Calendar

Apr
28
Fri
MA defense: Mirza Baig
Apr 28 @ 7:30 am – 8:30 am

Inverting the Radon Transform using Summability Kernels

Link to Master’s project

Abstract. We study an inversion technique of the Radon Transform using Summability Kernels and consider the problem of numerically implementing this algorithm. In doing so we investigate the tradeoff between the various analytical and discretization parameters involved and propose a simple framework using recent results in literature for integrating over $mathbb{S}^{n-1}$ to estimate the rate of convergence of our numerical implementation to the analytical inversion technique as well as offer a heuristic in parameter selection which would considerably reduce a brute force search over a large search space. We also discuss how the smoothness of the phantom to be estimated controls the convergence in the numerical inversion algorithm and have numerical experiments to validate our theoretical findings.

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Logic Seminar: Jack Yoon @ Keller 404
Apr 28 @ 2:30 pm – 3:30 pm

Jack Yoon will continue his explication of Proof Mining.

May
3
Wed
Colloquium: Yuriy Zinchenko (U. Calgary) @ Keller 401
May 3 @ 3:00 pm – 4:00 pm

TITLE
Dose-volume requirements modeling for radiotherapy optimization

ABSTRACT
Radiation therapy is an important modality in cancer treatment. To find a good treatment plan, optimization models
and methods are typically used, while dose-volume requirements play an important role in plan’s quality evaluation.
We compare four different optimization approaches to incorporate the so-called dose-volume constraints into the
fluence map optimization problem for intensity modulated radiotherapy. Namely, we investigate (1) conventional
emph{Mixed Integer Programming} (MIP) approach, (2) emph{Linear Programming} (LP) approach to partial volume
constraints, (3) emph{Constrained Convex Moment} (CCM) approach, and (4) emph{Unconstrained Convex Moment
Penalty} (UCMP) approach. The performance of the respective optimization models is assessed using anonymized data
corresponding to eight previously treated prostate cancer patients. Several benchmarks are compared, with the goal
to evaluate the relative effectiveness of each method to quickly generate a good initial plan, with emphasis on
conformity to DVH-type constraints, suitable for further, possibly manual, improvement.

BIO
Dr. Zinchenko received his PhD from Cornell University on 2005 under supervision of Prof. James Renegar.
From 2005 to 2008 he held a PDF position at the Advanced Optimization Lab at McMaster University, working
with Prof. Tamas Terlaky and Prof. Antone Deza, and spent portion of his fellowship with radiation oncology group
at the Princess Margaret Hospital in Toronto. Currently, Yuriy is an Associate Professor of Mathematics & Stat at the
University of Calgary.
Dr. Zinchenko’s primary research interest lies in convex optimization, and particularly, the curvature of the central path for
interior-point methods, and applications. Yuriy’s work on optimal radiotherapy design was recognized by 2008 MITACS
Award for Best Novel Use of Mathematics in Technology Transfer, and in 2012-2015 he served as one of the PIs for
PIMS Collaborative Research Group grant on optimization.

May
4
Thu
MA defense: Sean Sanford
May 4 @ 9:30 am – 10:30 am

Smooth Actions of Compact Lie Groups on $S^2$ are Smoothly Equivalent to Linear Actions:

Link to Master’s project

Abstract:

Mathematicians have been interested in group actions on spheres since before the algebraic description of a group was defined. The rotational and reflective symmetries of the circle and of S^2 were naturally among the first to be considered. When we restrict attention to a compact topological group, there is a classic theorem of Kerekjártó to the effect that for S^2 , these are essentially the only actions:

Theorem 1 (Kerekjártó, [3]). Every continuous, effective action of a compact topological group G on S^2 is topologically conjugate to a linear action (to the standard action of a subgroup of O(3) on S^2 as a subset of R^3 ).

Thus in the topological category, in order to understand all effective actions of compact groups on S^2 , it is enough to understand the subgroups G ≤ O(3) and their actions via matrix multiplication on S^2 ⊆ R^3 (so called linear actions).

The goal of this paper is to extend this result to the smooth category. In other words, we show that every smooth, effective action of a compact Lie group, on S^2 is smoothly conjugate (i.e. conjugate through a diffeomorphism) to a linear action.

The main theorem of the paper is helpful for studying the topology of the space of actions of compact Lie groups on S^2 and we present a corollary as an example of this. We also explicitly determine all compact subgroups of O(3) up to conjugacy within O(3), and use this information to construct explicit G-CW decompositions for preferred representatives of each of these conjugacy classes. The G-CW decompositions are useful in other areas for example in the classification of G-equivariant vector bundles over S^2, and in determining whether such bundles have algebraic models.
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May
5
Fri
Logic Seminar: David Webb
May 5 @ 1:00 pm – May 5 @ 2:00 pm

David Webb will speak at 1:00-2:00 in Keller 402

Title: Every Function Can be Computable

Abstract: I will relay an interesting result of Joel David Hamkins: that
there is an algorithm which can compute any function f of natural
numbers, if it is carried out in the right model of arithmetic
(corresponding to f). In particular, I will construct the necessary
models using Rosser sentences and describe the algorithm.

Colloquium: Andrei Jorza (Notre Dame) @ Keller 401
May 5 @ 2:30 pm – 3:30 pm

Speaker: Andrei Jorza (Notre Dame)

Title: p-adic interpolations and L-functions

Abstract: The classical Birch and Swinnerton-Dyer conjecture (one of the million dollar Millenium Problems) posits a relationship between the arithmetic and complex analytic behaviors of elliptic curves. Recent progress towards this conjecture relies on the p-adic analytic behavior of these elliptic curves, which can be thought of as an intermediary between arithmetic and complex analysis. I will describe what it means to p-adically interpolate complex analytic L-functions and some p-adic analogues of the Birch and Swinnerton-Dyer conjecture.

Colloquium: Pamela Harris (Williams)
May 5 @ 3:30 pm – 4:30 pm
May
9
Tue
Colloquium: Jakob Kotas (U. Washington-Seattle)
May 9 @ 3:30 pm – 4:30 pm

Speaker: Jakob Kotas, Applied Mathematics at the University of Washington–Seattle

Title: Response-guided dosing

Abstract: Within the broad field of personalized medicine, there has been a recent surge of clinical interest in the idea of response-guided dosing. Roughly speaking, the goal is to devise strategies that administer the right dose to the right patient at the right time. We will present stochastic models that attempt to formalize such optimal dosing problems. Theoretical results about the structure of optimal dosing strategies and associated solution methods rooted in convex optimization, stochastic dynamic programming, robust optimization, and Bayesian learning will be described. Computational results on rheumatoid arthritis will be discussed.