Graduate Student Seminar

Upcoming Talk

14 January 2022

Divergence is only the Beginning

Will Hoffer ๐ŸŒ

When Èmile Borel brought his pioneering work on the resummation of divergent series to Mittag-Leffler, the established mathematican put his hand on the complete works of his teacher, Karl Weierstrauss, and told Borel sternly in Latin, ``The Master forbids it." In this talk, we will delve into the very same methods that made mathematicians of yore tremble.

Borel's work essentially amounts to formally applying a term-by-term inverse Laplace transform, which divides each coefficient by a factorial. As such, it turns divergent series whose coefficients grow at most factorially into possibly convergent ones. Under suitable conditions, this new function admits a Laplace tranform to undo the first Borel transform, and this function is asymptotically equivalent to the original divergent series.

Today, we shall explore Borel's work on resumming divergent series and see that this is only the beginning of a much larger field of active research known as resurgence.

Scheduled Talks

Winter 2022

14 January 2022

Divergence is only the Beginning

Will Hoffer ๐ŸŒ

When Èmile Borel brought his pioneering work on the resummation of divergent series to Mittag-Leffler, the established mathematican put his hand on the complete works of his teacher, Karl Weierstrauss, and told Borel sternly in Latin, ``The Master forbids it." In this talk, we will delve into the very same methods that made mathematicians of yore tremble.

Borel's work essentially amounts to formally applying a term-by-term inverse Laplace transform, which divides each coefficient by a factorial. As such, it turns divergent series whose coefficients grow at most factorially into possibly convergent ones. Under suitable conditions, this new function admits a Laplace tranform to undo the first Borel transform, and this function is asymptotically equivalent to the original divergent series.

Today, we shall explore Borel's work on resumming divergent series and see that this is only the beginning of a much larger field of active research known as resurgence.

Fall 2021

19 November 2021

Sites & Bytes: Website Workshop!

Raymond Matson ๐ŸŒ and Will Hoffer ๐ŸŒ

Bring your laptop and youโ€™ll walk away with your own personal website! This week, weโ€™ll walk you through making a completely free website on a platform of your choice. Options include Google Sites, Wordpress, and Github pages. Weโ€™ll help you pick the right choice and then launch it to the internet during this interactive workshop.

If you have a website already, come join us too! You can show off, learn about new options or customizations, and/or help your peers join the club.

12 November 2021

Melting Snowflake Fractals

Will Hoffer ๐ŸŒ

The von Koch snowflake is one of the first studied fractals and thus a great starting point for investigating fractal and spectral geometry. In this talk, we'll study how much heat flows into a fractal shape from its heated boundary. To that end, we'll discuss von Koch fractals, the heat equation, and how to find the total amount of heat which flows into the fractal. In particular, we build and solve an approximate functional equation in order to estimate the total amount of heat in the fractal after small amounts of time.

5 November 2021

Colors, Strings, and Beads: Categories for Children

Christian Williams

There is a graphical language for formal category theory [Myers 2016]. The language empowers users to understand concepts and theorems with striking facility and clarity. There is also a logical interpretation: colors are types, strings are judgements, and beads are inferences. Together, category theory and logic can be taught in a simple, colorful, and engaging way. To demonstrate, we explore how the concepts of adjunction and extensions (in the category of relations) might be taught as early as elementary school.

29 October 2021

Boundaries at Infinity and Stable Subgroups

Jacob Garcia

Given a proper, geodesic, Gromov-hyperbolic metric space X, one can construct a boundary on the space called the visual boundary. The topology of this visual boundary is an invariant under a natural equivalence for hyperbolic spaces, and has been extensively studied with respect to quasi-convex subgroups of Iso(X). There has been much recent work towards constructing an analogue to this boundary in non-hyperbolic spaces.

In this talk, I'll relive the trauma of my oral exam slides as we review some of the known properties of the visual boundary, some of the issues this boundary has with non-hyperbolic spaces, and we'll introduce a new analogue which can be used instead.

22 October 2021

Six Operations on Diamond Topos

Shanna Dobson

We present preliminary work conjecturing six operations on the topos of a nonarchimedean object towards a new reciprocity law.

15 October 2021

Intro to Random Walks: From an electrifying problem to modeling forest fires

Neima Ghandian

This talk is a basic introduction into random walk theory, mainly focused on the Monte Carlo method for approximations and Markov Chains for exact solutions. Throughout the study of mathematics, physics has always been a huge source of problems and starting points of inquiry. However, as we generalize mathematics, we often leave the realm of โ€œreal worldโ€ application. BUT! There is a well-known connection between random walks and electric networks, and we will get to investigate some different techniques on arriving at a solution. Furthermore, if time permits, we explore a couple different ideas of how to apply the techniques from this talk, including the forest fire model mentioned in the title.

8 October 2021

This is not the title of this talk

Will Hoffer๐ŸŒ

You may have heard of the birthday paradox, but what about a birthday full of paradoxes? Join me as I begin my quarter life crisis with some mathematical anomalies! Can we find the person at Getaway such that if they're drinking, everyone else is drinking? Did you know that most of your friends probably have more friends than you? Were you aware that if this claim is true, you'll get a thousand bucks just for showing up?

In this talk, we'll fill half of the whiteboard with paradoxes, and then we'll fill half of the remaining boardspace with paradoxes, and then we'll fill half of the remaining boardspace with paradoxes, and then we'll...

1 October 2021

Top 5 Undecidable Problems: Number 4 will shock you!

Chris Grossack๐ŸŒ

Undecidable problems are problems which are, in a certain sense, provably unsolvable. In this talk weโ€™ll survey some undecidable problems in mathematics, both famous and surprising, and discuss how these problems witness the close ties between mathematical logic and other branches of math.

The UC Riverside Math Graduate Student Seminar (GSS) is brought to you by the UCR student chapter of the American Mathematics Society. GSS is organized by the officers of the chapter.

Officers

Will Hoffer, President

Jacob Garcia, Vice President

Alysha Toomey, Secretary

Raymond Matson, Treasurer

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