Landau-Ginzburg (LG) models have proven very useful in the study of mirror symmetry for Fano varieties. An LG model for a space X consists of a mirror space Y and a rational function on Y encoding certain enumerative data associated to X. In the particular case of homogeneous spaces, such as the Grassmannians Gr(k,n), Rietsch gave a Lie-theoretic formulation of an LG model. It is often easier to work with an LG model defined directly in terms of coordinates on Y, and in joint work with Peter Spacek, we study a Plucker coordinate construction of Rietsch's LG models for cominuscule homogeneous spaces. Along the way, we discuss several interesting connections to the geometry and representation theory of homogeneous spaces.
Dynamical algebraic combinatorics explores actions on sets of discrete combinatorial objects, many of which can be built up by small local changes, e.g., Schützenberger's promotion and evacuation, or the rowmotion map on order ideals. There are strong connections to the combinatorics of representation theory and with Coxeter groups. Birational liftings of these actions are related to the Y-systems of statistical mechanics, thereby to cluster algebras, in ways that are still relatively unexplored.
The term "homomesy" (fka "combinatorial ergodicity") describes the following widespread phenomenon: Given a group action on a set of combinatorial objects, a statistic on these objects is called "homomesic" if its average value is the same over all orbits. Along with its intrinsic interest as a kind of "hidden invariant", homomesy can be used to prove certain properties of the action, e.g., facts about the orbit sizes. Homomesy can often be found among the same dynamics that afford cyclic sieving. Proofs of homomesy often involve developing tools that further our understanding of the underlying dynamics, e.g., by finding an equivariant bijection.
This talk will be an introduction to these ideas, giving a number of examples of such actions and pointing out connections to other areas.
From Zeno's paradoxes to quantum physics, the question of the continuous nature of our world has been prominent and remains unanswered. From a mathematical point of view, discrete structures or models behave quite differently from continuous ones. The great success story of mathematics from the 18-th century has been the development of analysis. Discrete mathematics had a later start, with a large boost from computers.
However, these worlds are not as far apart as they seem. Computers force us to approximate continuous structures by finite ones; but perhaps more surprisingly, very large finite structures can be very well approximated by continuous structures, often getting rid of inconvenient details. These approaches cross-fertilize each other.
We show that for any non-negative quaternary quartic form $f$ there exists a product of two non-negative quadrics $q$ so that $qf$ is a sum of squares (s.o.s.) of quartics. This is a much better upper bound on the degree of a multiplier needed to demonstrate nonnegativity of $f$ via an s.o.s. decomposition than known previously; similar almost tight bounds are only known for ternary forms.As a step towards deciding whether it is sufficient to use a quadratic multiplier $q$, we show that there exist non-s.o.s. non negative ternary sextics $ac-b^2$, with $a$, $b$, $c$ of degrees 2, 3, 4, respectively.
Biologists frequently need to reconcile conflicting estimates of the evolutionary relationships between species by taking a ‘consensus’ of a set of phylogenetic trees. This is because different data and/or different methods can produce different trees. If we think of each tree as providing a ‘vote’ for the unknown true phylogeny, then we can view consensus methods as a type of voting procedure. Kenneth Arrow’s celebrated ‘impossibility theorem’ (1950) shows that no voting procedure can simultaneously satisfy seemingly innocent and desirable properties. We consider a similar axiomatic approach to consensus and asks what properties can be jointly achieved.
In the second part of the talk, we consider phylogenetic networks (which are more general than trees as they allow for reticulate evolution). The question ‘when is a phylogenetic network merely a tree with additional links between its edges?’ is relevant to biology and interesting mathematically. It has recently been shown that such ‘tree-based’ networks can be efficiently characterized. We describe some further results related to Dilworth’s theorem for posets (1950), and matching theory in bipartite graphs. In this way, one can obtain fast algorithms for determining when a network is tree-based and, if not, to calculate how ‘close’ to tree-based it is.
We give an introduction to random walks in random media and explain some basic techniques (electric network techniques for the reversible case, regeneration times). We give some results about the following models:1. Sinai's walk as a (one-dimensional) model for metastability 2. Random walks with drift on percolation clusters 3. Branching random walks in random environments.
We give an introduction to random walks in random media and explain some basic techniques (electric network techniques for the reversible case, regeneration times). We give some results about the following models:1. Sinai's walk as a (one-dimensional) model for metastability 2. Random walks with drift on percolation clusters 3. Branching random walks in random environments.
We give an introduction to random walks in random media and explain some basic techniques (electric network techniques for the reversible case, regeneration times). We give some results about the following models:1. Sinai's walk as a (one-dimensional) model for metastability 2. Random walks with drift on percolation clusters 3. Branching random walks in random environments.
Quantum Markov processes can be obtained from the quantized version of a road-coloured graph. This allows to bring in ideas from coding theory as well as from scattering theory. In particular, asymptotic completeness of a Markov process is closely related to the possibility of preparing states of an open system by prepararing its input states but it also allows a certain coding of the Markov process.
In this talk we give an overview over recent developments in this area. We discuss asymptotic completeness under topological and measure theoretical aspects, present new criteria for asymptotic completeness, which, in particular, cover the micro-maser system, and put asymptotic completeness into the context of coding theory.
Time permitting, a discussion of the implications of localization for Witten's (formal) representation of the Donaldson invariants as expectation values in TQFT
Rephrase the discrete S'-Localization theorem in terms of the equivariant Euler class. In this form the localization theorem extends to the nondiscrete case.
A Quick introduction to the Cartan model of equivariant cohomology, a statement of the discrete equivariant localization theorem and the proof of "equivariant localization implies (generalized) Duistermaat-Heckman"
Here I will try to describe the motivation behind the lecture series by discussing, quite informally, Cartan's generalization of the Chern-Weil construction of characteristic classes in equivariant cohomology, the Mathai-Quillen repesentatives of the Thom and Euler classes, the phenomenon of localization and a formal extension of these finite-dimensional theorems by Atiyah-Jeffrey to an infinite-dimensional context related to the Donaldson invariants and Witten'S (1988) topoloigical quantum field theory. The remaining lectures will provide a few more detials on the localization theorems.
The interaction between the geometric topology of low dimensional manifolds and classical and quantum field theories began with a few special results in the early 1980s. It has since developed into a very active area of research in both mathematics and physics. In Lecture I, we will review the early and by now well established results for dimensions 3 and 4. In the second lecture we will discuss the various homology theories associated to knots and links in 3-manifolds. If time permits, we will also consider the relation between WRT invariants and topological string amplitudes.
For some background, see my MPI webprints 45/2004, 53/2000, and references there.
The interaction between the geometric topology of low dimensional manifolds and classical and quantum field theories began with a few special results in the early 1980s. It has since developed into a very active area of research in both mathematics and physics. In Lecture I, we will review the early and by now well established results for dimensions 3 and 4. In the second lecture we will discuss the various homology theories associated to knots and links in 3-manifolds. If time permits, we will also consider the relation between WRT invariants and topological string amplitudes.
For some background, see my MPI webprints 45/2004, 53/2000, and references there.
This paper consists of two parts. The first one studies integral relations to which the solutions of the Navier-Stokes equations or Euler equations satisfy in the case of fluids filling the entire three-dimensional space. The existence of these relations is due to a rapid decrease of the velocity field at infinity (but not too rapid in order that the required asymptotic forms are reproduced with time). Of special interest are the integrals of motion whose density depends quadratically on the velocities or their derivative with respect to the coordinates. Such integrals (conservation laws) for the Navier-Stokes equations were recently found by Dobrokhotov and Shafarevich. In the present paper, new conservation laws are obtained, which are quadratic in the derivatives of the velocity and lead to identities that link the averaged and pulsation characteristics of free turbulent flows.
The second part is devoted to equations of rotationally symmetric motion of a viscous incompressible liquid. There is proposed a new approach to investigation of such kind of motions. It is shown that the projection of momentum equation on the axis of cylindrical coordinate system has a form of conservation law. This gives possibility to introduce a new unknown function instead of the pressure. The resulting system consists of an elliptic equation and two second-order parabolic equations for the stream function and peripheral component of velocity, which is weakly connected with the elliptic equation. Moreover, in comparison with traditional approach, boundary conditions for all sought functions are separated completely. The obtained system has a very rich group of symmetries. On the base of this group, the new exact solutions of the Navier-Stokes equations are constructed.
References:V.V.Pukhnachov. Integrals of motion of an incompressible fluid occupying the entire space. Journal of Applied Mechanics and Technical Physics, Vol. 45, No. 2, pp. 167-171, 2004.S.N.Aristov, V.V.Pukhnachov. On equations of the rotationally symmetric motion of a viscous incompressible liquid. Doklady Akademii Nauk, Vol. 394, No.5, pp. 611-614, 2004.
After explaining the basics of convenient Calculus in infinite dimensions (as in Kriegl-Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997. 618 pages) I shall present a Frobenius theorem (due to Teichmann) for finitely many vector fields admitting flows and derive the existence of flows for combinations of PDE's of the type of KdV, Burgers', Camassa-Holm, etc.