Department Seminars & Colloquia
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A chaotic expansion of subordination of L´evy process is developed. The chaotic ex- pansion is expressed in term of power jump processes like Nualart-Schoutens. We characterize the jump processes due to underlying process and subordinate. Considering different time scales, we decompose L2 space orthogonally according to different scales. Also, following Le´on, Malliavin derivative and Clark-Ocone formula for each subordi- nator is derived. Applications to several subordinations and hedging are studied.
Abstract:
This talk is designed for 3-hours crash course for graduate students
who may have backgrounds in probability and stochastic analysis, but not in stochastic control theory.
It consists of two parts.
The first part will be general introductions on two popular techniques in control theory: dynamic programming principle (DPP) with HJB equations, pontryagin maximal principle with FBSDE.
Classic Merton’s portfolio optimization will be used to show the utilisation of the above two methodologies.
The second part is deemed to be research oriented topic, in which some delicate measurable issues will be addressed in the proof of DPP. In this regard, the rigorous proof of DPP for the general control framework is still widely open. We then determine a weaker sufficient condition than that of Theorem 5.2.1 in the book [Fleming and Soner (2006)] for the continuity of the value functions, which eventually leads to the proof of DPP in stochastic exit time control problems. Some further possible development will be also discussed.
All talk will be focused on finite time horizon. Some part is taken from the paper
“On the continuity of stochastic exit time control problems” by E. Bayraktar, Q. Song, and J. Yang.
Let A and B be finite nonempty subsets of a multiplicative group G, and consider the product set AB = { ab | a in A and b in B }. When |G| is prime, a famous theorem of Cauchy and Davenport asserts that |AB| is at least the minimum of {|G|, |A| + |B| - 1}. This lower bound was refined by Vosper, who characterized all pairs (A,B) in such a group for which |AB| < |A| + |B|. Kneser generalized the Cauchy-Davenport theorem by providing a natural lower bound on |AB| which holds in every abelian group. Shortly afterward, Kemperman determined the structure of those pairs (A,B) with |AB| < |A| + |B| in abelian groups. Here we present a further generalization of these results to arbitrary groups. Namely we generalize Kneser’s Theorem, and we determine the structure of those pairs with |AB| < |A| + |B| in arbitrary groups.
Abstract:
This talk is designed for 3-hours crash course for graduate students
who may have backgrounds in probability and stochastic analysis, but not in stochastic control theory.
It consists of two parts.
The first part will be general introductions on two popular techniques in control theory: dynamic programming principle (DPP) with HJB equations, pontryagin maximal principle with FBSDE.
Classic Merton’s portfolio optimization will be used to show the utilisation of the above two methodologies.
The second part is deemed to be research oriented topic, in which some delicate measurable issues will be addressed in the proof of DPP. In this regard, the rigorous proof of DPP for the general control framework is still widely open. We then determine a weaker sufficient condition than that of Theorem 5.2.1 in the book [Fleming and Soner (2006)] for the continuity of the value functions, which eventually leads to the proof of DPP in stochastic exit time control problems. Some further possible development will be also discussed.
All talk will be focused on finite time horizon. Some part is taken from the paper
“On the continuity of stochastic exit time control problems” by E. Bayraktar, Q. Song, and J. Yang.
In this talk, I will present two different topics; minimax lower bound in normal mixtures, and global rates of convergence in a log-concave shape-constrained estimation.
The first half (part of my Ph.D. thesis, accepted in Bernoulli, 2013) deals with minimax rates of convergence for estimation of density functions on the real line. The densities are assumed to be location mixtures of normals, a global regularity requirement that creates subtle difficulties for the application of standard minimax lower bound methods. Using novel Fourier and Hermite polynomial techniques, we determine the minimax optimal rate|slightly larger than the parametric rate|under squared error loss.
In the second half, I will present recent results in log-concave density estimation (joint work with Richard Samworth, submitted to the Annals of Statistics, 2014). We study the performance of log- concave density estimators with respect to global (e.g. squared Hellinger) loss functions, and adopt a minimax approach....
Matt DeVos
Simon Franser U.
Lecture 4) 9. 23(Tue) PM 4:00 ~ 6:00 E6-1 Rm 1409
Graphs and Sumsets (Schrijver-Seymour)
Abstract: I intend to give an introduction to some of the wonderful topics in the world of additive combinatorics. This is a broad subject which features numerous different tools and techniques, and is presently a hotbed of exciting research. My focus will be on the combinatorics, and I will keep things as basic as possible (I will assume nothing more than a basic background in combinatorics). I’ll begin the tour with some of the classical theorems like Cauchy-Davenport and Erdos-Ginzburg-Ziv and I will exhibit some very clean proofs of these and other results such as the Theorems of Schrijver-Seymour, Green-Ruzsa, Dvir, and Elekes. We will also discuss (but not prove) some more recent results like the Breulliard-Green-Tao Theorem.
Matt DeVos
Simon Franser U.
Lecture 4) 9. 23(Tue) PM 4:00 ~ 6:00 E6-1 Rm 1409
Graphs and Sumsets (Schrijver-Seymour)
Abstract: I intend to give an introduction to some of the wonderful topics in the world of additive combinatorics. This is a broad subject which features numerous different tools and techniques, and is presently a hotbed of exciting research. My focus will be on the combinatorics, and I will keep things as basic as possible (I will assume nothing more than a basic background in combinatorics). I’ll begin the tour with some of the classical theorems like Cauchy-Davenport and Erdos-Ginzburg-Ziv and I will exhibit some very clean proofs of these and other results such as the Theorems of Schrijver-Seymour, Green-Ruzsa, Dvir, and Elekes. We will also discuss (but not prove) some more recent results like the Breulliard-Green-Tao Theorem.
Let a 3-dimensional smooth and bounded domain be given. We compare two problems arising in kinetic theory: the Vlasov-Poisson system and the Fokker-Planck equation. In the Vlasov-Poisson case, the existence of regular solutions is determined according to whether the boundary of the domain is convex or not. But, in the Fokker-Planck case, there is a smoothing effect due to the random force, solutions are expected to be more regular.
For each smooth del Pezzo surface S, we find ample divisors A on the surface S
such that S admits an A-polar cylinder and we present an eff ective divisor D that is Q-linearly
equivalent to A and such that the open set , the complement of Supp(D) is a cylinder.
Moreover using similar construction of cylinders, we prove that affine cones over any ample polarization of
del Pezzo surfaces with degree 4 are flexible.
The square $G^2$ of a graph G is the graph defined on V(G) such that two vertices u and v are adjacent in $G^2$ if the distance between u and v in G is at most 2. Let $chi(H)$ and $chi_{ell}(H)$ be the chromatic number and the list chromatic number of H, respectively. A graph H is called chromatic-choosable if $chi_{ell} (H) = chi(H)$. It is an interesting problem to find graphs that are chromatic-choosable.
Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that $G^2$ is chromatic-choosable for every graph G. Recently, Kim and Park showed that the List Square Coloring Conjecture does not hold in general by finding a family of graphs whose squares are complete multipartite graphs with partite sets of unbounded size. It is a well-known fact that the List Total Coloring Conjecture is true if the List Square Coloring Conjecture holds for special class of bipartite graphs. On the other hand, the counterexamples to the List Square Coloring Conjecture are not bipartite graphs. Hence a natural question is whether $G^2$ is chromatic-choosable or not for every bipartite graph G.
In this paper, we give a bipartite graph G such that $chi_{ell} (G^2) neq chi(G^2)$. Moreover, we show that the value $chi_{ell}(G^2) - chi(G^2)$ can be arbitrarily large. This is joint work with Boram Park.
Matt DeVos
Simon Franser U.
Lecture 3) 9. 18(Thu) PM 4:00 ~ 6:00 E6-1 Rm 3433
Sums and Products (Elekes and Dvir)
Abstract: I intend to give an introduction to some of the wonderful topics in the world of additive combinatorics. This is a broad subject which features numerous different tools and techniques, and is presently a hotbed of exciting research. My focus will be on the combinatorics, and I will keep things as basic as possible (I will assume nothing more than a basic background in combinatorics). I’ll begin the tour with some of the classical theorems like Cauchy-Davenport and Erdos-Ginzburg-Ziv and I will exhibit some very clean proofs of these and other results such as the Theorems of Schrijver-Seymour, Green-Ruzsa, Dvir, and Elekes. We will also discuss (but not prove) some more recent results like the Breulliard-Green-Tao Theorem.
Matt DeVos
Simon Franser U.
Lecture 3) 9. 18(Thu) PM 4:00 ~ 6:00 E6-1 Rm 3433
Sums and Products (Elekes and Dvir)
Abstract: I intend to give an introduction to some of the wonderful topics in the world of additive combinatorics. This is a broad subject which features numerous different tools and techniques, and is presently a hotbed of exciting research. My focus will be on the combinatorics, and I will keep things as basic as possible (I will assume nothing more than a basic background in combinatorics). I’ll begin the tour with some of the classical theorems like Cauchy-Davenport and Erdos-Ginzburg-Ziv and I will exhibit some very clean proofs of these and other results such as the Theorems of Schrijver-Seymour, Green-Ruzsa, Dvir, and Elekes. We will also discuss (but not prove) some more recent results like the Breulliard-Green-Tao Theorem.
A Noetherian ring is called quasi-Gorenstein if the ring is (locally) isomorphic to a canonical module. A Gorenstein ring is a Cohen-Macaulay quasi-Gorenstein ring. In general, a quasi-Gorenstein ring is not Gorenstein. In this talk, we show that certain classes of quasi-Gorenstein extended Rees algebras are Gorenstein.
Matt DeVos
Simon Franser U
Lecture 2) 9. 16(Tue) PM 4:00 ~ 6:00 E6-1 Rm 1409
Rough Structure (Green-Ruzsa)
Abstract: I intend to give an introduction to some of the wonderful topics in the world of additive combinatorics. This is a broad subject which features numerous different tools and techniques, and is presently a hotbed of exciting research. My focus will be on the combinatorics, and I will keep things as basic as possible (I will assume nothing more than a basic background in combinatorics). I’ll begin the tour with some of the classical theorems like Cauchy-Davenport and Erdos-Ginzburg-Ziv and I will exhibit some very clean proofs of these and other results such as the Theorems of Schrijver-Seymour, Green-Ruzsa, Dvir, and Elekes. We will also discuss (but not prove) some more recent results like the Breulliard-Green-Tao Theorem.
Matt DeVos
Simon Franser U
Lecture 2) 9. 16(Tue) PM 4:00 ~ 6:00 E6-1 Rm 1409
Rough Structure (Green-Ruzsa)
Abstract: I intend to give an introduction to some of the wonderful topics in the world of additive combinatorics. This is a broad subject which features numerous different tools and techniques, and is presently a hotbed of exciting research. My focus will be on the combinatorics, and I will keep things as basic as possible (I will assume nothing more than a basic background in combinatorics). I’ll begin the tour with some of the classical theorems like Cauchy-Davenport and Erdos-Ginzburg-Ziv and I will exhibit some very clean proofs of these and other results such as the Theorems of Schrijver-Seymour, Green-Ruzsa, Dvir, and Elekes. We will also discuss (but not prove) some more recent results like the Breulliard-Green-Tao Theorem.
E6-1, ROOM 1409
Discrete Math
Matt DeVos (Simon Fraser University, Canada)
Immersion in Graphs and Digraphs
E6-1, ROOM 1409
Discrete Math
Graph immersion is a natural containment relation like graph minors. However, until recently, graph immersion has received relatively little attention. In this talk we shall describe some recent progress toward understanding when a graph does not immerse a certain subgraph. Namely, we detail a rough structure theorem for graphs which do not have K_t as an immersion, and we discuss the precise structure of graphs which do not have K_{3,3} as an immersion. Then we turn our attention to a special class of digraphs, those forwhich every vertex has both indegree and outdegree equal to 2. Thesedigraphs have special embeddings in surfaces where every vertex has alocal rotation in which the inward and outward edges alternate. Itturns out that the nature of these embeddings relative to immersion isquite closely related to the usual theory of graph embedding and graphminors. Here we describe the complete list of forbidden immersionsfor (special) embeddings in the projective plane. These results are joint with various coauthors including Archdeacon,Dvorak, Fox, Hannie, Malekian, McDonald, Mohar, and Scheide.
Given a curve in a plane, we construct a factorization of a polynomial multiplied by an identity matrix into the product of two matrices, by counting certain polygons in a plane. Such correspondences between geometric objects (curves, polygons) and algebraic objects (matrix factorizations of a polynomial) are instances of homological mirror symmetry. We explain the generalization of the construction to higher dimensions, and its application to the proof of homological mirror symmetry conjecture for certain spaces.
자연과학동 E6-1, ROOM 1409
Discrete Math
Joonkyung Lee (University of Oxford, UK)
Some Advances in Sidorenko's Conjecture
자연과학동 E6-1, ROOM 1409
Discrete Math
Sidorenko's conjecture states that for every bipartite graph H on {1,...,k} $$ int prod_{(i,j)in E(H)} h(x_i, y_j) dmu^{|V(H)|} ge left( int h(x,y) ,dmu^2 right)^{|E(H)|} $$ holds, where $mu$ is the Lebesgue measure on [0,1] and h is a bounded, non-negative, symmetric, measurable function on [0,1]^2. An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph H to a graph G is asymptotically at least the expected number of homomorphisms from H to the Erdos-Renyi random graph with the same expected edge density as G.In this talk, we will give an overview on known results and new approaches to attack Sidorenko's conjecture. This is a joint work with Jeong Han Kim and Choongbum Lee.
Matt DeVos
Simon Franser U.
Lecture 1) 9. 2(Tue) PM 4:00 ~ 6:00 E6-1 Rm 1409
Sumsets and Subsequence Sums (Cauchy-Davenport, Kneser, and Erdos-Ginzburg-Ziv)
Abstract: I intend to give an introduction to some of the wonderful topics in the world of additive combinatorics. This is a broad subject which features numerous different tools and techniques, and is presently a hotbed of exciting research. My focus will be on the combinatorics, and I will keep things as basic as possible (I will assume nothing more than a basic background in combinatorics). I’ll begin the tour with some of the classical theorems like Cauchy-Davenport and Erdos-Ginzburg-Ziv and I will exhibit some very clean proofs of these and other results such as the Theorems of Schrijver-Seymour, Green-Ruzsa, Dvir, and Elekes. We will also discuss (but not prove) some more recent results like the Breulliard-Green-Tao Theorem.
Matt DeVos
Simon Franser U.
Lecture 1) 9. 2(Tue) PM 4:00 ~ 6:00 E6-1 Rm 1409
Sumsets and Subsequence Sums (Cauchy-Davenport, Kneser, and Erdos-Ginzburg-Ziv)
Abstract: I intend to give an introduction to some of the wonderful topics in the world of additive combinatorics. This is a broad subject which features numerous different tools and techniques, and is presently a hotbed of exciting research. My focus will be on the combinatorics, and I will keep things as basic as possible (I will assume nothing more than a basic background in combinatorics). I’ll begin the tour with some of the classical theorems like Cauchy-Davenport and Erdos-Ginzburg-Ziv and I will exhibit some very clean proofs of these and other results such as the Theorems of Schrijver-Seymour, Green-Ruzsa, Dvir, and Elekes. We will also discuss (but not prove) some more recent results like the Breulliard-Green-Tao Theorem.
자연과학동 E6-1, ROOM 1409
Discrete Math
Suil O (Georgia State University, USA)
Finding a spanning Halin subgraph in 3-connected {K_{1,3}, P_5}-free graphs
자연과학동 E6-1, ROOM 1409
Discrete Math
A Halin graph is constructed from a plane embedding of a tree whose non-leaf vertices have degree at least 3 by adding a cycle through its leaves in the natural order determined by the embedding. In this talk, we prove that every 3-connected {K_{1,3},P_5} -free graph has a spanning Halin subgraph. This result is best possible in the sense that the statement fails if K_{1,3} is replaced by K_{1,4} or P_5 is replaced by P_6. This is a joint work with Guantao Chen, Jie Han, Songling Shan, and Shoichi Tsuchiya.
E6, Room 1409
KMRS Seminar
Imre Barany (Hungarian Academy of Sciences & University College)
Random points and lattice points in convex bodies
E6, Room 1409
KMRS Seminar
Assume K is a convex body in R^d and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong toX? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of X) approximate K? In this lecture I will talk about these questions mainly in two cases. The first is when X is a random sample of n uniform, independent points from K. In this case motivation comes from Sylvester's famous four-point problem and from the theory of random polytopes. The second case is when X is the set of lattice points contained in K and the questions come from integer programming and geometry of numbers. Surprisingly (or not so surprisingly), the answers in the two cases are rather similar. The methods are, however, very different.
E6, Room 1409
KMRS Seminar
Imre Barany (Hungarian Academy of Sciences & University College)
Random points and lattice points in convex bodies
E6, Room 1409
KMRS Seminar
Assume K is a convex body in R^d and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong toX? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of X) approximate K? In this lecture I will talk about these questions mainly in two cases. The first is when X is a random sample of n uniform, independent points from K. In this case motivation comes from Sylvester's famous four-point problem and from the theory of random polytopes. The second case is when X is the set of lattice points contained in K and the questions come from integer programming and geometry of numbers. Surprisingly (or not so surprisingly), the answers in the two cases are rather similar. The methods are, however, very different.
The dynamics of plasma can be simulated using kinetic or fluid models. Kinetic models are valid over most of the spatial and temporal scales that are of physical relevance in many application problems; however, they are computationally expensive due to the high-dimensionality of phase space. Fluid models have a more limited range of validity, but are generally computationally more tractable than kinetic models. One critical aspect of fluid models is the question of what assumptions to make in order to close the fluid moment hierarchy. A theoretically ideal but practically difficult to achieve model would combine positive aspects from both the kinetic and the fluid descriptions.
In this work we discuss efforts to develop efficient finite element schemes for solving the kinetic Vlasov-Poisson system and its fluid approximations.
We describe a discontinuous Galerkin (DG) approach for the kinetic system using semi-Lagrangian time-stepping. Results on several standard test cases are shown. Next we consider fluid approximations of Vlasov-Poisson and develop a high-order DG scheme on a class of fluid models derived from quadrature-based moment closures. Finally we describe ongoing efforts to hybridize fluid and kinetic models to simultaneously improve computational efficiency and model validity.
The theory of complex multiplication allows one to construct explicit class fields and cryptographic curves of genus g=1 or g=2. Essential to this is are special values of the j-invariant (g=1) or absolute Igusa invariants (g=2). Class invariants are special values of arbitrary modular functions that lie in the same field as the aforementioned values. Such class invariants can replace the j-invariant or Igusa invariants in applications, which speeds op these applications when the class invariants have small height. Schertz gave a systematic way of creating class invariants using modular functions for the group Gamma_0(N) in the case g=1. I will show how to generalize Schertz's method to modular functions on higher-dimensional moduli spaces. This is joint work with Andreas Enge.
We condsider the axisymmetric initial data (M,g,k) for the Einstein equations, with asimply connected Riemanninan manifold having two ends, one asymptotically flat and the other either asymptotically flat or asymptotically cylindrical. Penrose's heuristic arguments relate the ADM mass and the angular momentum of the intial data via the angular momentum-mass inequality. This has been proven when the initial data is maximal(tr k=0) and vacuum(Dain), and extended thereafter.
Here we show how to reduce the general formulation of the angular momentum-mass inequality for the non-maximal initial data, to the known maximal case, whenever a geometrically motivated system of two elliptic equations admits a solution. This procedure is based on a certain deformation of the intial data which preserves the relevant geometry, while achieving the maximal condition and its implied inequality for the scalar curvature. Each equation in the system is analyzed in detail individually, and it is shown that the appropriate existence/uniqueness of results holds with the solution satisfying thedesired asymptotics.
The functions of living cells are regulated by the complex biochemical network, which consists of stochastic interactions among genes and proteins. However, due to the complexity of biochemical networks and the limit of experimental techniques, identifying entire biochemical interaction network is still far from complete. On the other hand, output of the networks, timecourses of genes and proteins can be easily acquired with advances in technology. I will describe how to use oscillating timecourse data to reveal biochemical network structure by using a fixed-point criteria. Moreover, I will describe how mathematical modeling can be used to understand the dynamics and functions of complex biochemical networks with an example of circadian clock. Finally, in biochemical networks, reactions occur on disparate timescale. This timescale separation has been used to project deterministic models of biochemical networks onto lower-dimensional slow manifolds with quasi-steady state approximation (QSSA). I will discuss whether this reduction technique for deterministic systems can be used for stochastic systems. Specifically, I will show when macroscopic rate functions derived with QSSA (e.g. Hill functions) can be used to derive the propensity functions for microscopic rates.
Abstract:
In this talk, we consider the Carlitz multiple polylogarithms (CMPLs) at algebraic points. We show that they form a graded algebra defined over the base rational function field. We further show that any multiple zeta value (MZV) defined by Thakur can be expressed as a linear combination of CMPLs at algebraic points, which is a generalization of the work of Anderson-Thakur on the depth one case.
As a conseqence, we obtain a function field version of Goncharov's conjecture
for MZVs.
The Lojasiewicz exponent is an analytic invariant of hypersurfaces with isolated singularities. It is an open question whether it is a topological invariant as well: so far this has been proven to be the case for weighted homogeneous isolated singularities of curves and surfaces. In this talk I will focus on the Lojasiewicz exponent of normal rational singularities of complex surfaces in comparison to its counterpart associated to the ideals of the local ring. Via this comparison I will give a bound for the case of ADE singularities. This is a joint work with Meral Tosun and Gülay Kaya.
Title: Numerical Computing with Chebfun
Chebfun is a Matlab-based system for numerical computing with functions as opposed to just numbers. This talk will describe some of the algorithms behind Chebfun and demonstrate it in action, including the extension to two dimensions known as Chebfun2.
자연과학동 (E6-1), ROOM 3433
Discrete Math
Suho Oh (University of Michigan/Texas State University, USA)
Fun with wires
자연과학동 (E6-1), ROOM 3433
Discrete Math
Wiring diagrams are widely used combinatorial objects that are mainly used to describe reduced words of a permutation. In this talk, I will mention a fun property I recently found about those diagrams, and then introduce other results and problems related to this property.