Dept. of Mathematics
http://hdl.handle.net/2142/16339
Fri, 22 Sep 2017 02:48:42 GMT2017-09-22T02:48:42ZA look at T1 and Tb theorems on non-homogeneous spaces through time-frequency analysis
http://hdl.handle.net/2142/95539
A look at T1 and Tb theorems on non-homogeneous spaces through time-frequency analysis
Klamsakul, Natawat
One of the widely studied topics in singular integral operators is T1 theorem. More precisely, it asks if one can extend a Calder\'on-Zygmund operator to a bounded operator on $L^p$. In addition, Tb theorem was raised when one asks if the T1 theorem remains true if the function $1$ is substituted by some bounded function $b$. In this dissertation, we apply time-frequency analysis to T1 theorem and Tb theorem. In particular, the theory of tiles and trees is used to prove T1 theorem on non-homogeneous spaces. This provides an alternative and a more visualized point of view to some parts of the proof. We also verify estimates from $L^p\times L^q$ to $L^r$ for the paraproducts appeared in T1 theorem. Although the paraproduct is specific, the method is applicable to this kind of study. Lastly, an extension to the proof of Tb theorem is established via a different tree from T1 theorem.
Calderon-Zygmund operator; T1 theorem; Tb theorem; time-frequency analysis
Thu, 25 Aug 2016 00:00:00 GMThttp://hdl.handle.net/2142/955392016-08-25T00:00:00ZKlamsakul, NatawatOn some problems in reconstruction
http://hdl.handle.net/2142/95378
On some problems in reconstruction
Spinoza, Hannah R
A graph is {\it reconstructible} if it is determined by its {\it deck} of unlabeled subgraphs obtained by deleting one vertex; a {\it card} is one of these subgraphs. The {\it Reconstruction Conjecture} asserts that all graphs with at least three vertices are reconstructible.
In Chapter $2$ we consider $k$-deck reconstruction of graphs. The {\it $k$-deck} of a graph is its multiset of $k$-vertex induced subgraphs. We prove a generalization of a result by Bollob\'as concerning the $k$-deck reconstruction of almost all graphs, showing that when $\ell \le (1-\epsilon)\frac{n}{2}$, the probability than an $n$-vertex graph is reconstructible from some $\binom{\ell+1}{2}$ of the graphs in the $(n-\ell)$-deck tends to $1$ as $n$ tends to $\infty$.
We determine the smallest $k$ such that all graphs with maximum degree $2$ are $k$-deck reconstructible. We prove for $n\ge 26$ that whether a graph is connected is determined by its $(n-3)$-deck. We prove that if $G$ is a complete $r$-partite graphs, then $G$ is $(r+1)$-deck reconstructible (the same holds for $\overline{G}$).
In Chapter $3$ we consider degree-associated reconstruction. An $(n-1)$-vertex induced subgraph accompanied with the degree of the missing vertex is called a {\it dacard}. The {\it degree-associated reconstruction number} of a graph $G$ is the fewest number of dacards needed to determine $G$. We provide a tool for reconstructing some graphs from two dacards. We prove that certain families of trees and disconnected graphs can be reconstructed from two dacards. We also determine the degree-associated reconstruction number for complete multipartite graphs and their complements. For such graphs, we also determine the least $s$ such that {\it every} set of $s$ dacards determine the graph.
In Chapter $4$ we consider the reconstruction of matrices from principal submatrices. A $(n-\ell)$-by-$(n-\ell)$ principal submatrix is a submatrix formed by deleting $\ell$ rows and columns symmetrically. The {\it matrix reconstruction threshold} $mrt(\ell)$ is the minimum integer $n_0$ such that for $n\ge n_0$ all $n$-by-$n$ matrices are reconstructible from their deck of $(n-\ell)$-by-$(n-\ell)$ principal submatrices. We prove $mrt(\ell) \leq \frac{2}{\ln 2}\ell^2+3\ell$.
Graph theory; Reconstruction
Thu, 01 Dec 2016 00:00:00 GMThttp://hdl.handle.net/2142/953782016-12-01T00:00:00ZSpinoza, Hannah RSome properties of Markoff chains
http://hdl.handle.net/2142/94732
Some properties of Markoff chains
Blackwell, David Harold
Markov processes; Markoff chains; Mathematics
Wed, 01 Jan 1941 00:00:00 GMThttp://hdl.handle.net/2142/947321941-01-01T00:00:00ZBlackwell, David HaroldIntrinsic contractivity for some non-symmetric Lévy processes with non-local operators
http://hdl.handle.net/2142/93052
Intrinsic contractivity for some non-symmetric Lévy processes with non-local operators
Lu, Qu
In this thesis we consider two types of non-symmetric processes, which are similar to the symmetric α-stable process. We derive sharp estimates for the eigenfunctions of the Feynman- Kac semigroups of these two types of processes and established their intrinsic contractivities. Our methods are mainly probabilistic and depend essentially on the sharp estimates of heat kernels.
intrinsic contractivity; non-symmetric semigroups; non-local operators; ground state domination
Tue, 12 Jul 2016 00:00:00 GMThttp://hdl.handle.net/2142/930522016-07-12T00:00:00ZLu, QuEmbedding problems and Ramsey-Turán variations in extremal graph theory
http://hdl.handle.net/2142/93041
Embedding problems and Ramsey-Turán variations in extremal graph theory
Sharifzadeh, Maryam
In this dissertation, we will focus on a few problems in extremal graph theory. The first chapter consists of some basic terms and tools.
In Chapter 2, we study a conjecture of Mader on embedding subdivisions of cliques. Improving a bound by Mader, Bollobás and Thomason, and independently Komlós and Szemerédi proved that every graph with average degree d contains a subdivision of K_[Ω(√d)]. The disjoint union of complete bipartite graph K_(r,r) shows that their result is best possible. In particular, this graph does not contain a subdivision of a clique of order w(r). However, one can ask whether their bound can be improved if we forbid such structures. There are various results in this direction, for example Kühn and Osthus proved that their bound can be improved if we forbid a complete bipartite graph of fixed size. Mader proved that that there exists a function g(r) such that every graph G with ẟ(G) ≥ r and girth at least g(r) contains a TK_(r+1). He also asked about the minimum value of g(r). Furthermore, he conjectured that C_4-freeness is enough to guarantee a clique subdivision of order linear in average degree. Some major steps towards these two questions were made by Kühn and Osthus, such as g(r) ≤ 27 and g(r) ≤ 15 for large enough r. In an earlier result, they proved that for C_4-free graphs one can find a subdivision of a clique of order almost linear in minimum degree. Together with József Balogh and Hong Liu, we proved that every C_(2k)-free graph, for k ≥ 3, has such a subdivision of a large clique. We also proved the dense case of Mader's conjecture in a stronger sense.
In Chapter 3, we study a graph-tiling problem. Let H be a fixed graph on h vertices and G be a graph on N vertices such that h|n. An H-factor is a collection of n/h vertex-disjoint copies of H in G. The problem of finding sufficient conditions for a graph G to have an H-factor has been extensively studied; most notable is the celebrated Hajnal-Szemerédi Theorem which states that every n-vertex graph G with ẟ(G) ≥ (1-1/r)n has a K_r-factor. The case r=3 was proved earlier by Corrádi and Hajnal. Another type of problems that have been studied over the past few decades are the so-called Ramsey-Turán problems. Erdős and Sós, in 1970, began studying a variation on Turán's theorem: What is the maximum number of edges in an n-vertex, K_r-free graph G if we add extra conditions to avoid the very strict structure of Turán graph. In particular, what if besides being K_r-free, we also require α(G) = o(n) . Since the extremal example for the Hajnal-Szemerédi theorem is very similar to the Turán graph, one can similarly ask how stable is this extremal example. With József Balogh and Theodore Molla, we proved that for an n-vertex graph G with α(G) = o(n), if ẟ(G) ≥ (1/2+o(1))n then G has a triangle factor. This minimum degree condition is asymptotically best possible. We also consider a fractional variant of the Corrádi-Hajnal Theorem, settling the triangle case of a conjecture of Balogh, Kemkes, Lee, and Young.
In Chapter 4, we first consider a Ramsey-Turán variant of a theorem of Erd\Ho s. In 1962, he proved that for any r > l ≥ 2, among all K_r-free graphs, the (r-1)-partite Turán graph has the maximum number of copies of K_l. We consider a Ramsey-Turán-type variation of Erdős's result. In particular, we define RT(F,H,f(n)) to be the maximum number of copies of F in an H-free graph with n-vertices and independence number at most f(n). We study this function for different graphs F and H. Recently, Balogh, Hu and Simonovits proved that the Ramsey-Turán function for even cliques experiences a jump. We show that the function RT(K_3,H,f(n)) has a similar behavior when H is an even clique. We also study the sparse analogue of a theorem of Bollobás and Gy\Ho ri about the maximum number of triangles that a C_5-free graph can have. Finally, we consider a Ramsey-Turán variant of a function studied by Erdős and Rothschild about the maximum number of edge-colorings that an n-vertex graph can have without a monochromatic copy of a given graph.
Combinatorics; Graph; Ramsey-Turan
Thu, 14 Jul 2016 00:00:00 GMThttp://hdl.handle.net/2142/930412016-07-14T00:00:00ZSharifzadeh, MaryamA construction of topological field theories
http://hdl.handle.net/2142/92901
A construction of topological field theories
Aramyan, Nerses
We give an explicit construction of extended topological field theories over a manifold taking values in the deloopings of U(1) from the data of differential forms on the manifold. More specifically, for a manifold M, using a version of the Dold-Kan construction, we create from the Deligne complex a Kan complex, |D_n|^+(M), and show that there is a natural map into (Fun)^⊗(Bord_n^or (M),B^nU(1)). The main theorem asserts that thismap becomes a weak equivalence after restricting to the framed bordisms. The construction is on the point- set level, so immediately can be refined to the setting where the Deligne complex is considered discretely, which is known as a model for the differential cohomology.
As a part of the proof we show that the geometric realization of the bordism (∞,n)-category is the n-fold delooping of the infinite loopspace of the Madsen-Weiss spectrum MTO(n). The direct attempts to generalize the Galatius-Madsen-Tillmann-Weiss proof of n = 1 case fail due the globularity restriction on the bordism (∞,n)-category. To overcome this, we use the abstract transversality argument of Bökstedt and Madsen, Gromov's theory of microflexible sheaves and Rezk's argument on realization fibrations.
topological field theories, cobordism hypothesis, Deligne complex, Dold-Kan construction
Tue, 21 Jun 2016 00:00:00 GMThttp://hdl.handle.net/2142/929012016-06-21T00:00:00ZAramyan, NersesA small presentation for Morava E-theory power operations
http://hdl.handle.net/2142/92811
A small presentation for Morava E-theory power operations
Nelson, Peter D.
Let E denote a Morava E-theory at a prime p and height h. We characterize the power operations on π0 of a K(h)-local E∞-E-algebra in terms of a small amount of algebraic data. This involves only the E-cohomology of two groups, namely the symmetric groups on p and p2 letters. Along the way, we also define and explore a notion of coquadratic comonad.
Power Operations; Morava E-Theory
Wed, 13 Jul 2016 00:00:00 GMThttp://hdl.handle.net/2142/928112016-07-13T00:00:00ZNelson, Peter D.Extremal problems on cycle structure and colorings of graphs
http://hdl.handle.net/2142/92769
Extremal problems on cycle structure and colorings of graphs
Santana, Michael L
In this Thesis, we consider two main themes: conditions that guarantee diverse cycle structure within a graph, and the existence of strong edge-colorings for a specific family of graphs.
In Chapter 2 we consider a question closely related to the Matthews-Sumner conjecture, which states that every 4-connected claw-free graph is Hamiltonian. Since there exists an infinite family of 4-connected claw-free graphs that are not pancyclic, Gould posed the problem of characterizing the pairs of graphs, {X,Y}, such that every 4-connected {X,Y}-free graph is pancyclic. In this chapter we describe a family of pairs of graphs such that if every 4-connected {X,Y}-free graph is pancyclic, then {X,Y} is in this family. Furthermore, we show that every 4-connected {K_(1,3),N(4,1,1)}-free graph is pancyclic. This result, together with several others, completes a characterization of the family of subgraphs, F such that for all H in ∈, every 4-connected {K_(1,3), H}-free graph is pancyclic.
In Chapters and 4 we consider refinements of results on cycles and chorded cycles. In 1963, Corrádi and Hajnal proved a conjecture of Erdös, showing that every graph G on at least 3k vertices with minimum degree at least 2k contains k disjoint cycles. This result was extended by Enomoto and Wang, who independently proved that graphs on at least 3kvertices with minimum degree-sum at least 4k - 1 also contain k disjoint cycles. Both results are best possible, and recently, Kierstead, Kostochka, Molla, and Yeager characterized their sharpness examples. A chorded cycle analogue to the result of Corrádi and Hajnal was proved by Finkel, and a similar analogue to the result of Enomoto and Wang was proved by Chiba, Fujita, Gao, and Li. In Chapter 3 we characterize the sharpness examples to these statements, which provides a chorded cycle analogue to the characterization of Kierstead et al.
In Chapter 4 we consider another result of Chiba et al., which states that for all integers r and s with r + s ≥ 1, every graph G on at least 3r + 4s vertices with ẟ(G) ≥ 2r+3s contains r disjoint cycles and s disjoint chorded cycles. We provide a characterization of the sharpness examples to this result, which yields a transition between the characterization of Kierstead et al. and the main result of Chapter 3.
In Chapter 5 we move to the topic of edge-colorings, considering a variation known as strong edge-coloring. In 1990, Faudree, Gyárfás, Schelp, and Tuza posed several conjectures regarding strong edge-colorings of subcubic graphs. In particular, they conjectured that every subcubic planar graph has a strong edge-coloring using at most nine colors. We prove a slightly stronger form of this conjecture, showing that it holds for all subcubic planar loopless multigraphs.
graphs; cycles; strong edge-colorings
Thu, 07 Jul 2016 00:00:00 GMThttp://hdl.handle.net/2142/927692016-07-07T00:00:00ZSantana, Michael LA new computation of the Bergman kernel and related techniques
http://hdl.handle.net/2142/92749
A new computation of the Bergman kernel and related techniques
Huo, Zhenghui
We introduce a technique for obtaining the Bergman kernel on certain Hartogs domains. To do so, we apply a differential operator to a known kernel function on a domain in lower dimensional space. We rediscover some known results and we obtain new explicit formulas. Using these formulas, we analyze the boundary behavior of the kernel function on the diagonal. Our technique also leads us to results about a cancellation of singularities for generalized hypergeometric functions and weighted Bergman kernels. Finally, we give an alternative approach to obtain explicit bases for complex harmonic homogeneous polynomial spaces.
Bergman Kernel; Bergman Projection; Boundary Behavior; Generalized Hypergeometric Function
Fri, 08 Jul 2016 00:00:00 GMThttp://hdl.handle.net/2142/927492016-07-08T00:00:00ZHuo, ZhenghuiK-theoretic Schubert calculus and applications
http://hdl.handle.net/2142/92740
K-theoretic Schubert calculus and applications
Pechenik, Oliver A
A central result in algebraic combinatorics is the Littlewood-Richardson rule that governs products in the cohomology of Grassmannians. A major theme of the modern Schubert calculus is to extend this rule and its associated combinatorics to richer cohomology theories.
This thesis focuses on K-theoretic Schubert calculus. We prove the first Littlewood-Richardson rule in torus-equivariant K-theory. We thereby deduce the conjectural rule of H. Thomas and A. Yong, as well as a mild correction to the conjectural rule of A. Knutson and R. Vakil. Our rule manifests the positivity established geometrically by D. Anderson, S. Griffeth and E. Miller, and moreover in a stronger 'squarefree' form that resolves an issue raised by A. Knutson. Our work is based on the combinatorics of genomic tableaux, which we introduce, and a generalization of M.-P. Schuetzenberger's jeu de taquin. We further apply genomic tableaux to obtain new rules in non-equivariant K-theory for Grassmannians and maximal orthogonal Grassmannians, as well as to make various conjectures relating to Lagrangian Grassmannians. This is joint work with Alexander Yong.
Our theory of genomic tableaux is a semistandard analogue of the increasing tableau theory initiated by H. Thomas and A. Yong. These increasing tableaux carry a natural lift of M.-P. Schuetzenberger's promotion operator. We study the orbit structure of this action, generalizing a result of D. White by establishing an instance of the cyclic sieving phenomenon of V. Reiner, D. Stanton and D. White. In joint work with J. Bloom and D. Saracino, we prove a homomesy conjecture of J. Propp and T. Roby for promotion on standard tableaux, which partially generalizes to increasing tableaux. In joint work with K. Dilks and J. Striker, we relate the action of K-promotion on increasing tableaux to the rowmotion operator on plane partitions, yielding progress on a conjecture of P. Cameron and D. Fon-der-Flaass. Building on this relation between increasing tableaux and plane partitions, we apply the K-theoretic jeu de taquin of H. Thomas and A. Yong to give, in joint work with Z. Hamaker, R. Patrias and N. Williams, the first bijective proof of a 1983 theorem of R. Proctor, namely that that plane partitions of height k in a rectangle are equinumerous with plane partitions of height k in a trapezoid.
Schubert calculus; K-theory; genomic tableau; cyclic sieving; homomesy; plane partition; resonance; doppelganger
Wed, 06 Jul 2016 00:00:00 GMThttp://hdl.handle.net/2142/927402016-07-06T00:00:00ZPechenik, Oliver A