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Math Calendar

Upcoming Events

November 3, 2020 (Tuesday), 11:00 am

Number Theory Seminar

Talk Title TBA

Madeline Locus Dawsey, The University of Texas at Tyler
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November 4, 2020 (Wednesday), 4:10 pm

Computational Analysis Seminar

Stochastic Hodgkin-Huxley Models and Noise Decomposition

Shusen Pu, Vanderbilt University
Visit Computational Analysis Seminar page for Zoom meeting information

In this talk, we will present a natural 14-dimensional Langevin model for the Hodgkin-Huxley (HH) conductance-based neuron model in which each directed edge in the ion channel state transition graph acts as an independent noise source, leading to a 14 × 28 noise coefficient matrix.  We show that

  1. the corresponding 14D mean-field ordinary differential equation system is consistent with the classical 4D representation of the HH system;
  2. the 14D representation leads to a noise coefficient matrix that can be obtained cheaply on each timestep, without requiring a matrix decomposition;
  3. sample trajectories of the 14D representation are pathwise equivalent to trajectories of several existing Langevin models, including one proposed by Fox and Lu in 1994;
  4. our 14D representation give the most accurate interspike-interval distribution, not only with respect to moments but under both the L1 and L metric-space norms; and
  5. the 14D representation gives an approximation to exact Markov chain simulations that are as fast and as efficient as all equivalent models.

We combine the stochastic shielding (SS) approximation, introduced by Schmandt and Galán in 2012, with Langevin versions of the HH model to derive an analytic decomposition of the variance of the interspike intervals (ISI), based on the mean–return-time oscillator phase. We prove in theory, and demonstrate numerically, that in the limit of small noise, the variance of the ISI decomposes linearly into a sum of contributions from each directed edge. Unlike prior analyses, our results apply to current clamp rather than voltage clamp conditions. Under current clamp, a stochastic conductance-based model is an example of a piecewise-deterministic Markov process. Our theory is exact in the limit of small channel noise.  Through numerical simulations we demonstrate its applicability over a range from small to moderate noise levels. We show numerically that the SS approximation has a high degree of accuracy even for larger, physiologically relevant noise levels.

November 4, 2020 (Wednesday), 5:10 pm

Topology & Group Theory Seminar

Outer Space for Right-Angled Artin Groups

Ruth Charney, Brandeis
Visit Topology & Group Theory Seminar page for Zoom meeting information

Right-angled Artin groups (RAAGs) span a range of groups from free groups to free abelian groups. Thus, their (outer) automorphism groups range from Out(F_n) to GL(n,Z). Automorphism groups of RAAGs have been well-studied over the past decade from a purely algebraic viewpoint. To allow for a more geometric approach, one needs to construct a contractible space with a proper action of the group. I will talk about joint work with Bregman and Vogtmann in which we construct such a space, namely, an analogue of Culler-Vogtmann’s Outer Space for arbitrary RAAGs.

November 6, 2020 (Friday), 4:10 pm

Subfactor Seminar

Fusion, positivity, and finite-index subfactors

James Tener, Australian National University
Zoom Meeting ID: 975 6315 9768
Email Dietmar Bisch to request pass code

In the 90’s, Wassermann showed that the subfactors of index less than 4 of type A were realized via conformal field theories corresponding to SU(2) at level N. This landmark work relied on an analysis of the “Connes’ fusion” of bimodules via quantum fields and differential equations. In this talk we will discuss a generalization of these ideas in terms of a recent ‘geometric realization’ of subfactors arising in conformal field theory. In addition, we will discuss how to use this approach to obtain finite index subfactors from Lie groups of all types.

November 9, 2020 (Monday), 10:00 am

Number Theory Seminar

Talk Title TBA

William Craig, The University of Virginia
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November 11, 2020 (Wednesday), 5:10 pm

Topology & Group Theory Seminar

Talk Title TBA

Alex Margolis, Technion and Vanderbilt University
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November 12, 2020 (Thursday), 4:10 pm

Colloquium

Talk Title TBA

Kirsten Wickelgren, Duke University
Zoom Meeting ID: 998 6775 5871
Email mathcolloquium@vanderbilt.edu to request pass code

(Contact Person: Anna Marie Bohmann)

November 13, 2020 (Friday), 4:10 pm

Subfactor Seminar

Talk Title TBA

Pieter Spaas, UCLA
Zoom Meeting ID: 975 6315 9768
Email Dietmar Bisch to request pass code

November 18, 2020 (Wednesday), 5:10 pm

Topology & Group Theory Seminar

Talk Title TBA

Alexander J. Rasmussen, University of Utah
Click “Topology & Group Theory Seminar” to visit seminar site for Zoom link and pass code

November 20, 2020 (Friday), 4:10 pm

Subfactor Seminar

Talk Title TBA

Luca Giorgetti, Vanderbilt University
Zoom Meeting ID: 975 6315 9768
Email Dietmar Bisch to request pass code

December 2, 2020 (Wednesday), 5:10 pm

Topology & Group Theory Seminar

Talk Title TBA

Rémi Coulon, CNRS Rennes
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December 3, 2020 (Thursday), 4:10 pm

Colloquium

Talk Title TBA

Richard Montgomery, University of California, Santa Cruz
Zoom Meeting ID: 998 6775 5871
Email mathcolloquium@vanderbilt.edu to request pass code

(Contact Person: Marcelo Disconzi)

December 4, 2020 (Friday), 11:00 am

Subfactor Seminar

Talk Title TBA

Remi Boutonnet, CNRS, Universite de Bordeaux
Zoom Meeting ID: 975 6315 9768
Email Dietmar Bisch to request pass code

December 4, 2020 (Friday), 4:10 pm

PDE Seminar

Talk Title TBA

Nestor Guillen, Texas State University
Zoom Meeting ID: TBA

December 9, 2020 (Wednesday), 5:10 pm

Topology & Group Theory Seminar

Dehn Functions of Finitely Presented Metabelian Groups

Wenhao Wang, Vanderbilt University
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The Dehn function was introduced by computer scientists Madlener and Otto to describe the complexity of the word problem of a group, and also by Gromov as a geometric invariant of finitely presented groups. In this talk, I will show that the upper bound of the Dehn function of finitely presented metabelian group G is 2n2k, where k is the torsion-free rank of the abelianization Gab, answering the question that if the Dehn functions of metabelian groups are uniformly bounded. I will also talk about the relative Dehn function of finitely generated metabelian group and its relation to the Dehn function.