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 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, we address the problem of local well-posedness of two types of abelian Chern-Simons gauge theories, namely the Chern-Simons-Dirac (CSD) and the Chern-Simons-Higgs (CSH) equations, under the Lorenz gauge condition. One of our main contributions is the uncovering of a special structure (null structure) of (CSD) which, combined with the standard machinery of $X^{s,b}$ spaces, allows us to obtain local well-posedness of (CSD) for initial data $a_{mu}, psi in H^{1/4+epsilon}_{x}$. Moreover, it is observed that the same techniques applied to (CSH) lead to an improvement over the previous result due to Selberg-Tesfahun (2012). This is a joint work with H. Huh.

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 first application of Szemeredi’s regularity method was the following celebrated Ramsey-Turan result proved by Szemeredi in 1972: any K4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollobas and Erdos gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K4-free graph on N vertices with independence number o(N) and (1/8 – o(1)) N^2 edges. Starting with Bollobas and Erdos in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this work, we give nearly best-possible bounds, solving the various open problems concerning this critical window.

More generally, it remains an important problem to determine if, for certain applications of the regularity method, alternative proofs exist which avoid using the regularity lemma and give better quantitative estimates. In their survey on the regularity method, Komlos, Shokoufandeh, Simonovits, and Szemeredi surmised that the regularity method is likely unavoidable for applications where the extremal graph has densities in the regular partition bounded away from 0 and 1. In particular, they thought this should be the case for Szemeredi’s result on the Ramsey-Turan problem. Contrary to this philosophy, we develop new regularity-free methods which give a linear dependence, which is tight, between the parameters in Szemeredi’s result on the Ramsey-Turan problem.

Joint work with Jacob Fox and Yufei Zhao.

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.

Bridson, Martin R.; Haefliger, André. Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften 319. Springer-Verlag, Berlin, 1999의 Chapter I.5, "Some Basic Constructions"에 대해 발표하는 김상현 교수님 지도 대학원생 랩세미나입니다. 발표 내용은 다음과 같은 대상에 대한 정의 및 기본적인 성질입니다.

- κ-cones

- Spherical joins

- Quotient pseudometrics

- Gluing along isometric subspaces

- Gromov-Hausdorff convergence, Gromov's theorem

- Ultralimits and asymptotic cones

발표자는 대학원 1학년차인 이돈성입니다.

The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2ⁿ vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of order N contains the cube graph Q_n or an independent set of order s. Burr and Erdős in 1983 asked whether the simple lower bound r(Q_n, K_s) ≥ (s-1)(2ⁿ -1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.
Joint work w/ David Conlon, Jacob Fox, and Benny Sudakov

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.

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 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.

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.