MOTIVATION Haplotypes play a crucial role in genetic analysis and have many applications such as gene disease diagnoses, association studies, ancestry inference and so forth. The development of DNA sequencing technologies makes it possible to obtain haplotypes from a set of aligned reads originated from both copies of a chromosome of a single individual. This approach is often known as haplotyp...

Let V be the standard two-dimensional representation of the algebraic group G = SL(2,C), and write Vn = SymnV for the irreducible (n+1)-dimensional representation of G on the nth symmetric tensor power of V . Also write Wn = V ⊗n for the nth tensor power of V (which has dimension 2n). Then, as virtual representations, it is known that each Vn can be written in terms of W0, . . . , Wn as Vn = Wn...

In the course of evolution, genes traverse the nucleotide sequence space, which translates to a trajectory of changes in the protein sequence in protein sequence space. The correspondence between regions of the nucleotide and protein sequence spaces is understood in general but not in detail. One of the unexplored questions is how many sequences a protein can reach with a certain number of nucl...

We define S1-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo. We show that it is related to the usual symplectic homology by a Gysin exact sequence. As an important ingredient of the proof, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian func...

A function complex is introduced on the category of flows so that the model category of flows becomes a simplicial model category. This allows us to show that the homotopy branching space of the cone of a morphism of flows is homotopy equivalent to the cone of its image by the homotopy branching space functor. The crux of the proof is that the homotopy branching space of the terminal flow is co...

We construct a long exact sequence involving the homology of an FI-module. Using the long exact sequence, we give two methods to bound the Castelnuovo–Mumford regularity of an FI-module which is generated and related in finite degree. We also prove that for an FImodule which is generated and related in finite degree, if it has a nonzero higher homology, then its homological degrees are strictly...

These notes are based on the Number Theory seminar I gave at Purdue University on September 19, 2013 titled “Hilbert’s theorem 90 and generalization”. The proof of Hilbert’s 90 is taken from an answer I found on MathOverflow at http://mathoverflow.net/a/21117. In the seminar, Professor Goins asked an interesting question about the isomorphism classes of rank one tori which I have blogged here. ...

The classical Stallings-Stammbach five-term exact sequence associated with a short exact sequence of groups is shown to exist in a general categorical context. This makes it possible to establish a new result relating low-dimensional homology and central series, which extends a well known result from the theory of groups.

We develop an analogue of the Birman exact sequence for the Torelli subgroup of AutpFnq. This builds on earlier work of the authors who studied an analogue of the Birman exact sequence for the entire group AutpFnq. These results play an important role in the authors’ recent work on the second homology group of the Torelli group.

The identification of sequence motifs is a fundamental method for suggesting good candidates for biologically functional regions such as promoters, splice sites, binding sites, etc. We investigate the following approach to identifying motifs: given a collection of orthologous sequences from multiple species related by a known phylogenetic tree, search for motifs that are well conserved (accordi...