Title: Stability for the multidimensional rigid body and singular curves

Date: 09/26/2016

Time: 2:30 PM - 3:20 PM

Place: C304 Wells Hall

A classical result of Euler says that the rotation of a torque-free 3-dimensional rigid body about the short or the long axis is stable, while the rotation about the middle axis is unstable. I will present a multidimensional generalization of this result and explain how it can be proved by studying line bundles over singular algebraic curves.

Title: Boundary value problem and the Ehrhard inequality.

Date: 09/26/2016

Time: 4:02 PM - 5:00 PM

Place: C517 Wells Hall

I will present a new proof of the Ehrhard inequality. In fact I will talk about a more general result and the Ehrhard inequality will be consequence of it. The idea of the method is similar to Brascamp-Lieb's approach to Prekopa-Leindler inequality via sharp reverse Young's inequality for convolutions. Indeed, we shall rewrite essential supremum as a limit of Lp norms but with very specially chosen test functions and measures. Next rewriting Lp norm by duality as a scalar product the question boils down to an estimate of double integral of compositions of test functions by the mass of these functions. To verify the last estimate which looks like Jensen's inequality we will use a subtle inequality, a ``modified Jensen's inequality', which in its turn boils down to the fact that a corresponding quadratic form has a definite sign, and this is the main technical part of the method. If time allows we will show that in the class of even probability measures with smooth strictly positive density Gaussian measure is the only one which satisfies the functional form of the Ehrhard inequality on the real line with their own distribution function.

Title: Affine Cluster Variables As Generalized Minors

Date: 09/27/2016

Time: 1:00 PM - 1:50 PM

Place: C304 Wells Hall

A fundamental result in the theory of cluster algebras due to Berenstein, Fomin, and Zelevinsky is the existence of (upper) cluster algebra structures on the coordinate rings of the double Bruhat cells in semi-simple algebraic groups, later this was extended to cover all Kac-Moody groups by Williams. In every case the initial cluster consists of a collection of generalized minors and generalizations of the classic Jacobi-Desnanot identity for minors of a matrix provide many of the exchange relations. However, most non-initial cluster variables are yet undocumented functions on the group and, in particular, they are usually expected not to be generalized minors. In this talk I will describe a special class of double Bruhat cells where all cluster variables turn out to be generalized minors and then discuss new identities among generalized minors resulting from these observations.

Speaker: Qing Nie, University of California, Irvine

Title: Data-Driven multiscale modeling of cell fate dynamics

Date: 09/27/2016

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

Fates of cells are not preordained. Cells make fate decisions in response to different
and dynamic environmental and pathological stimuli. Recently, there
has been an explosion of experimental data at various biological scales, including
gene expression and epigenetic measurements at the single cell level,
lineage tracing, and live imaging. While such data provide tremendous detail
on individual elements, many gaps remain in our knowledge and understanding
of how cells make their dynamic decisions in complex environments. In
addition to developing new models to analyze data at each scale, we are working
on multiscale modeling challenges in analyzing single-cell molecular data
(data-rich scale) and their connections with spatial tissue dynamics (datapoor
scale). Our approach requires new and challenging mathematical and
computational tools in machine learning, stochastic analysis and simulations,
and PDEs with moving boundaries. We then use our novel data-driven multiscale
modeling approach to uncover new principles for cell fate dynamics in
development, regeneration, and disease.

Let p = p_1 ... p_n be a permutation in the symmetric group S_n written as a sequence. The descent set of p is the set of indices i such that p_i > p_{i+1}. A classic result of MacMahon states the the number of permutations in S_n with a given descent set is a polynomial in n. But little work seems to have been done concerning the properties of these polynomials. The peak set of p is the set of indices i such that p_{i+1} < p_i > p_{i+1}. Recently Billey, Burdzy, and Sagan proved that the number of permutations in S_n with given peak set is a polynomial in n times a power of two. I will survey what is known about these two polynomials, including their degrees, roots, coefficients, and analogues for other Coxeter groups.

Title: The Riemann-Roch theorem and its application to group structures of a non-singular cubic curve in P^2

Date: 09/28/2016

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

A non-singular projective curve of degree 3 in
$\mathbb{P}^2_k$ and its point of inflection determines a unique additive group structure on the curve. The Riemann-Roch theorem
relates the dimension of the space of meromorphic functions on the curve with prescribed zeroes and poles to the genus of the curve.
Among many useful applications of the Riemann-Roch theorem, one can see the law of associativity using the degree-genus formula.

In this talk we will be discussing the knot concordance invariant Upsilon, derived from knot Floer homology. After introducing the definition and basic properties, we will discuss how the work of Hedden and Van Cott on tau-invariant of cable knots can be used to obtain a similar result in Upsilon, which is an inequality relating the Upsilon invariant of a knot and that of its cable. We will also see some applications of this result.

Title: Computing without subtracting (and/or dividing)

Date: 09/29/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Algebraic complexity of a rational function can be defined as the minimal number of arithmetic operations required to compute it. Can restricting the set of allowed arithmetic operations dramatically increase the complexity of a given function (assuming it is still computable in the restricted model)? In particular, what can happen if we disallow subtraction and/or division? This is joint work with D.
Grigoriev and G. Koshevoy.