Mathew Ellison
Harmonic and Symmetric Polynomials
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A polynomial in $n$ variables is harmonic if it vanishes under the Laplace operator, $\partial_1^2 + ... + \partial_n^2$, and symmetric when invariant under permutations of variables. We'll use an unusual inner product to uncover structure in these polynomials. This talk will follow one of the same name from Lang's Math Talks for Undergraduates --- but should be interesting for graduates too!
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Jonathan Lindbloom
Oblique Projections and Low Rank Structure in Inverse Problems
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This talk will cover some linear algebra topics relevant to the solution of inverse problems: oblique projectors and low-rank structure. We introduce the oblique projection operator, which projects vectors onto a subspace along another subspace. We then demonstrate how oblique projectors can be applied to standard form transformations for regularized least-squares problems. Next, we introduce the matrix determinant lemma and the Sherman-Morrison-Woodbury (SMW) matrix identity. We conclude by using the SMW identity to motivate an algorithm for approximately sampling high-dimensional Gaussians arising in statistical inverse problems.
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Ryan Maguire
Legendre Polynomials and Saturn
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Fresnel transforms are a commonly used tool in the study of
Fourier optics, but lead to a difficult inverse problem.
The original Fresnel approximation reduces the kernel of the
transform to an exponentiated quadratic, which has a closed-form
solution for the inverse. By modifying Fresnel's derivation
one can get a new approximation in terms of Legendre polynomials.
In this talk we'll discuss these approximations and how they have
been used to study the radio science data for the Cassini mission
to Saturn.
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Steve Fan
Arithmetic Combinatorics: Integer Partitions and Sequences
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Arithmetic combinatorics concerns combinatorial estimates for objects equipped with certain arithmetic structures. Particularly, topics related to estimates of the sums, differences, and products of finite sets and to objects such as arithmetic progressions have been extensively studied. One remarkable feature of the field is the use of tools from many diverse fields of mathematics, which makes it a rich and fascinating subject. In this talk, I will highlight this feature by discussing a few interesting results concerning partitions of the set of integers and integer sequences of special interests, whose proofs take advantage of ideas from graph theory, probability theory, and ergodic theory.
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Beth Anne Catellano
The Lindström-Gessel-Viennot Lemma
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The Lindström-Gessel-Viennot Lemma is a fundamental theorem in combinatorics that connects lattice paths and determinants (and is much deeper than "lemma" gives it credit for, I promise!). We'll explore applications of this result to several fun areas of combinatorics, including the Catalan numbers, plane partitions, tilings, and the Jacobi-Trudi identities for symmetric functions. This talk will be accessible to everyone who knows what a determinant is and likes pretty pictures.
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Brian Mintz
Discussion Session: Problems and Solutions in the Department
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What is this?
I will moderate a group conversation about how things are going in the department - we can talk about specific ideas you might have, specific problems you might have (ex. "still haven't got reimbursed from conference travel"), and amorphous large worries (ex. "I don't feel like I know what is going on/expected of me").
What is the point?
1. To bring to light any concerns/problems/solutions that you might have so that I can relay that info to the GPC (graduate program committee, the math faculty committee that makes decisions about graduate students). (Hopefully they can solve some of the problems, or at least keep those concerns in mind when making decisions about the grad program.)
2. To talk about issues people are having so we can realize that a lot of us struggle with similar things.
3. To talk about issues people are having in hopes that another grad student might know a good solution.
4. To talk about issues that other grad students are noticing, and group brainstorm ways to prevent a future problem / stop an existing problem from getting worse.
What's important to keep in mind during this discussion?
Don't be dismissive - just because something isn't a problem for you, that doesn't mean it isn't a problem for someone else. We all experience things differently.
I am not expecting us to actually solve all of our problems in this meeting, or even during this next year (sorry) - but it is still worth discussing - I think there are a lot of reasonable changes we could request, and the ones that might not seem so reasonable now can become more reasonable over time if we keep talking about them.
This meeting can be a great forum in which we can discuss grad student problems, but it is not the only chance to do so. So if you think of something later you wish would have been said, you can always talk to me or each other in the hallway another time.
Hope to see as many of you as possible there!
(If you can't come today but have something you'd like to be brought up in the meeting, please send me an email or come by my office -- I have a dentist appointment but will be in at least by 1pm. And even if you CAN come to meeting, if for whatever reason there's an issue you want brought up but you don't want to be the one to say it, send me an email or come talk to me.)
Brian
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