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| Monday 22 |
3:10-4 pm, room 1432. Graph Theory and Combinatorics Seminar. Mike Plummer, Vanderbilt University. Recent progress in matching extension. Let G be a graph with at least 2n+2 vertices, where n is a
non-negative integer. The graph G is said to be n-extendable if every
matching of size n in G extends to (i.e., is a subset of) a perfect
matching. The study of this concept began in earnest in the 1980's,
although it was born out of the study of canonical matching decompositions
carried out in the 1970's and before. As is often the case, in retrospect
it is apparent that there are roots of this topic to be found even earlier. In the present talk, we will begin with a brief history of the subject
and then concentrate on reviewing results on n-extendability and closely
related areas obtained in the last ten - fifteen years. The areas to be
discussed include restricted matching extension, bricks and braces,
matching extension in embedded graphs and Pfaffian graphs. (This talk was given originally at the 60th birthday conference for
Laszlo Lovasz in Budapest in August of this year.)
4:10-5 pm, room 1431. NCGOA Research Training Group Seminar. Vladimir Chaynikov, Vanderbilt University. Introduction to K_0 for rings. This lecture will be based on Section 1 of Milnor's Introduction to Algebraic K-theory. |
| Tuesday 23 |
4:10-5 pm, room 1432. Noncommutative Geometry Seminar. Mrinal Raghupathi, Vanderbilt University. Constrained Nevanlinna-Pick Interpolation. Given points $z_1,\ldots,z_n, w_1,\ldots,w_n$ in the unit disk, the Nevanlinna-Pick theorem gives a characterization for the existence of a holomorphic map $f$ that maps the disk to the disk and "interpolates" $z_i$ to $w_i$. Problems of this type are usually called Nevanlinna-Pick problems. In this talk I will provide some background and describe the classical setting. I will then look at a simple variation on the original problem that arises from studying the interpolation problem for curves embedded in the bidisk. If time permits I will talk about the interpolation problem on Riemann surfaces. (Some of the results presented here were obtained in joint work with Ken Davidson, Vern Paulsen and Dinesh Singh).
4:10-5:00 pm, room 1312. Computational Analysis Seminar. Akram Aldroubi, Vanderbilt University. Invariance of shift-invariance spaces. A shift-invariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. We will characterize those shift-invariant subspaces S that are also invariant under additional (non-integer) translations. For the case of finitely generated spaces, these spaces are characterized in terms of the generators of the space. As a consequence, it is shown that principal shift-invariant spaces with a compactly supported generator cannot be invariant under any non-integer translations. 4:30-5:30 pm, room 1308. Universal Algebra and Logic Seminar. George Metcalfe, Vanderbilt University. Proof Theory and Algebra I. Classes of algebraic structures, such as groups, lattices, Boolean algebras etc. are often presented using a list of identities (equations) that specify when an algebra belongs to the class. However, this presentation does not help much in testing whether a particular identity holds for all members of the class. In this talk, I will introduce alternative algorithmic or "proof-theoretic" presentations of classes of algebras, and explain some of the main results, applications, and challenges in the field. 7-8 pm, room 1206. Undergraduate Seminar in Mathematics. Mikil Taylor, Vanderbilt University. The Golden Ratio. What is the Golden Ratio? In this talk, we will explore this mysterious number "phi" throughout its history. We will explain how to derive it, and how it naturally arises in all sorts of surprising situations, including geometry, music, art, poetry, and pineapples! Free pizza. |
| Wednesday 24 | 4:10 pm, room 1310. Topology & Group Theory Seminar. Denis Osin, Vanderbilt University. Outer automorphisms of groups with infinitely many ends. A well-known theorem of Baumslag states that the automorphism group of a finitely generated residually finite group is itself residually finite. In general, the analogous property does not hold for groups of outer automorphisms. However the Baumslag's theorem does have an 'outer' analogue for groups with infinitely many ends. The proof essentially uses the theory of relatively hyperbolic groups. This is a joint work with A. Minasyan. |
| Thursday 25 | 4:10-5 pm, room 5211. Colloquium. Jeff Brock, Brown University. Beyond the geometrization conjecture: models, bounds, and effective rigidity for hyperbolic 3-manifolds. Though Perelman's solution of Thurston's geometrization conjecture raises the possibility of extracting geometric information from a purely topological description of a 3-manifold, it does not directly produce it. In this talk, I will begin with the simple combinatorial data of a certain type of Heegaard splitting of a 3-manifold and extract geometric estimates with constants depending only on the genus of the splitting. This gives rise to a kind of "effective" rigidity theory where one can produce not only the existence of a negatively curved metric but estimates on its shape. Tea at 3:30 pm in SC 1425. |
| Friday 26 |
4:10-5:30 pm, room 1310.
Subfactor Seminar. Thomas Sinclair, Vanderbilt University. Superrigidity of Bernoulli Actions of Some Product Groups. Following Sorin Popa ("On the Superrigidity of Malleable Actions with Spectral Gap," available on arXiv) I will demonstrate that the Bernoulli action of G = H x K is cocycle superrigid, where H is a nonamenable group and K is an arbitrary infinite group. As a consequence, any group L which has an ergodic action orbit equivalent to the Bernoulli action of G is isomorphic to G and the actions are conjugate.
4:10 pm, room 1307. Partial Differential Equations Seminar. Joanna Pressley, Vanderbilt University. Response dynamics of integrate-and-fire neuron models. The brain is a complex network composed of more than 100 billion neurons, each making thousands of connections with other neurons. Neurons use pulse-like electrical signals called action potentials or "spikes" to encode information. The rate of spikes, or the firing rate, is believed to contain a majority of the pertinent information transferred in a spike train. One of the fundamental problems in neuroscience is characterizing the transfer function that converts noisy synaptic inputs into output firing rates. Using Fokker-Planck formalism, we describe the firing rate response dynamics of integrate-and-fire neuron models. Model dynamics are complex, with significant nonlinearities and heightened responses at certain frequencies of input. Elucidating the response properties of simplified stochastic models is a crucial step toward understanding how the brain performs reliable, temporally precise computations using circuits composed of noisy neurons. |