Colloquia scheduled for Tuesdays and Thursdays take place at 4:30-5:30 pm in LN 2205 with refreshments served from 4:00-4:25 pm in the Anderson Memorial Reading Room, while colloquia scheduled for Mondays, Wednesdays and Fridays normally take place at 4:40-5:40 pm in LN 2205 with refreshments served from 4:10-4:35 pm in the Anderson Memorial Reading Room.
Unless stated otherwise, colloquia are scheduled for Thursdays.
Here you find some directions to Binghamton University and the Department of Mathematical Sciences.
DATE: Monday, January 17, 2011 (Special Date)
Speaker:
Nam
Quang Le (Columbia
University)
Title:
Regularity results for the mean curvature flow
Time:
4:40 - 5:40 pm
Room: LN-2205
Abstract: Mean curvature flow is the evolution of a hypersurface moving with normal velocity equal to its mean curvature vector.
In this talk, we will present some recent regularity results on the mean curvature flow. We will show that if the flow is a type I flow then the mean curvature controls the flow in the sense that the blow up of the second fundamental form cannot occur if the mean curvature is uniformly bounded. In the case of surfaces, we will show that the mean curvature controls the flow provided that either the Multiplicity One Conjecture holds or the Gaussian density is less than two. When the mean curvature of our flow blows up, we will also give its (sharp) blow-up rate.
DATE: Friday, January 21, 2011 (Special Date)
Speaker:
Brian Weber (Courant
Institute of Mathematical Sciences)
Title:
F-structures and the Geometry of Collapse
Time:
3:00pm- 4:00 pm (followed by coffee/tea in the Anderson Reading
Room)
Room: LN-2205
Abstract: Collapsing in differential geometry, the phenomenon of a manifold `appearing' to lose a dimension, interferes with our geometric understanding of everything from geometric flows to string theory. At root, the problem is that the analytic techniques (for instance Moser iteration and Lipschitz convergence) we use to understand Riemannian manifolds become less precise as injectivity radii degenerate, and lose all meaning in the limit. However, as disks provide local models for Riemannian manifolds with controlled geometric characteristics, F-structures (and N-structures, their cousins) provide local models for collapsing manifolds. Interest in these structures is on the rise, in part due to their role in powerful new techniques for 4- manifolds with canonical geometries (manifolds with Einstein, extremal Kaehler, or half-conformally flat metrics). In these cases the interaction between curvature and topology is particularly strong, and F-structures are indispensable in the taming of their structures. In this talk I will describe F-structures themselves, discuss some new techniques and their use in studying the structure of manifolds with canonical metrics (including new work on extremal Kaehler metrics I have been involved in), and indicate directions for further progress in this rich field.
DATE: Tuesday, January 25, 2011
Speaker:Erik
Van Erp (Dartmouth
College)
Title:
Noncommutative techniques in topology
Time:
4:40 - 5:40 pm
Room: LN-2205
Abstract: I discuss how the methods of noncommutative geometry can be applied to study questions in "classical" topology or geometry. I will focus on the solution of a specific topological problem (an index problem) to illustrate the interplay between noncommutative algebras and topological spaces. While the final solution of the problem can be stated in purely topological terms, all the steps in the proof rely crucially on the use of noncommutative algebras. Specifically, there exists a cohomology theory (K-theory) that applies to both categories. This provides a powerful vehicle for going back and forth between the two.
DATE: Monday Feb. 7, 2011 (Special Date)
Speaker:Rustam
Sadykov (University of Toronto)
Title:
Homotopy theoretic approach to solving differential relations.
Time:
4:40 - 5:40 pm
Room: LN-2205, preceded by
coffee etc. in Anderson Room
Abstract: The h-principle is a general observation that differential geometry problems can often be reduced to problems in (unstable) homotopy theory. For example, by a general theorem of Gromov, the existence problems for
-- a symplectic/contact structure,
-- a foliation of a given codimension, and
-- a Riemannian metric of positive/negative scalar curvature
on an given open manifold can be reduced to homotopy theory problems (which can be approached by standard homotopy theory methods).
Similarly, the b-principle is a general observation that differential geometry problems can often be reduced to problems in stable homotopy theory. The known instances of the b-principle type phenomena include the Barratt-Priddy-Quillen theorem, Eliashberg theorem on Legendrian immersions, and the Mumford conjecture/Madsen-Weiss theorem on moduli spaces of Riemann surfaces. I show that the b-principle phenomenon occurs in a fairly general setting. As an application I determine new invariants of singular sets of smooth maps. Surprisingly these invariants can be non-trivial and well defined even for singular sets that possess no fundamental class.
DATE: Wednesday Feb. 9, 2011 (Special Date)
Speaker:
R. Inanc Baykur
(Brandeis University)
Title: Smooth
four-manifolds, surgeries along tori, and exotica
Time:
4:40 - 5:40 pm
Room: LN-2205, preceded by
coffee etc. in Anderson Room
Abstract: In this talk, we will demonstrate the novel role of surgeries along embedded tori in four-manifolds both in
(1) producing new infinite families of pairwise non-diffeomorphic four-manifolds within the same homeomorphism class, and in
(2) relating homeomorphic but not diffeomorphic four-manifolds.
Meanwhile, we are going to unfold the strong affiliation of round handles with smooth four-manifolds.
DATE: Thursday Feb. 10, 2011
Speaker:Jose-Manuel
Gomez (University of British Columbia)
Title:
Stable splittings, commuting and almost commuting elements in Lie
groups
Time: 4:30 - 5:30 pm
Room:
LN-2205, preceded by coffee etc. in Anderson Room
Abstract: The goal of this talk is to study the space of commuting elements in Lie groups through a homotopical point of view. I will show how this study naturally leads to the concept of almost commuting elements. Then I will explain a stable splitting that holds for these spaces. Some explicit computations will be provided.
DATE: Tuesday March 1st, 2011 (Special
Date)
Speaker:Magdalena
Czubak, University of Toronto
Title:
Non-uniqueness of the Navier-Stokes equation in the hyperbolic
setting
Time: 4:30 - 5:30 pm
Room:
LN-2205, preceded by coffee etc. in Anderson Room
Abstract: The Navier-Stokes equation is one of the fundamental equations of fluid mechanics. Due to the work of Leray in 1930's and Hopf in 1950's, we have the Leray-Hopf weak solutions, which are of crucial importance in the study of the Navier-Stokes equation. In particular, the Leray-Hopf solutions are known to be unique in two dimensions in the euclidean setting. After some background, we consider the Navier-Stokes equation on a two dimensional hyperbolic space and prove non-uniqueness of smooth solutions of the Leray-Hopf type. As a corollary we show a lack of the Liouville theorem in the hyperbolic setting both in two and three dimensions. This is joint work with Chi Hin Chan.
DATE: Thursday March 3rd, 2011
Speaker:
Nathaniel
Eldredge (Cornell University)
Title:
Sub-Riemannian geometry and hypoelliptic heat kernels
Time:
4:30 - 5:30 pm
Room: LN-2205, preceded by
coffee etc. in Anderson Room
Abstract: Hypoelliptic operators occupy an interesting boundary region between "nice" elliptic operators on one side and utter degeneracy on the other. I will begin with some elementary examples in sub-Riemannian geometry, which is a natural context for such objects, then describe some operators of interest from analytic and probabilistic viewpoints. I will conclude by describing some results on estimates for heat kernels in this setting.
DATE: Friday March 4th, 2011
(Special Date)
Speaker: Ye
Li (Duke University)
Title:
Penrose Inequality in General Relativity: Quantifying the
Distribution of Mass in Universe
Time:
4:40 - 5:40 pm
Room: LN-2205, preceded by
coffee etc. in Anderson Room
Abstract: In 1973, Penrose presented an argument that the total mass of a spacetime which contains black holes with event horizons of total area A should be bounded from below by a constant depending on A. The failure of this inequality implies the failure of Cosmic Censorship Conjecture, which says that God abhors naked singularity. The existence of naked singularity leads to the fact that general relativity could not be a deterministic theory. In this talk, I will introduce a mixed flow approach to the Penrose inequality in spacetime and discuss the progress we've made. This is a joint work with Hubert Bray.
April
28, 2011 (Dean's Speaker Series in Geometry/Topology)
Speaker:
Eyal Goren (McGill University)
Title: Recent
developments in the theory of complex multiplication
Time:
4:30 - 5:30 pm
Room: LN-2205
Abstract:
Our story begins more than a century ago with Kronecker's
Jugendtraum and Hilbert's 12th problem, where complex multiplication
appears as a way to understand Galois extensions of number fields -
a problem still at the heart of number theory. Our lecture will have
a strong historical flavour; we shall attempt to survey the
development of the theory of complex multiplication and the
philosophy behind it. On this background, we will present some
exciting recent results, some of which build in an essential way on
Borcherds' theory. Finally, we shall sketch some of the key
challenges of the theory of complex multiplication today and future
directions.
May 5, 2011 (Dean's Speaker Series in Geometry/Topology)
Speaker: Rostuslav Grigorchuk (Texas A&M University)
Title: To be announced
Time: 4:30-5:30 pm
Room: LN-2205
Abstract: This will be a survey of known (and may be not very known) results, with discussion of some open problems.
May 12, 2011 (Dean's Speaker Series in Geometry/Topology)
Speaker: Michael Larsen (Indiana University )
Title: To be announced
Time: 4:30-5:30 pm
Room: LN-2205
Abstract: Every word w in the
free group on d generators defines, for every group G, a natural
function $G^d\to G$, called a "word map". Typically, when
G is a simple group, word maps have large image, and often they are
surjective. I will talk about recent progress in our understanding
of word maps on finite simple groups.