Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, September 1, 2016.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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30 31

Thursday, September 1, 2016

11:00 am in 241 Altgeld Hall,Thursday, September 1, 2016

#### A lower bound for the least prime in an arithmetic progression

###### Junxian Li (Illinois Math)

Abstract: Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek lower bounds for $P(k)$. In particular, we show that for almost every $k$ one has $P(k) \gg \phi(k) \log k \log_2 k \log_4 k / \log_3 k,$ answering a question of Ford, Green, Konyangin, Maynard, and Tao. We rely on their recent work on large gaps between primes. Our main new idea is to use sieve weights to capture not only primes, but also small multiples of primes. We also give a heuristic which suggests that $\liminf_{k} \frac{P(k)}{ \phi(k) \log^2 k} = 1$. This is joint work with Kyle Pratt and George Shakan.

12:00 pm in Altgeld Hall 243,Thursday, September 1, 2016

#### Gauge theory for webs and foams

###### Tomasz Mrowka (Massachusetts Institute of Technology)

Abstract: I will discuss how to use Instanton Floer theory ideas to construct invariants of Webs (knotted trivalent graphs) in the three manifolds. These invariants are functorial for Foam cobordisms. The talk will the discuss some properties of these invariants and some computational tools. A particular variant of this story might be applicable lead to a non-computer aided proof of the four color theorem.

3:00 pm in 243 Altgeld Hall,Thursday, September 1, 2016

#### Organizational Meeting

3:00 pm in 441 Altgeld Hall,Thursday, September 1, 2016

#### ADE classifications in Slodowy slices

###### Josh Wen (UIUC Math)

Abstract: Among other things, the ADE Dynkin diagrams classify: finite subgroups of $SL_2(\mathbb{C})$ (McKay correspondence); isolated surface singularities with multiplicity 2 that can be resolved by successive blowups (Kleinian/du Val singularities); and some complex simple Lie algebras (Killing-Cartan/Dynkin classfication). I'll tell you a story about how these three families are tied together in the geometry of the Grothendieck-Springer resolution and Slodowy slices.

4:00 pm in 245 Altgeld Hall,Thursday, September 1, 2016

#### An approach to the Four Color Theorem via Gauge Theory

###### Tomasz Mrowka (Massachusetts Institute of Technology)

Abstract: This is the 40th anniversary of the Appel and Haken proof that four colors suffice to color any planar maps. Their 1976 proof was particularly notable for its reliance on computers in a serious manner. A few years ago Peter Kronheimer and I realized that some of the invariants of knots in three manifolds coming from gauge theory had a generalization to invariants of knotted trivalent graphs. One of these invariants plausibly can be used to attack the four color theorem. In this talk I will sketch in rather broad terms some of these ideas. These invariants connect the four color problem to an important line of work combining gauge theory and 3 dimensional topology (where Haken's work provided an important foundation.)