Department of

# Mathematics

Seminar Calendar
for events the day of Friday, March 29, 2019.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, March 29, 2019

3:00 pm in 245 Altgeld Hall,Friday, March 29, 2019

#### How to Become a Liberated Mathematician in 13+3 Painful Years

###### Piper Harron (University of Hawai'i at Mānoa )

Abstract: Piper H never wanted to be liberated. She would have much preferred to be conventionally successful, living by other people's standards. Though she tried, she couldn't make herself fit. You can say she has some complaints. Some people want to spread her message, other people think she needs to go away forever. In this talk Piper lets you in on the secret that actually she's just a very tired person trying to find more time for naps.

4:00 pm in 145 Altgeld Hall,Friday, March 29, 2019

#### Geometric ideas in number theory

###### Robert Dicks (UIUC)

Abstract: Jurgen Neukirch in 1992 wrote that Number Theory is Geometry. At first glance, it seems nothing could be further from the truth, but it turns out that tools such as vector bundles, cohomology, sheaves, and schemes have become indispensable for understanding certain chapters of number theory in recent times. The speaker aims to discuss an analogue in the context of number fields of the classical Riemann-Roch theorem, which computes dimensions of spaces of meromorphic functions on a Riemann surface in terms of its genus. The aim is for the talk to be accessible for any graduate student; we'll find out what happens.

4:00 pm in 345 Altgeld Hall ,Friday, March 29, 2019

#### Generalized sum-product phenomenon for polynomials

###### Souktik Roy (UIUC Math)

Abstract: Suppose $P(x,y)$ and $Q(x,y)$ are real polynomials with non-trivial dependence on $x$ and $y$, and $\epsilon$ is any positive constant. If, for a sufficiently large $n$-element set $A$ of real numbers, both $|P(A,A)|$ and $|Q(A,A)|$ are simultaneously smaller than $n^{5/4-\epsilon}$, then we shall prove that either $P(x,y) = f(u(x)+Cu(y)) \text{ and } Q(x,y) = g(u(x)+Du(y)),$ or $P(x,y) = f(u(x)u^{c}(y)) \text{ and } Q(x,y) = g(u(x)u^{d}(y)),$ where $f,g,u$ are polynomials and $C,D,c,d$ are constants. As a corollary, we obtain a strengthening of a classic result of Elekes and Rónyai in a symmetric setting of natural interest. The proof combines ideas from incidence geometry and o-minimality in model theory. This is joint work with Yifan Jing (UIUC) and Minh Chieu Tran (UIUC).

4:00 pm in 245 Altgeld Hall,Friday, March 29, 2019

#### Ants on pants

###### Agnès Beaudry   [email] (University of Colorado, Boulder)

Abstract: In this talk, I will give an introduction to manifolds and cobordism. What are manifolds? An ant living on a very large circle wouldn't know that it isn't living on the (flat) real line. In analogy, a d-manifold is a geometric object which, from an ant's perspective, looks flat like Euclidean space R^d, but which, from a bird's-eye view, can look curved or otherwise interesting, like the unit sphere in R^(d+1). What is cobordism? Think of a 2-dimensional surface that looks like a pair of empty pants. If the waist is the large circle which is the ant's universe, then the pants represent a transformation of the ant's world into a two circle universe. In analogy, a cobordism is a d+1 manifold with boundary which transforms one d-manifold into another. Two manifolds are cobordism equivalent if such a transformation exists. An interesting and difficult question is that of classifying manifolds. A raw classification in arbitrary dimensions is nearly impossible, and for this reason, mathematicians often settle for less precise answers. For example, can one classify manifolds up to cobordism equivalence? Come to my talk and find some answers to the ants on pants conundrum.