Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Topological Data Analysis - Multiple Speakers

Monday, November 18th, 2019

Time: 2:30 p.m. Place: Goodes Hall 120

Speaker: Jordan Kokocinski, Arne Kuhrs, Catherine Pfaff, David Riegert Luke Steverango

Topics:  The graduate students have a well-prepared presentation on the Bubenik worksheet. This is highly recommended, even if you've missed a few meetings &/or are confused about homology. With leftover time Pfaff will provide some supplement to last week's presentation.

All are welcome!

Geometry & Representation - Gregory G. Smith

Monday, November 18th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Gregory G. Smith (Queen's University)

Title: Smooth Hilbert schemes.

Abstract:  In algebraic geometry, Hilbert schemes are the prototypical parameter spaces: their points correspond to closed subschemes in a projective space with a fixed Hilbert polynomial. After surveying some of their known features, we will present new numerical conditions on the polynomial that completely characterize when the associated Hilbert scheme is smooth. In this smooth situation, our explicit description of the subschemes being parametrized also provides new insights into the global geometry of Hilbert schemes. This talk is based on joint work with Roy Skjelnes (KTH).

Dynamics, Geometry, & Groups - Elizabeth Field (UIUC)

Friday, November 15th, 2019

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Elizabeth Field (University of Illionois Urbana-Champagn)

Title: Trees, dendrites, and the Cannon-Thurston map.

Abstract: When 1 -> H -> G -> Q -> 1 is a short exact sequence of three word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point z in the Gromov boundary of Q an ``ending lamination'' on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(F_n), one can identify the resultant quotient space with a certain R-tree in the boundary of Culler-Vogtmann's Outer space.

Department Colloquium - Alexei Novikov (Penn State)

Alexei Novikov (Penn State)

Friday, November 15th, 2019

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Alexei Novikov (Penn State)

Title: The Noise Collector for sparse recovery in high dimensions.

Abstract: The ability to detect sparse signals from noisy high-dimensional data is a top priority in modern science and engineering. A sparse solution of the linear system $Ax=b$ can be found efficiently with an $l_1$-norm minimization approach if the data is noiseless. Detection of the signal's support from data corrupted by noise is still a challenging problem, especially if the level of noise must be estimated. We propose a new efficient approach that does not require any parameter estimation. We introduce the Noise Collector (NC) matrix $C$ and solve an augmented system $Ax+Cy=b+e$, where $e$ is the noise. We show that the $l_1$-norm minimal solution of the augmented system has zero false discovery rate for any level of noise and with probability that tends to one as the dimension of $b$ increases to infinity. We also obtain exact support recovery if the noise is not too large, and develop a Fast Noise Collector Algorithm which makes the computational cost of solving the augmented system comparable to that of the original one. I'll introduce this new method and give its geometric interpretation.

Prof. Alexei Novikov obtained his Ph.D.~from Stanford in 1999 and then held postdoctoral positions at the IMA and at CalTech before joining the Pennsylvania State University where he is now a Professor in the Department of Mathematics. Prof. Novikov specializes in applied analysis and probability. His research has been supported by the NSF since 2006, as well as by the US--Israel Binational Science Foundation from 2005-2009.

CYMS Seminar - Oswaldo Sevilla

Thursday, November 14th, 2019

Time: 2:30 p.m Place: Jeffery Hall 319

Speaker: Oswaldo Sevilla (Fields Institute and Centro de Investigacion en Matematicas A.C.)

Title: Calabi Yau Threefolds arising from certain root lattices.

Abstract: I'll show my work on the construction of Calabi Yau threefolds that are constructed from the C_3 and C_4 root systems, using a construction by H. Verrill (Root lattices and pencils o f varieties, 1996) based on a paper of V. Batyrev (Dual Polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties,1994).

Number Theory Seminar - Mike Roth (Queen's University)

Thursday, November 14th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Mike Roth (Queen's University)

Title: A measure of positivity of a line bundle along a subschemes, and a simpler proof of the Ru-Vojta arithmetic theorem.

Abstract: Diophantine geometry seeks to link properties of rational solutions of a set of equations to the geometric properties of the variety they define. One of the main tools in Diophantine geometry is Diophantine approximation — results bounding how the complexity of a rational point must grow as it approaches a subvariety. In this talk I will discuss a somewhat recent new measure of the positivity of an ample line bundle along a subscheme, and show how its formal properties give a simple proof of a theorem of Ru-Vojta on Diophantine approximation. This is joint work with David McKinnon at Waterloo.

Geometry & Representation - Chris Brav (Higher School of Economics, Moscow)

Monday, November 11th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Chris Brav (Higher School of Economics, Moscow)

Title: Cartan's magic without the formulas.

Abstract:  The Cartan calculus concerns vector fields on a smooth variety X acting on differential forms via Lie derivative and contraction, with Cartan's magic formula expressing the relation between the two actions: Lie derivative is the graded commutator of the de Rham differential with contraction. On a smooth variety, the magic formula can be checked in local coordinates, while for singular schemes (and more general prestacks) it is necessary to work with the tangent complex, where it is no longer feasible to give explicit local formulas. Interpreting the magic formula as giving Griffiths transversality for the Gauss-Manin connection of the universal infinitesimal deformation of X, we are able to construct a formula-free, chain level Cartan calculus using the tangent complex of a singular scheme, and to establish the compatibility of this calculus with the noncommutative calculus of Hochschild cochains acting on Hochschild chains. This is joint work with Nick Rozenblyum.

Department Colloquium - Ari Arapostathis (UT Austin)

Ari Arapostathis (UT Austin)

Friday, November 8th, 2019

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Ari Arapostathis (UT Austin)

Title: Lower bounds on the rate of convergence for heavy-tailed driven SDEs motivated by large scale stochastic networks.

Abstract: We show that heavy-tailed Levy noise can have a dramatic effect on the rate of convergence to the invariant distribution in total variation. This rate deteriorates from the usual exponential to strictly polynomial under the presence of heavy-tailed noise. To establish this, we present a method to compute a lower bound on the rate of convergence. We should keep in mind that standard Foster-Lyapunov theory furnishes only an upper bound on this rate. To motivate the study of such systems, we describe how L\'evy driven stochastic differential equations arise in the study of stochastic queueing networks. This happens when the arrival process is heavy-tailed, or the system suffers asymptotically negligible service interruptions. We identify conditions on the parameters in the drift, the Levy measure and/or covariance function which result in subexponential and/or exponential ergodicity, and we show that these conditions are sharp. In addition, we show that for the queueing models described above with no abandonment, the rate of convergence to the stationary distribution in total variation is polynomial, and we provide a sharp quantitative characterization of this rate via matching upper and lower bounds. We conclude by presenting analogous results on convergence in the Wasserstein distance.

This talk is based on joint work with Hassan Hmedi, Guodong Pang and Nikola Sandric.

Ari Arapostathis is a professor in the Department of Electrical and Computer Engineering at The University of Texas at Austin, and holds the Texas Atomic Energy Research Foundation Centennial Fellowship in Electrical Engineering. He received his BS from MIT and his PhD from U.C. Berkeley, in 1982. He is a Fellow of the IEEE, and was a past Associate Editor of the IEEE Transactions on Automatic Control and the Journal of Mathematical Systems and Control. His research has been supported by several grants from the National Science Foundation, the Air-Force Office of Scientific Research, the Army Research Office, the Office of Naval Research, DARPA, the Texas Advanced Research/Technology Program, Samsung, and the Lockheed-Martin Corporation.