Seminars, Colloquia, and Conferences
Colloquiua
The colloquium meets on Fridays at 4:00pm in
BromfieldPearson 101, unless otherwise indicated.
Spring 2017
February 10, 2017
From Homogeneous Metric Spaces to Lie Groups
Sebastiano Nicolussi Golo (University of Jyväskylä & University of Trento)
February 24, 2017
Diophantine and tropical geometry
David ZureickBrown (Emory University)
March 3, 2017
Reflection Positivity:
Representation Theory meets Quantum Field Theory
Gestur Olafsson (LSU)
March 10, 2017
Diffusion of
Lorentz gas on scatterers with flat point
Hongkun Zhang (UMass Amherst)
March 17, 2017
Numerical difficulties
in the simulation of flow in deformable porous media
Carmen Rodrigo (University of Zaragoza, Spain)
March 31, 2017
Equidistribution
of Shapes of Number Fields of degree 3, 4, and 5
Piper Harron (University of Hawaii)
Abstract:
In her talk, Piper Harron will introduce the ideas that there are number fields,
that number fields have shapes, and that these shapes are everywhere you want them
to be. This result is joint work with Manjul Bhargava and uses his counting methods
which currently we only have for cubic, quartic, and quintic fields. She will sketch
the proof of this result and leave the rest as an exercise for the audience.
(Check your work by downloading her thesis!)
April 7, 2017
Fun with Finite Covers of 3Manifolds:
Connections between Topology, Geometry, and Arithmetic
Nathan Dunfield (University of Illinois at UrbanaChampagne)
Abstract:
From the revolutionary work of Thurston and Perelman, we know
that the topology of 3manifolds is deeply intertwined with their geometry.
In particular, hyperbolic geometry, the nonEuclidean geometry of
constant negative curvature, plays a central role. In turn, hyperbolic
geometry opens the door to applying tools from number theory,
specifically automorphic forms, to what might seem like purely
topological questions.
After a passing wave at the recent breakthrough results of Agol, I will
focus on exciting new questions about the geometric and arithmetic
meaning of torsion in the homology of finite covers of hyperbolic
3manifolds, motivated by the recent work of Bergeron, Venkatesh, Le,
and others. I will include some of my own results in this area that are
joint work with F. Calegari and J. Brock.
April 14, 2017
Crossing Matrices for Braids
Zbigniew Nitecki (Tufts University)
Abstract TBA:
Abstract: Given a geometric braid with N strands, one can codify the crossing information in an N by N matrix
whose ij entry is the algebraic number of crossings of strand i over strand j, that is, the number of
lefttoright crossings minus the number of righttoleft ones. This is the same for any two representatives
of the same braid (which is a homotopy class); this gives a matrixvalued “derivation” on the braid group
(at least Mo tells me that is what it is called). Mauricio Gutierrez and I, together with Pep Burillo and
Sava Krstic, published a paper about 15 years ago about these matrices, but one question was left open,
and has occupied Mo and me for the last too many years.
It is easy (using some general results on the braid group) to characterize which matrices occur as crossing
matrices for some braid, but the more interesting question is which ones occur for positive braids
(all crossings in the same direction) because then all the crossings appear (there is no cancellation).
The "obvious" answer to the positive realization question, adapting the answer for not necessarily positive braids,
is false, and we have been trying to find a characterization for crossing matrices of positive braids without
success for some time. We do, however, have an algorithmic answer, and a (limited) conjecture.
I plan to quickly introduce the basic notions of braids and the algebra of crossing matrices, the connection
between these and the ThurstonGarside normal form for braids, then try to explain what we know about the
characterization problem, as well as the algorithm we have which exhibits all possible realizations (if any exist)
of an appropriate matrix.
April 21, 2017
Inhibitionbased theta resonance in a hippocampal network
Horacio Rotstein (New Jersey Institute of Technology)
Abstract TBA:
Abstract: A crucial issue in the understanding of neuronal oscillations is to elucidate the microcircuits
that are the substrate to these rhythms in the different brain areas. This raises the question of whether
rhythmic activity results solely from the properties of the network connectivity (e.g., excitation and inhibition)
and topology or it involves the interplay of the latter with the intrinsic properties (e.g., ionic currents) of the
participating neurons. In this project we address this issue theoretically in the context of the hippocampal area
CA1 microcircuits, which include excitatory (PYR) and inhibitory (INT) cells. It has been observed that PYR exhibit
a preferred subthreshold frequency response to oscillatory inptus at (4  10 Hz) frequencies (resonance) 'in vitro.'
Contrary to expectation, these cells do not exhibit spiking resonance in response to 'in vivo' direct oscillatory
optogenetic activation, but, surprisingly, spiking resonance in PYR occurs when INT are activated. We combine
dynamical systems tools, biophysical modeling and numerical simulations to understand the underlying mechanisms of
these rather unexpected results. We show that the lowpass filter results form a combination of postinhibitory
rebound (the ability of a cell to spike in response to inhibition) and the intrinsic properties of PYR.
The bandpass filter requires additional timing mechanisms that prevent the occurrence of spikes at low frequencies.
We discuss various possible, conceptually different scenarios. These results and tools contribute to building a
general theoretical and conceptual framework for the understanding of preferred frequency responses to oscillatory
inputs in neuronal networks.
