MSCS Seminar Calendar

The Hodge numbers represent the dimensions of certain cohomology spaces (the Dolbeault cohomology) and are often used in the classification theory of complex manifolds. In this talk we will review inequalities involving Hodge numbers, starting from the classical ones and continuing through to the most recent ones. We will introduce all the notions necessary to the understanding of the problem and plenty of time will be devoted to examples. Though all are welcome to attend, this talk is intended to be accessible for graduate students of all levels; no background knowledge of Hodge theory is expected.

The theory of free knots/links invented by V. O. Manturov is a simplification of virtual knots theory and is useful in the study of various knots invariants. The Markov theorem gives precisely the condition when two braids have isotopic closure. We will present a Markov theorem of free links (http://arxiv.org/abs/1112.4061). This joint work with V. O. Manturov is proved using the L-move method introduced by Kauffman and Lambropoulou in "virtual braids and the L-move" in 2006.

We discuss the first two sections of Laskowski's a simpler axiomatization of the Shelah-Spencer almost sure theories.

Abstract: Let K be a proper (i.e., closed, pointed, full and convex) cone in $R^n$. We consider an n by n matrix $A$ which is K-primitive. That is, there exists a positive integer $l$ such that $A^l x$ is in interior of $K$ for every nonzero $x$ in $K$. The smallest such $l$ is called the exponent of A, denoted by $\gamma(A)$. For a polyhedral cone K, the maximum value of $\gamma(A)$, taken over all K-primitive matrices A, is denoted by $\gamma(K)$. Our main result is that for any positive integers $m, n$ such that $3 \le n \le m$, the maximum value of $\gamma(K)$, as K runs through all n-dimensional polyhedral cones with m extreme rays, equals $(n - 1)(m - 1) + 0.5(1+(-1)^{(n-1)m})$. We will consider various uniqueness issues related to the main result as well as its connections to known results. This talk is based on a joint work with Micha Perles and Bit-Shun Tam.

Study of measuring agreement is mainly aimed to answer one question, whether the readings from one instrument/method agree with the ones from another instrument/method. In this talk, we are going to present a general method to assess agreement for a wide range of data types with repeated measurements using linear and generalized linear mixed models. Likelihood-based approaches are developed to estimate all the within- and between-instrument agreement statistics. and asymptotic properties of these agreement estimates are discussed for different data structures. Furthermore, our method has the merit of handling missing values and covariates naturally. And a new set of restricted agreement statistics is proposed in order to capture the true random variations and between-instrument effects rather than the covariate effects. Simulations and several case studies, involving method comparison and bioequivalence, are used to show the accuracy and effectiveness of our method.

A classical way to study a line bundle L is to analyze the map defined by its sections. I will show how to construct a map that instead reflects the numerical properties of L. This map is in many ways better behaved; in particular, it has interesting ramifications for the minimal model program.

Continuing from last week, this talk will describe work of Satoh and of Rourke on the relationship between welded links and embeddings of tori in four-space.

I will discuss recent results concerning the analysis of incompressible fluid flows at very low viscosity in the presence of walls. Dating back to the pioneering work of Prandtl, such flows are modeled as inviscid away from the walls, but near the walls viscosity effects cannot be neglected, which give rise to a boundary layer where the flow is potentially violent and vorticity is created. Understanding the viscous boundary layer is of fundamental importance in many physical phenomena, for instance it creates lift around flying objects. A rigorous analysis of boundary layers is still lacking except in special cases. I will present a few classes of flows where such analysis is possible.

Vertebrate hemoglobin and myoglobin were the first proteins to have their structures and sequences determined about 50 years ago. In the subsequent pregenomic period, numerous related proteins came to light in plants, invertebrates, microbial eukaryotes and bacteria, that shared the highly conserved secondary structure consisting of eight $\alpha$-helices A-H. The myoglobin fold is described as a 3-on-3 $\alpha$-helical sandwich forming a hydrophobic cavity, within which the axial positions of the bound heme group are coordinated to the side-chain groups of residues within the E and F helices. It is underpinned by a consistent set of over 30 conserved amino acid residues, and is preserved even in cases of <20% identity. Some 15 years ago, a truncated 2-on-2 $\alpha$-helical fold with a shortened or missing helix A and a loop substituting for most of helix E, was observed in algal, ciliate and some plant and bacterial globins. The list of known globins was greatly expanded by the rapid accretion of genomic information over the last 15 years, demonstrating their existence in all three domains of life, ranging from close to 100% in multicellular eukaryote genomes, to about 60% bacterial and 10% of archaeal genomes. In bacteria, all globins occur in three families: the F (flavohemoglobin) and S (sensor) families that exhibit the canonical 3/3 $\alpha$-helical fold, and the T (truncated 3/3 fold) globins characterized by the abbreviated 2/2 $\alpha$-helical fold. Chimeric and single-domain globins are found in all three globin families. Vertebrate globins now include in addition to the familiar $\alpha$- and $\beta$-globins and myoglobins, the equally ubiquitous neuroglobins and cytoglobins, and several additional globin lineages with a more limited distribution, such as GlbY, GlbX and GbE. Very recently, we have found a new metazoan globin lineage comprising large, ca. 1600 residues, chimeric proteins with an N-terminal cysteine protease domain and a central globin domain, named androglobins, because of their specific expression in testis tissue (Hoogewijs et al., 2011). All metazoan globins, including symbiotic and nonsymbiotic plant globins and many globins found in microbial eukaryotes have the 3/3 $\alpha$-helical fold and have sequences that homologous to the F family globins. T family group 1 and 2 globins occur in microbial eukaryotes (ciliates, stramenopiles, oomycets, opisthokonts, etc.) and in plants. Fungi are unique in having F and S family globins. We have proposed that eukaryote globins evolved from the respective bacterial lineage via horizontal gene transfer resulting from one or both of the accepted endosymbiotic events responsible for the origin of mitochondria and chloroplastids, involving an $\alpha$-proteobacterium and a cyanobacterium, respectively (Vinogradov et al., 2007). Within this framework, it appears evident that the F family globins that had one or more enzymatic functions in the early bacteria, evolved in multicellular eukaryotes to have new properties, including reversible binding of important diatomic ligands, namely oxygen, nitric oxide and sulfide, that permitted the development of transport and storage functions (Vinogradov and Moens, 2008). Furthermore, it is also clear that the evolutionary success of the F family has far outstripped that of the other two globin families. Currently, we are engaged in unraveling the evolutionary history of globins in metazoans. The first step has been the identification of at least four distinct globin paralogs in the deuterostome common ancestor (Hoffmann et al., 2012).

I will introduce the Cayley graph, which is a geometric tool for studying groups. I will give an explicit example showing a connection between the coarse geometric properties of the Cayley graph and the algebraic properties of the group. Though all are welcome to attend, this talk is intended to be accessible for graduate students of all levels; no background knowledge of geometric group theory is expected.

We discuss section three of Laskowski's "A simpler axiomatization of the Shelah-Spencer almost sure theories".

Section rings of arbitrary line bundles on toric varieties are polytopal semigroup rings and thus always finitely generated. A related question is whether the section ring of the Serre line bundle on the projectivization of a toric vector bundle is always finitely generated. It turns out that this is not the case. We show this by finding toric vector bundles whose Cox ring is a polynomial ring over the Cox ring of the blow up of points in projective space. The latter is well known not to be finitely generated in general. This is joint work with José González, Sam Payne and Hendrik Süss.

We will discuss three-dimensional water wave models in situations where the wave motion does not tend to zero as a lateral variable tends to infinity. Local and long-time well-posedness results will be presented.

Among all existing alignment-free methods for comparing biological sequences, the sequence graphical representation provides a simple approach to view, sort, and compare gene structures. The aim of graphical representation is to display DNA or protein sequences graphically so that we can easily find out visually how similar or how different they are. Of course, only the visual comparison of sequences is not enough for the follow-up research work. We need more accurate comparison. This leads us to develop the application of the graphical representation for biological sequences. I will talk about two contributions for this direction. (1) We construct a protein map with the help of our proposed new graphical representation for protein sequences. Each protein sequence can be represented as a point in this map, and cluster analysis of proteins can be performed for comparison between the points. This protein map can be used to mathematically specify the similarity of two proteins and predict properties of an unknown protein based on its amino acid sequence. (2) We construct a novel genome space with biological geometry, which is a subspace in R^N. In this space each point corresponds to a genome. The natural distance between two points in the genome space reflects the biological distance between these two genomes. The genome space will provide a new powerful tool for analyzing the classification of genomes and their phylogenetic relationships.

I will present a construction of a nef divisor for every moduli space of Bridgeland stable complexes on an algebraic variety. In the case of K3 surfaces, we can use it to prove projectivity of the moduli space, generalizing a result of Minamide, Yanagida and Yoshioka. Its dependence on the stability condition gives a systematic explanation for the compatibility of wall-crossing of the moduli space with its birational transformations; this phenomenon had first been observed by Arcara-Bertram. This is based on joint work with Emanuele Macrì.

Matching theory is a large field with many directions of research, both in practical algorithms and combinatorial theory. In this talk I will aim to show some of the breadth of the subject, and some recent advances on matching theory for hypergraphs (joint work with Richard Mycroft). Informally speaking, we show that the obstructions to perfect matchings are geometric, and are of two distinct types: `space barriers' from convex geometry, and `divisibility barriers' from arithmetic lattice-based constructions. We apply our theory to the solution of two open problems on hypergraph packings: the minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemeredi Theorem.

de la Garza Phenomenon relates to the Information Matrix in the context of a standard Gauss-Markov Linear Model. It works well in the framework of approximate or continuous designs. For discrete designs, one has to be careful in extracting its full spirit. We propose to discuss some features of this highly fascinating area of research.

Much progress has been made in recent years in existence theory for initial value problems in interfacial fluid dynamics. We will introduce two other existence problems: the problem of global weak solutions for interfacial flows with surface tension, and the problem of time-periodic interfacial flows. We will report on progress for these problems, which includes both analytical and numerical work. This is joint work with Milton Lopes Filho, Helena Nussenzveig Lopes, Walter Strauss, and Jon Wilkening.

In family studies with multiple continuous phenotypes, we are interested in finding linear combinations of the phenotypes with large heritabilities, which can be considered as new phenotypes for genetic analysis. The problem can be recast as linear discriminant analysis (LDA). When the number of phenotypes is large, LDA is not appropriate for two reasons: the standard estimate for the within-family covariance matrix is singular, and it is difficult to interpret the newly defined phenotypes. Here we propose a novel version of penalized LDA, with an $L_1^2$ penalty in the denominator of the Rayleigh quotient. Besides overcoming the above two problems, the proposed method has at least three advantages compared with the existing regularization methods. First, it solves the singularity problem and achieves the sparsity property simultaneously. Second, the method is scale-invariant. Third, the consistency can be proved. We evaluate the performances of the method using simulations and two family studies.

We show that the self-intersection of a homology class defined by an unbounded averaging sequence for a foliation $\mathcal F$ of a lamination $\Lambda$ embedded in a compact manifold M always vanishes. The leaves of the lamination $\Lambda$ are assume to be smoothly embedded submanifolds of M, but no assumption on the transverse differentiability of the holonomy maps for $\mathcal F$ is required. The main result has applications to the study of Anosov diffeomorphisms.

The concentration of measure phenomenon and logarithmic Sobolev inequalities are closely related. In this talk, I will present some recent results in this direction for stochastic differential equations(SDEs) driven by fractional Brownian motions. In particular, as a consequence of the concentration property, we obtain a Gaussian upper bound for the density of solution to such SDEs. The presentation is based on a joint work with F. Baudoin and S. Tindel.

TBA

The F-signature of a local ring R of characteristic p > 0 is a real number which reflects the severity of the singularities of R. It was introduced explicitly by C. Huneke and G. Leuschke building upon work of K. Smith and M. Van den Bergh. In this talk, I will explain some of its history, and its recent generalization to the context of pairs. I will also explain connections to the minimal log discrepancy and how the F-signature might be useful for studying geometric questions in the future. Most of what is discussed is joint work with Manuel Blickle and Kevin Tucker.