Many problems of spherical discrete and metric geometry may be reformulated as energy minimization problems and require techniques that stem from harmonic analysis, potential theory, optimization etc. We shall discuss several such problems as well of applications of these ideas to combinatorial geometry, discrepancy theory, signal processing etc.
The likelihood of HIV infection following risky contact is believed to be low. This suggests that the infection process is stochastic and governed by rare events. I will present mathematical branching process models of early infection and show how we have used them to gain insights into the duration of the undetectable phase of HIV infection, the likelihood of success of pre- and post-exposure prophylaxis, and the effects of prior infection with HSV-2. Although I will describe quite a bit of theory, I will try to keep giant and incomprehensible formulae to a minimum.
Khovanov homology and knot Floer homology are two knot invariants which are defined using very different techniques, with Khovanov homology having its roots in representation theory and knot Floer homology in symplectic geometry. However, they seem to contain a lot of the same topological data about knots. Rasmussen conjectured that this similarity stems from a spectral sequence from Khovanov homology to knot Floer homology. In this talk I will give a construction of this spectral sequence. The construction utilizes a recently defined knot homology theory HFK_2 which provides a framework in which the two theories can be related.
If a finite group $G$ acts on a Cohen-Macaulay ring $A$, and the order of $G$ is a unit in $A$, then the invariant ring $A^G$ is Cohen-Macaulay as well, by the Hochster-Eagon theorem. On the other hand, if the order of $G$ is not a unit in $A$ then the Cohen-Macaulayness of $A^G$ is a delicate question that has attracted research attention over the last several decades, with answers in several special cases but little general theory. In this talk we show that the statement that $A^G$ is Cohen-Macaulay is equivalent to a statement quantified over the inertia groups for the action of G$ on $A$ acting on strict henselizations of appropriate localizations of $A$. In a case of long-standing interest—a permutation group acting on a polynomial ring—we show how this can be applied to find an obstruction to Cohen-Macaulayness that allows us to completely characterize the permutation groups whose invariant ring is Cohen-Macaulay regardless of the ground field. This is joint work with Sophie Marques.
We present a convenient joint generalization of mixing and the local
central limit theorem which we call MLLT. We review results on the MLLT
for hyperbolic maps and present new results for hyperbolic flows. Then
we apply these results to prove global mixing properties of some
mechanical systems. These systems include various versions of the
Lorentz gas (periodic one; locally perturbed; subject to external
fields), the Galton board and pingpong models. Finally, we present
applications to random walks in deterministic scenery. This talk is
based on joint work with D. Dolgopyat and partially with M. Lenci.
We consider a triangular gap of side two in a 90 degree angle on the triangular lattice with mixed boundary conditions: a constrained, zig-zag boundary along one side, and a free lattice line boundary along the other. We study the interaction of the gap with thecorner as the rest of the angle is completely filled with lozenges. We show that the resulting correlation is governed by the product of the distances between the gap and its three images in the sides of the angle. This, together with a few other results we worked out previously, provides evidence for a unified way of understanding the interaction of gaps with the boundary under mixed boundary conditions, which we present as a conjecture. Our conjecture is phrased in terms of the steady state heat flow problem in a uniform block of material in which there are a finite number of heat sources and sinks. This new physical analogy is equivalent in the bulk to the electrostatic analogy we developed in previous work, but arises as the correct one for the correlation with the boundary.The starting point for our analysis is an exact formula we prove for the number of lozenge tilings of certain trapezoidal regions with mixed boundary conditions, which is equivalent to a new, multi-parameter generalization of a classical plane partition enumeration problem (that of enumerating symmetric, self-complementary plane partitions).
Problems from enumerative geometry have Galois groups. Like those from field extensions, these Galois groups reflect the internal structure of the original problem. The Schubert calculus is a class of problems in enumerative geometry that is very well understood, and may be used as a laboratory to study new phenomena in enumerative geometry.I will discuss this background, and sketch a picture that is emerging from a sustained study of Schubert problems from the perspective of Galois theory. This includes a conjecture concerning the possible Schubert Galois groups, a partial solution of the inverse Galois problem, as well as glimpses of the outline of a possible classification of Schubert problems for their Galois groups.
I will introduce briefly the notion of Berkovich analytic spaces and certain metric graphs associated to them called the skeleton. Then we will describe divisors on metric graphs and a lifting theorem that allows us to find tropicalizations of curves in P^3. This is joint work with Philipp Jell.