Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Fall 2014 Talks are at noon on Monday in room B660 of University Hall For more information, or to receive an email announcement of each week's seminar, contact Nathan Ng < ng AT cs DOT uleth DOT ca > or Dave Morris .
 The next talk: Sept 22 at noon in UHall B660 Farzad Aryan (University of Lethbridge) On Binary and Quadratic Divisor Problem Let $$d(n)=\sum_{d|n} 1$$. This is known as the divisor function. It counts the number of divisors of an integer. Consider the following shifted convolution sum $$\sum_{an-m=h}d(n) \, d(m) \, f(an, m),$$ where $$f$$ is a smooth function which is supported on $$[x, 2x]\times[x, 2x]$$ and oscillates mildly. In 1993, Duke, Friedlander, and Iwaniec proved that $$\sum_{an-m=h}d(n) \, d(m) \, f(an, m) = \textbf{Main term}(x)+ \mathbf{O}(x^{0.75}).$$ Here, we improve (unconditionally) the error term in the above formula to $$\mathbf{O}(x^{0.61})$$, and conditionally, under the assumption of the Ramanujan-Petersson conjecture, to $$\mathbf{O}(x^{0.5}).$$ We will also give some new results on shifted convolution sums of functions coming from Fourier coefficients of modular forms.

 Talks in the series this semester: (Click on any title for more info, including the abstract. Then click on it again to hide the info.)

 Date Speaker Title Sept 8 everyone Open problem session at noon in UHall B660 Please bring your favourite (math) problems. Anyone with a problem to share will be given about 5 minutes to present it. We will also choose most of the speakers for the rest of the semester. Sept 15 Joy Morris Colour-permuting automorphisms of Cayley graphs at noon in UHall B660 University of Lethbridge A Cayley graph $$\mathrm{Cay}(G;S)$$ has the elements of $$G$$ as its vertices, with $$g \sim gs$$ if and only if $$s \in S$$. There is a natural colouring of the edges of any such graph, by assigning colour $$s$$ to an edge if it came from the element $$s$$ of $$S$$. For a Cayley digraph, any graph automorphism that preserves this colouring has to be a group automorphism of $$G$$. For a Cayley graph, this is not the case. I will present examples of Cayley graphs that have automorphisms that do not correspond to group automorphisms of $$G$$. I will also show that for some families of groups, such examples are not possible. I will also discuss the more general problem of automorphisms that permute the colours, rather than necessarily preserving all of them. Sept 22 Farzad Aryan On Binary and Quadratic Divisor Problem at noon in UHall B660 University of Lethbridge Let $$d(n)=\sum_{d|n} 1$$. This is known as the divisor function. It counts the number of divisors of an integer. Consider the following shifted convolution sum $$\sum_{an-m=h}d(n) \, d(m) \, f(an, m),$$ where $$f$$ is a smooth function which is supported on $$[x, 2x]\times[x, 2x]$$ and oscillates mildly. In 1993, Duke, Friedlander, and Iwaniec proved that $$\sum_{an-m=h}d(n) \, d(m) \, f(an, m) = \textbf{Main term}(x)+ \mathbf{O}(x^{0.75}).$$ Here, we improve (unconditionally) the error term in the above formula to $$\mathbf{O}(x^{0.61})$$, and conditionally, under the assumption of the Ramanujan-Petersson conjecture, to $$\mathbf{O}(x^{0.5}).$$ We will also give some new results on shifted convolution sums of functions coming from Fourier coefficients of modular forms. Sept 29 Adam Felix Title TBA at noon in UHall B660 University of Lethbridge Abstract TBA Oct 6 Nathan Ng Title TBA at noon in UHall B660 University of Lethbridge Abstract TBA Oct 20 Sean Fitzpatrick Title TBA at noon in UHall B660 University of Lethbridge Abstract TBA Oct 27 Kevin Henriot Title TBA at noon in UHall B660 UBC Abstract TBA Nov 3 Amir Akbary Title TBA at noon in UHall B660 University of Lethbridge Abstract TBA Nov 10 James Parks Title TBA at noon in UHall B660 University of Lethbridge Abstract TBA Nov 17 Manoj Kumar Title TBA at noon in UHall B660 University of Lethbridge Abstract TBA Nov 24 Soroosh Yazdani Title TBA at noon in UHall B660 Google (Waterloo, ON) Abstract TBA Dec 1 Speaker TBA Title TBA at noon in UHall B660 University of Lethbridge Abstract TBA
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