random walk gives the physics of the thermal diffusion as clearly as the classical homogeneous case. As a result, a molecular level theory of thermal diffusion is finally obtained in this paper as a generalization of Einstein’s diffusion theory.

A Meissner state of bulk superconductors of type 2 loses its stability as the applied magnetic field increases to a critical field H_S. A Meissner state is described by a solution of a partial differential system in the 3-dimensional space, which is called a Meissner solution of the system. In this talk we shall show the existence of the Meissner solutions, and examine the convergence of the Meissner solutions to a solution of the limiting system as the Ginzburg-Landau parameter increases to infinity. Our results suggest that in order to determine the critical field H_S one needs to measure the maximum value of the tangential component of the induced magnetic field on the surface of the superconductor.

In this talk, I would like to introduce an extended magnetostatic Born-Infeld model, which involves functionals with operator curl. And some results on existence of C^{2,alpha} solutions of corresponding quasilinear degenerate systems will be shown.

Brooks’ Theorem states that for a graph G with maximum degree Δ(G) at least 3, the chromatic number is at most Δ(G) when the clique number is at most Δ(G). Vizing proved that the list chromatic number is also at most Δ(G) under the same conditions. Borodin and Kostochka conjectured that a graph G with maximum degree at least 9 must be (Δ(G)-1)-colorable when the clique number is at most Δ(G)-1; this was proven for graphs with maximum degree at least 10¹⁴ by Reed. We prove an analogous result for the list chromatic number; namely, we prove that a graph G with Δ(G)≥ 10²⁰ is (Δ(G)-1)-choosable when the clique number is at most Δ(G)-1. This is joint work with H. A. Kierstead, L. Rabern, and B. Reed.

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

In this talk, the diagrams of affine permutations and their balanced labellings will be introduced. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the column strict balanced labellings is the affine Stanley symmetric function defined by Lam. The affine Stanley symmetric function is the object of active research in the field of Schubert calculus. It is the affine counterpart of the Stanley symmetric function which is the limit of Schubert polynomials. Our construction is a natural tableau-theoretic realization of this function. We also give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we will give a necessary and sufficient condition for a diagram to be an affine permutation diagram. If time allows, we will
introduce some conjectures about when the affine Stanley symmetric functions coincide. This talk is based on the joint work with Taedong Yun.

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

LIST OF SPEAKERS

1. 11AM-12PM Tomáš Kaiser (University of West Bohemia, Czech Republic) Applications of the matroid-complex intersection theorem
2. 1:30PM-2:30PM Paul Seymour (Princeton University, USA) Graphs, Tournaments, Colouring and Containment
3. 2:45PM-3:45PM Maria Chudnovsky (Columbia University, USA) Excluding paths and antipaths
4. 4:30PM-5:30PM Daniel Kráľ (University of Warwick, UK) Quasirandomness and property testing of permutations (This is a department colloquium at Room 1501)

ABSTRACTS

11AM Applications of the matroid-complex intersection theorems (Tomáš Kaiser)

The Matroid intersection theorem of Edmonds gives a formula for the maximum size of a common independent set in two matroids on the same ground set. Aharoni and Berger generalized this theorem to the `topological’ setting where one of the matroids is replaced by an arbitrary simplicial complex. I will present two applications of this result to graph-theoretical problems. The first application is related to the existence of spanning 2-walks in tough graphs, the other one is more recent and gives a bound on the fractional arboricity of a graph G ensuring that G can be covered by k forests and a matching. In both cases, slightly better results can be obtained by other methods, but there seems to be room for improvement on the topological side as well.

1:30PM Graphs, tournaments, colouring and containment (Paul Seymour)

Some tournaments H are heroes; they have the property that all tournaments not containing H as a subtournament have bounded chromatic number (colouring a tournament means partitioning its vertex-set into transitive subsets). In joint work with eight authors, we found all heroes explicitly. That was great fun, and it would be nice to find an analogue for graphs instead of tournaments. The problem is too trivial for graphs, if we only exclude one graph H; but it becomes fun again if we exclude a finite set of graphs. The Gyarfas-Sumner conjecture says that if we exclude a forest and a clique then chromatic number is bounded. So what other combinations of excluded subgraphs will give bounded chromatic (or cochromatic) number? It turns out (assuming the Gyarfas-Sumner conjecture) that for any finite set S of graphs, the graphs not containing any member of S all have bounded cochromatic number if and only if S contains a complete multipartite graph, the complement of a complete multipartite graph, a forest, and the complement of a forest. Proving this led us to the following: for every complete multipartite graph H, and every disjoint union of cliques J, there is a number n with the following property. For every graph G, if G contains neither of H,J as an induced subgraph, then V(G) can be partitioned into two sets such that the first contains no n-vertex clique and the second no n-vertex stable set. In turn, this led us (with Alex Scott) to the following stronger result. Let H be the disjoint union of H_1,H_2, and let J be obtained from the disjoint union of J_1,J_2 by making every vertex of J_1 adjacent to every vertex of J_2. Then there is a number n such that for every graph G containing neither of H,J as an induced subgraph, V(G) can be partitioned into n sets such that for each of them, say X, one of H_1,H_2,J_1,J_2 is not contained in G|X. How about a tournament analogue of this? It exists, and the same (short) proof works; and this leads to a short proof of the most difficult result of the heroes paper that we started with. There are a number of other related results and open questions. Joint work with Maria Chudnovsky.

2:45PM Excluding paths and antipaths (Maria Chudnovsky)

The Erdos-Hajnal conjecture states that for every graph H, there exists a constant delta(H)>0, such that every n-vertex graph with no induced subgraph isomorphic to H contains a clique or a stable set of size at least n^delta(H). This conjecture is still open. We consider a variant of the conjecture, where instead of excluding a single graph H as an induced subgraph, a family of graphs is excluded. We prove this modified conjecture for the case when the five-edge path and its complement are excluded. Our second result is an asymmetric version of this: we prove that for every graph G such that G contains no induced six-edge path, and the complement of G contains no induced four-edge path, G contains a polynomial-size clique or stable set. This is joint work with Paul Seymour.

4:30PM Quasirandomness and property testing of permutations (Daniel Kráľ) (This is a department colloquium at Room 1501)

A systematic study of large combinatorial objects has recently led to discovering many connections between discrete mathematics and analysis. In this talk, we explore the analytic view of large permutations. We associate every sequence of permutations with a measure on a unit square and show the following: if the density of every 4-element subpermutations in a permutation p is 1/4!+o(1), then the density of every k-element subpermutation is 1/k!+o(1). This solves a question of Graham whether quasirandomness of a permutation is captured by densities of its 4-element subpermutations. At the end of the talk, we present a result related to an area of computer science called property testing. A property tester is an algorithm which determines (with a small error probability) properties of a large input object based on a small sample of it. Specifically, we prove a conjecture of Hoppen, Kohayakawa, Moreira and Sampaio asserting that hereditary properties of permutations are testatble with respect to the so-called Kendal’s tau distance. The results in this talk were obtained jointly with Tereza Klimosova or Oleg Pikhurko.

A systematic study of large combinatorial objects has recently led to discovering many connections between discrete mathematics and analysis. In this talk, we explore the analytic view of large permutations. We associate every sequence of permutations with a measure on a unit square and show the following: if the density of every 4-element subpermutations in a permutation p is 1/4!+o(1), then the density of every k-element subpermutation is 1/k!+o(1). This solves a question of Graham whether quasirandomness of a permutation is captured by densities of its 4-element subpermutations. At the end of the talk, we present a result related to an area of computer science called property testing. A property tester is an algorithm which determines (with a small error probability) properties of a large input object based on a small sample of it. Specifically, we prove a conjecture of Hoppen, Kohayakawa, Moreira and Sampaio asserting that hereditary properties of permutations are testable with respect to the so-called Kendal's tau distance. The results in this talk were obtained jointly with Tereza Klimosova or Oleg Pikhurko.

we consider the boltzmann equation in a bounded domain with several boundary conditions. we discuss the well-posedness of the weak solution for both steady problems and dynamic problems as well as their dynamical stability. the pointwise estimate of the steady problem has an application to the validity of the fourier law. further we discuss the regularity properties of the boltzmann solution in a various domains. we can observe the formation and propagation of discontinuities in the case of non-convex domains. in the case of convex domains we establish the regularity estimate in some weighted sobolev space and weighted c^1 space. this talk is based on the following papers :

Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric motives over a perfect field, which has many of the properties expected of the derived category of the conjectural abelian category of motives. Recently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky's category of effective geometric motives, assuming resolution of singularities. Soon after, Levine extended this idea to construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme $S$ generated by the motives of smooth projective S-schemes, assuming that S is itself smooth over a perfect field. In both constructions, the tensor structure requires the rational coefficients.  I will  describe the category of smooth motives defined by Levine and talk about a ``pseudo-tensor structure'' on the DG category of motives and show that when $S$ is semi-local and essentially smooth over a field of characteristic zero,  it provides a tensor structure on Levine's category of smooth motives, making it a tensor triangulated subcategory of the category of motives over S.

Equilibrium business cycle models will be introduced to students: these are indispensable tools in modern macroeconomics.

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

I’ll explain how to construct, given a set system, an inclusion-exclusion formula valid for that set system and with size sensitive to the size of the Venn diagram of the set system. The main ingredient of the proof are random simplicial complexes. The talk will start from first principle.
This is joint work with J. Matousek, M. Tancer, Z. Safernova and P. Patak from Charles university.

In this talk I will recall briefly the definition of the category of Chow motives and explain the meaning of Rost nilpotence. Finally I will present some results of the last years in this direction, in particular I will sketch a proof the Rost nilpotence is true for surfaces over fields of characteristic zero.

The field of Symbolic Dynamics which, according to Morse, is an algebra and geometry of recurrence will be explored along with an introduction to Measurable Dynamics and Topological Dynamics.

In random matrix theory, it is believed that local statistics of eigenvalues of random matrices are universal in the sense that it depends only on the symmetric class of the matrix ensembles and not on the distribution of individual entries. The distribution of the largest eigenvalue is given by Tracy-Widom law for a very general class of Wigner matrices, which is called the edge universality. In this talk, a necessary and sufficient condition for Tracy-Widom law will be explained. Basic methods and tools to study random matrices will also be covered.

We present origin of the Hamilton-Jacobi-Bellman equation, which is used to investigate optimization problems in financial mathematics, from the perspective of mathematical physics.

The Black-Scholes parabolic equation is the foremost important partial differential equation in option pricing theory. It is a special case of a parabolic differential equation, and in this talk we present existence conditions for the solutions of the parabolic equations with general growth conditions.

Let F be a field and T a torus over F (not assumed to be split) and X an affine smooth integral F-variety. Assuming some conditions on X, which are e.g. satisfied for a non singular affine quadric, or the variety of a simply connected algebraic group, I show that the pull-back morphisms Br (T) -> Br (X  x_F T) and Br (F) -> Br (X) are isomorphisms away from char. F-torsion. As an application we get some computations of Brauer groups of reductive groups.

we establish new bounds of the sobolev norms of solutions of
some semilinear wave equations for data lying in the hs, s<1, closure of
compactly supported data inside a ball of radius r, with r a positive
number. in order to do that we perform an analysis in a neighborhood of
the cone, using an almost shatah-struwe estimate, an almost conservation
law and some estimates for localized functions: this allows to prove a
decay estimate and establish an estimate of the low frequency component
of the position of the solution. then, in order to establish an
estimate of the high frequency component of the solution and the velocity,
we use this decay estimate and another almost conservation law.

In this talk we present some recent results contained in the paper “Roll-Over Parameters and Options Pricing”, and also discuss legal difficulties in hedging derivatives based on the speaker’s own personal experience.

Since 2008 bankruptcy of Lehman Brothers, the counter-party credit risk became more important in the financial market. Banks generally make profit by selling their credits and the different credit rating between banks causes a lot of unclear risks. We use collaterals to remove these unclear risks and a collateral had been used as a sub-instrument to strengthen credits. However this sub-instrument is still composed of some valuable instruments and banks start to recognize that it may be used as a profitable instrument. We investigate how banks make more profit with this sub-instrument and try to find methods to avoid these kinds of unfair arbitrage.

The finite time blow-up problem of the 3D incompressible Euler equations is one of the most
outstanding open problems in the partial differential equations. In this lecture we introduce the problem, and discuss the current status. We also discuss the studuies on the closely related equations, namely the compressible Euler equations, quasi-geostropic equations,
 and the Boussinesq equations.

3차원 오일러방정식의 해가 유한시간에 폭발하는지에 관한 문제는 편미분방정식의 가장 중요한 난제중의 하나이다. 이 강연에서는 이 문제를 소개하고 현재까지의 연구 현황을 얘기하고자 한다. 이와 더불어 이와 관련된 유체역학의 다른 방정식들의 연구 결과들도 소개한다.

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

The concept of time-changed Brownian motions has been used as one of the standard tools to construct parametric Levy models in finance.

 In Lecture 1, we review briefly the theory on time-changed Brownian motions, and then discuss the existing models which can be defined as time-changed Brownian motions.

 In Lecture 2, we propose a new model, called the Bessel tempered stable model, and then describe it as a time-changed Brownian motion.

For a hypergraph G and a positive integer c, let χ_ℓ(G,c) be the minimum value of ℓ such that G is L-colorable from every list L with |L(v)|=ℓ for each v∈V(G) and |L(u)∩L(v)|≤c for all e=uv∈E(G). This parameter was studied by Kratochvíl, Tuza and Voigt for various kinds of graphs. In this talk, we present the asymptotics of χ_ℓ(G,c) for complete graphs, complete multipartite graphs and hypergraphs.

This is a joint work with Z. Füredi and A. Kostochka.

 The concept of time-changed Brownian motions has been used as one of the standard tools to construct parametric Levy models in finance.

 In Lecture 1, we review briefly the theory on time-changed Brownian motions, and then discuss the existing models which can be defined as time-changed Brownian motions.

 In Lecture 2, we propose a new model, called the Bessel tempered stable model, and then describe it as a time-changed Brownian motion.

서브 프라임 모기지로 시작된 금융위기는 금융의 각 분야에 큰 변화를 가져오고 있습니다. 특히 파생상품 시장에 도입되는 새로운 규제와 거래 방법들은 월가에서 일하는 퀀트들의 업무에도 직접적인 영향을 미치고 있습니다. 이러한 변화들이 퀀트들에게는 기회일지 위기일지, 그리고 한국에서 퀀트를 꿈꾸는 학생들에게는 어떤 의미가 있을지에 대해서 얘기를 나눠 보겠습니다.

Geometric invariant theory (GIT) is a powerful tool to construct moduli spaces as quotients of the Hilbert schemes. GIT was very successful to construct the moduli space of stable curves and the moduli space of vector bundles on curves. But in general it is very hard to understand explicitly (semi-)stable objects in the sense of GIT. Along the development of the minimal model program (MMP), birational geometry including MMP is again a powerful tool to construct compactified moduli space for varieties of general type. In the talk, I will introduce several notions of stabilities and discuss their relations. Recent developments of log MMP of the moduli space of stable curves will be also discussed.

We investigate the existence and the time-asymptotic stability of boundary layer solutions for a one-dimensional shallow water model. Under the subsonic condition on the far-field equilibrium state, we show that there exists a one-parameter family of boundary layer solutions satisfying the outflow constant-flux boundary condition. We also show that there is a unique time-asymptotic boundary layer pro file for a given initial data, and prove its stability using standard energy methods, provided that the initial perturbation is sufficiently small. Our stability result is obtained without the zero mass condition nor "artificial" condition on the shock strength. Rather the time-asymptotic profile is determined alone by the initial perturbation from the end state so that it satisfies the zero mass condition and its amplitude is appropriately small simultaneously.

We consider partitions of the vertices of a graph into a fixed number
of labelled cells such that (i) the subgraph induced by each cell
belongs to a family that depends on the cell, and (ii) edges joining
vertices in different cells are allowed only if the cells correspond
to adjacent vertices in a given pattern graph H. Both polynomiality
and NP-completeness results are presented. Tha focus is on methods
that give polynomial time algorithms.
This is a joint work with Peter Dukes and Steve Lowdon.

The ample divisors on a variety are defined geometrically as the divisors such that the linear system of its multiple defines an embedding of the variety into some projective space. There are also numerical, cohomological, and asymptotic cohomological characterizations of ample divisors. By relaxing the conditions in these characterizations we obtain the notions of so-called partially ample divisors, that are slightly different from each other. We will study their properties and show that they coincide on Fano varieties and Mori dream spaces.

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

Based on the newly understood dichotomy of sparse and dense
structures, we provide the general framework for study of structural
(mostly graph) limits. Our approach uses both model theoretic and
analytic tools and uses structural theory of bounded expansion
classes.