# Spring 2017 Colloquia

### February 10, 2017

#### Sebastiano Nicolussi Golo (University of Jyväskylä & University of Trento)

##### From Homogeneous Metric Spaces to Lie Groups

### February 24, 2017

#### David Zureick-Brown (Emory University)

##### Diophantine and tropical geometry

### March 3, 2017

#### Gestur Olafsson (LSU)

##### Reflection Positivity: Representation Theory meets Quantum Field Theory

### March 10, 2017

#### Hongkun Zhang (UMass Amherst)

##### Diffusion of Lorentz gas on scatterers with flat point

### March 17, 2017

#### Carmen Rodrigo (University of Zaragoza, Spain)

##### Numerical difficulties in the simulation of flow in deformable porous media

### March 31, 2017

#### Piper Harron (University of Hawaii)

##### Equidistribution of Shapes of Number Fields of degree 3, 4, and 5

**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

#### Nathan Dunfield (University of Illinois at Urbana-Champagne)

##### Fun with Finite Covers of 3-Manifolds: Connections between Topology, Geometry, and Arithmetic

**Abstract:** From the revolutionary work of Thurston and Perelman, we know that the topology of 3-manifolds is deeply intertwined with their geometry. In particular, hyperbolic geometry, the non-Euclidean 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 3-manifolds, 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

#### Zbigniew Nitecki (Tufts University)

##### Crossing Matrices for Braids

**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 left-to-right crossings minus the number of right-to-left ones. This is the same for any two representatives of the same braid (which is a homotopy class); this gives a matrix-valued “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 Thurston-Garside 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

#### Horacio Rotstein (New Jersey Institute of Technology)

##### Inhibition-based theta resonance in a hippocampal network

**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 low-pass filter results form a combination of post-inhibitory rebound (the ability of a cell to spike in response to inhibition) and the intrinsic properties of PYR. The band-pass 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.