Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, February 5, 2015.

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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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Thursday, February 5, 2015

11:00 am in 241 Altgeld Hall,Thursday, February 5, 2015

#### Larger gaps between consecutive prime numbers

###### Kevin Ford   [email] (UIUC Math)

Abstract: We show how to improve quantitatively the best known bounds for large gaps between consecutive prime numbers by a factor of $\log\log\log x$. More precisely, if $G(x)$ is the largest gap between consecutive prime numbers which are at most $x$, then we show that $$G(x) \ge c \log x \frac{\log\log x \log\log\log\log x}{\log\log\log x}$$ for some positive $c$ and large enough $x$. This improves upon the recent bounds of the speaker (in joint work with B. Green, S. Konyagin and T. Tao) and J. Maynard, who independently solved a long-standing conjecture of Erdos on the growth of $G(x)$. The new work, which is joint with Ben Green, Sergei Konyagin, James Maynard and Terence Tao, combines ideas from both of the aforementioned papers and also introduces some new ideas from probabilistic graph theory (hypergraph packing, Rodl nibble).

1:00 pm in Altgeld Hall 243,Thursday, February 5, 2015

#### The primitivity index function for a free group, and untangling closed curves on surfaces

###### Neha Gupta (UIUC Math)

Abstract: A theorem of Scott shows that any closed geodesic on a surface lifts to an embedded loop in a finite cover. Our motivation is to find a worst-case lower bound for the degree of this cover, in terms of the length of the original loop. We establish, via probabilistic methods, lower bounds for certain analogous functions, like the Primitivity Index Function and the Simplicity Index Function, in a free group. These lower bounds, when applied in a suitable way to the surface case, give us some lower bounds for our motivating question. This is joint work with Ilya Kapovich. View talk at http://youtu.be/-RS1SNhj3yA

2:00 pm in 140 Henry Administration Bldg,Thursday, February 5, 2015

#### Open problems in Number Theory, II

###### Bruce Berndt (UIUC Math)

4:00 pm in 245 Altgeld Hall,Thursday, February 5, 2015

#### Around Maxwell's conjecture

###### Boris Shapiro (Stockholm University)

Abstract: In his famous book ”A Treatise on Electricity and Magnetism” first published in 1867, J.C.Maxwell made a claim that any configuration of $N$ fixed point charges in $R^3$ creates no more than $(N-1)^2$ points of equilibrium. He provided this claim with an incomplete proof containing all elements of Morse theory to be created 60 years later. We will discuss what is known at present about his claim which is still open even for 3 charges in $R^3$. No preliminary knowledge of the subject is necessary.

Prof. Shapiro comes to our campus as an INSPIRE scholar. INSPIRE is to establish a transnational partnership alliance between the University of Illinois at Urbana-Champaign (Illinois) and three leading research universities in Stockholm, Sweden.