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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link="#0563C1" vlink="#954F72"><div class=WordSection1><p class=MsoNormal>INSTITUTE FOR ADVANCED STUDY<o:p></o:p></p><p class=MsoNormal>School of Mathematics<o:p></o:p></p><p class=MsoNormal>Princeton, NJ 08540<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><b>Mathematics Seminars<o:p></o:p></b></p><p class=MsoNormal><b>Week of March 25, 2019<o:p></o:p></b></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>--------------<o:p></o:p></p><p class=MsoNormal><u>Please note</u>: <o:p></o:p></p><p class=MsoListParagraph style='text-indent:-.25in;mso-list:l0 level1 lfo3'><![if !supportLists]><span style='font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'> </span></span></span><![endif]>There will be no Seminar on Theoretical Machine Learning on Monday, March 25. <o:p></o:p></p><p class=MsoListParagraph style='text-indent:-.25in;mso-list:l0 level1 lfo3'><![if !supportLists]><span style='font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'> </span></span></span><![endif]>Emerging Topic Working Group sessions will be held in the West Building Lecture Hall on Tuesday (March 26) and Wednesday (March 27).<o:p></o:p></p><p class=MsoNormal>--------------<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><b>Monday, March 25<o:p></o:p></b></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Computer Science/Discrete Mathematics Seminar I<o:p></o:p></p><p class=MsoNormal>Topic: On the Approximation Resistance of Balanced Linear Threshold Functions<o:p></o:p></p><p class=MsoNormal>Speaker: Aaron Potechin, University of Chicago<o:p></o:p></p><p class=MsoNormal>Time/Room: 11:00am - 12:00pm/Simonyi Hall 101<o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=128915"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=128915</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal>Seminar on Theoretical Machine Learning<o:p></o:p></p><p class=MsoNormal>Speaker: No Seminar<o:p></o:p></p><p class=MsoNormal>Time/Room: 12:15pm - 1:45pm/No Seminar<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Members' Seminar<o:p></o:p></p><p class=MsoNormal>Topic: To Be Announced<o:p></o:p></p><p class=MsoNormal>Speaker: Amie Wilkinson, University of Chicago<o:p></o:p></p><p class=MsoNormal>Time/Room: 2:00pm - 3:00pm/Simonyi Hall 101<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Symplectic Dynamics/Geometry Seminar<o:p></o:p></p><p class=MsoNormal>Topic: To be announced<o:p></o:p></p><p class=MsoNormal>Speaker: To be announced<o:p></o:p></p><p class=MsoNormal>Time/Room: 3:30pm - 5:00pm/Simonyi Hall 101<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Joint IAS/Princeton University Algebraic Geometry Seminar<o:p></o:p></p><p class=MsoNormal>Topic: Singular Hodge theory of matroids<o:p></o:p></p><p class=MsoNormal>Speaker: Jacob Matherne, Member, School of Mathematics<o:p></o:p></p><p class=MsoNormal>Time/Room: 5:00pm - 6:00pm/Simonyi Hall 101<o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=142001"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=142001</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal><b>Tuesday, March 26<o:p></o:p></b></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Computer Science/Discrete Mathematics Seminar II<o:p></o:p></p><p class=MsoNormal>Topic: Factors of sparse polynomials: structural results and some algorithms<o:p></o:p></p><p class=MsoNormal>Speaker: Shubhangi Saraf, Member, School of Mathematics<o:p></o:p></p><p class=MsoNormal>Time/Room: 10:30am - 12:30pm/Simonyi Hall 101<o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=129151"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=129151</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal>Emerging Topics working group<o:p></o:p></p><p class=MsoNormal>Topic: Coherence, planar boundaries, and the geometry of subgroups<o:p></o:p></p><p class=MsoNormal>Speaker: Genevieve Walsh, Tufts University<o:p></o:p></p><p class=MsoNormal>Time/Room: 11:00am - 12:00pm/West Building Lecture Hall<o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=142986"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=142986</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal>Variational Methods in Geometry Seminar<o:p></o:p></p><p class=MsoNormal>Topic: $\alpha-harmonic$ maps between spheres<o:p></o:p></p><p class=MsoNormal>Speaker: Tobias Lamm, Karlsruhe Institute of Technology<o:p></o:p></p><p class=MsoNormal>Time/Room: 1:00pm - 3:00pm/Simonyi Hall 101<o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=141233"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=141233</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal>Variational Methods in Geometry Seminar<o:p></o:p></p><p class=MsoNormal>Topic: A mountain pass theorem for minimal hypersurfaces with fixed boundary<o:p></o:p></p><p class=MsoNormal>Speaker: Rafael Montezuma, Princeton University<o:p></o:p></p><p class=MsoNormal>Time/Room: 3:30pm - 5:30pm/Simonyi Hall 101<o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=141230"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=141230</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal>Emerging Topics working group<o:p></o:p></p><p class=MsoNormal>Topic: One-relator groups, non-positive immersions and coherence<o:p></o:p></p><p class=MsoNormal>Speaker: Henry Wilton, Cambridge University<o:p></o:p></p><p class=MsoNormal>Time/Room: 4:00pm - 5:00pm/West Building Lecture Hall<o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=142989"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=142989</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal><b><span lang=FR><o:p> </o:p></span></b></p><p class=MsoNormal><b>Wednesday, March 27<o:p></o:p></b></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>BYOP@Lunch Working Group<o:p></o:p></p><p class=MsoNormal>Time/Room: 12:30pm - 1:30pm/Dilworth Room<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Working Group on Geometric Applications of the Langlands Correspondence<o:p></o:p></p><p class=MsoNormal>Speaker: Kiran Kedlaya, University of California, San Diego<o:p></o:p></p><p class=MsoNormal>Time/Room: 3:30pm - 5:30pm/Simonyi Hall Classroom 114<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Emerging Topics working group<o:p></o:p></p><p class=MsoNormal>Topic: Coherence and lattices<o:p></o:p></p><p class=MsoNormal>Speaker: Matthew Stover, Temple University<o:p></o:p></p><p class=MsoNormal>Time/Room: 4:00pm - 5:00pm/West Building Lecture Hall<o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=142992"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=142992</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal>Mathematical Conversations<o:p></o:p></p><p class=MsoNormal>Topic: The general case?<o:p></o:p></p><p class=MsoNormal>Speaker: Amie Wilkinson, University of Chicago<o:p></o:p></p><p class=MsoNormal>Time/Room: 6:00pm - 7:30pm/Dilworth Room<o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=136654"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=136654</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal><b>Thursday, March 28<o:p></o:p></b></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Venkatesh Working Group<o:p></o:p></p><p class=MsoNormal>Time/Room: 10:00am - 12:00pm/Simonyi Hall 101<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Analysis Seminar<o:p></o:p></p><p class=MsoNormal>Speaker: No Seminar<o:p></o:p></p><p class=MsoNormal>Time/Room: 1:00pm - 2:00pm/No Seminar<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Working Seminar in Algebraic Number Theory<o:p></o:p></p><p class=MsoNormal>Topic: Hida theory and Ohta's canonical comparison map<o:p></o:p></p><p class=MsoNormal>Speaker: Giada Grossi, University College London<o:p></o:p></p><p class=MsoNormal>Time/Room: 2:15pm - 4:15pm/<b>Princeton University, Fine 1201</b><o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=140061"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=140061</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal>Joint IAS/Princeton University Number Theory Seminar<o:p></o:p></p><p class=MsoNormal>Topic: Stronger Arithmetic Equivalence<o:p></o:p></p><p class=MsoNormal>Speaker: Andrew Sutherland, MIT<o:p></o:p></p><p class=MsoNormal>Time/Room: 4:30pm - 5:30pm/<b>Princeton University, Fine Hall 214</b><o:p></o:p></p><p class=MsoNormal><span lang=FR>Abstract Link: </span><a href="http://www.math.ias.edu/seminars/abstract?event=139918"><span lang=FR>http://www.math.ias.edu/seminars/abstract?event=139918</span></a><span lang=FR><o:p></o:p></span></p><p class=MsoNormal><span lang=FR><o:p> </o:p></span></p><p class=MsoNormal>1 On the Approximation Resistance of Balanced Linear Threshold Functions <br> Aaron Potechin <br><br>Constraint satisfaction problems (CSPs) are a central topic of study in computer science. A fundamental question about CSPs is as follows. Given a CSP where each constraint has the form of some predicate P and almost all of the constraints can be satisfied, is there a randomized polynomial time algorithm which is guaranteed to do significantly better in expectation than a random assignment? If so, then we say that the predicate P is approximable. If not, then we say that P is approximation resistant. <o:p></o:p></p><p class=MsoNormal>In 2008, Raghavendra proved a dichotomy theorem for the hardness of CSPs. Either a standard semidefinite program (SDP) gives a better approximation ratio than a random assignment or it is unique games hard to do so. However, for any given CSP it may be extremely hard to decide which case holds. In fact, it is not even known whether it is decidable! <o:p></o:p></p><p class=MsoNormal>The main result of this work is that there exists a balanced linear threshold function (LTF) which is unique games hard to approximate, refuting a conjecture of Austrin, Benabbas, and Magen. This work also shows that the almost monarchy predicate on k variables is approximable for sufficiently large k. <o:p></o:p></p><p class=MsoNormal>In this talk, I will describe Raghavendra's dichotomy theorem and illustrate it with two important examples, 3-SAT and the majority predicate. I will then describe how to construct a predicate which is a balanced LTF and is unique games hard to approximate.<o:p></o:p></p><p class=MsoNormal> <a href="http://www.math.ias.edu/seminars/abstract?event=128915">http://www.math.ias.edu/seminars/abstract?event=128915</a><o:p></o:p></p><p class=MsoNormal><br><br>2 Singular Hodge theory of matroids <br> Jacob Matherne <o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Kazhdan–Lusztig (KL) polynomials for Coxeter groups were introduced in the 1970s, providing deep relationships among representation theory, geometry, and combinatorics. In 2016, Elias, Proudfoot, and Wakefield defined analogous polynomials in the setting of matroids. In this talk, I will compare and contrast KL theory for Coxeter groups with KL theory for matroids. I will also associate to any matroid a certain ring whose Hodge theory can conjecturally be used to establish the positivity of the KL polynomials of matroids as well as the "top-heavy conjecture" of Dowling and Wilson from 1974 (a statement on the shape of the poset which plays an analogous role to the Bruhat poset). Examples involving the geometry of hyperplane arrangements will be given throughout. This is joint work with Tom Braden, June Huh, Nick Proudfoot, and Botong Wang.<o:p></o:p></p><p class=MsoNormal><a href="http://www.math.ias.edu/seminars/abstract?event=142001">http://www.math.ias.edu/seminars/abstract?event=142001</a><o:p></o:p></p><p class=MsoNormal><br><br>3 Factors of sparse polynomials: structural results and some algorithms <br> Shubhangi Saraf <br><br>Are factors of sparse polynomials sparse? This is a really basic question and we are still quite far from understanding it in general. <o:p></o:p></p><p class=MsoNormal>In this talk, I will discuss a recent result showing that this is in some sense true for multivariate polynomials when the polynomial has each variable appearing only with bounded degree. Our sparsity bound uses techniques from convex geometry, such as the theory of Newton polytopes and an approximate version of the classical Caratheodory's Theorem. Using our sparsity bound, we then show how to devise efficient deterministic factoring algorithms for sparse polynomials of bounded individual degree. Along the way we will see many of the ideas used in the classical randomized blackbox factoring algorithm of Kaltofen and Trager, which we will show to derandomize in our setting. <o:p></o:p></p><p class=MsoNormal>The talk is based on joint work with Vishwas Bhargav and Ilya Volkovich.<o:p></o:p></p><p class=MsoNormal> <a href="http://www.math.ias.edu/seminars/abstract?event=129151">http://www.math.ias.edu/seminars/abstract?event=129151</a><o:p></o:p></p><p class=MsoNormal><br><br>4 Coherence, planar boundaries, and the geometry of subgroups <br> Genevieve Walsh <o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Abstract: This will be a broad talk about coherence of groups, and how it relates to conjectures about hyperbolic groups with planar boundaries. A group is coherent if every finitely generated subgroup is finitely presented. This is a property enjoyed by the fundamental groups of 3-manifolds. A natural stepping stone to proving that certain hyperbolic groups are 3-manifold groups is to prove that they are coherent. A stronger property that some hyperbolic groups possess is "local quasi-convexity", in that every finitely generated subgroup is quasi-convex. It is conjectured that a hyperbolic group whose Gromov boundary is a Sierpinski carpet is locally quasi-convex. I'll also discuss several examples of coherent and incoherent groups that have 3-manifold like properties. <o:p></o:p></p><p class=MsoNormal><a href="http://www.math.ias.edu/seminars/abstract?event=142986">http://www.math.ias.edu/seminars/abstract?event=142986</a><o:p></o:p></p><p class=MsoNormal><br><br>5 $\alpha-harmonic$ maps between spheres <br> Tobias Lamm <br><br>In a famous paper, Sacks and Uhlenbeck introduced a perturbation of the Dirichlet energy, the so-called $\alpha$-energy $E_\alpha$, $\alpha > 1$, to construct non-trivial harmonic maps of the two-sphere in manifolds with a non-contractible universal cover. The Dirichlet energy corresponds to $\alpha = 1$ and, as $\alpha$ decreases to 1, critical points of $E_\alpha$ are known to converge to harmonic maps in a suitable sense. <o:p></o:p></p><p class=MsoNormal>However, in a joint work with Andrea Malchiodi and Mario Micallef, we show that not every harmonic map can be approximated by critical points of such perturbed energies. Indeed, we prove that constant maps and the rotations of $S^2$ are the only critical points of $E_\alpha$ for maps from $S^2$ to $S^2$ whose $\alpha$-energy lies below some threshold, which is independent of $\alpha$ (sufficiently close to 1). In particular, nontrivial dilations (which are harmonic) cannot arise as strong limits of $\alpha$-harmonic maps. We also show the optimality of our threshold assumption. <a href="http://www.math.ias.edu/seminars/abstract?event=141233">http://www.math.ias.edu/seminars/abstract?event=141233</a><o:p></o:p></p><p class=MsoNormal><br><br>6 A mountain pass theorem for minimal hypersurfaces with fixed boundary <br> Rafael Montezuma <o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>In this talk, we will be concerned with the existence of a third embedded minimal hypersurface spanning a closed submanifold B contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound B. For this purpose, we will study min-max methods similar to those of the work of De Lellis and Ramic, adapted to the discrete setting of Almgren and Pitts. It is expected that these methods can be applied in the construction of unbounded minimal surfaces in certain asymptotically flat 3-manifolds. We give some evidence that this can actually be done by analyzing related aspects in the case of the exact Riemannian Schwarzschild metric.<o:p></o:p></p><p class=MsoNormal><a href="http://www.math.ias.edu/seminars/abstract?event=141230">http://www.math.ias.edu/seminars/abstract?event=141230</a><o:p></o:p></p><p class=MsoNormal><br><br>7 One-relator groups, non-positive immersions and coherence <br> Henry Wilton <o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Abstract: There seems to be an analogy between the classes of fundamental groups of compact 3-manifolds and of one-relator groups. (Indeed, many 3-manifold groups are also one-relator groups.) For instance, Dehn’s Lemma for 3-manifolds (proved by Papakyriakopoulos) can be seen as analogous to Magnus’ Freiheitssatz for one-relator groups. But the analogy is still very incomplete, and since there are deep results on each side that have no analogue on the other, there is a strong incentive to flesh it out.<br><br>Coherence is one property for which the analogy remains unknown. A group is *coherent* if every finitely generated subgroup is finitely presented. A famous theorem of Scott asserts that 3-manifold groups are coherent; Baumslag asked in 1974 if one-relator groups are coherent, and the question remains open.<br><br>In this talk, I’ll describe some recent progress towards Baumslag’s problem, which centres around Wise’s notion of *non-positive immersions*. We will see that one-relator groups are homologically coherent, that one-relator groups with torsion are coherent, and that low-rank subgroups of one-relator groups are always free. <o:p></o:p></p><p class=MsoNormal><a href="http://www.math.ias.edu/seminars/abstract?event=142989">http://www.math.ias.edu/seminars/abstract?event=142989</a><o:p></o:p></p><p class=MsoNormal><br><br>8 Coherence and lattices <br> Matthew Stover <o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Abstract: I will survey (in)coherence of lattices in semisimple Lie groups, with a view toward open problems and connections with the geometry of locally symmetric spaces. Particular focus will be placed on rank one lattices, where I will discuss connections with reflection groups, "algebraic" fibrations of lattices, and analogies with classical low-dimensional topology. <o:p></o:p></p><p class=MsoNormal><a href="http://www.math.ias.edu/seminars/abstract?event=142992">http://www.math.ias.edu/seminars/abstract?event=142992</a><o:p></o:p></p><p class=MsoNormal><br><br>9 The general case? <br> Amie Wilkinson <o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>In the early 1930's, the Ergodic theorems of von Neumann and Birkhoff put Boltzmann's Ergodic Hypothesis in mathematical terms, and the natural question was born: is ergodicity the "general case" among conservative dynamical systems? Oxtoby and Ulam tackled this question early on and showed that the answer to this question is "yes" for continuous dynamical systems. The work of Kolmogorov Arnol'd and Moser beginning in the 1950's showed that the answer to this question is "no" for C^infty dynamical systems. I will discuss recent work with Artur Avila and Sylvain Crovisier that addresses what happens for C^1 dynamical systems.<o:p></o:p></p><p class=MsoNormal><a href="http://www.math.ias.edu/seminars/abstract?event=136654">http://www.math.ias.edu/seminars/abstract?event=136654</a><o:p></o:p></p><p class=MsoNormal><br><br>10 Hida theory and Ohta's canonical comparison map <br> Giada Grossi <o:p></o:p></p><p class=MsoNormal><br>From Romyar's AWS notes: <o:p></o:p></p><ul style='margin-top:0in' type=disc><li class=MsoNormal style='mso-list:l1 level1 lfo1'>Sections 4.1 and 4.2. <o:p></o:p></li></ul><p class=MsoNormal>Regarding the proof of Ohta's comparison isomorphism: <o:p></o:p></p><ul style='margin-top:0in' type=disc><li class=MsoNormal style='mso-list:l2 level1 lfo2'>Discuss Fukaya-Kato Section 1.7, which explains some p-adic Hodge theory background. <o:p></o:p></li><li class=MsoNormal style='mso-list:l2 level1 lfo2'>Discuss some elements of Ohta's proof, from Sections 3.3-3.4 of "On p-adic Eichler-Shimura". <o:p></o:p></li></ul><p class=MsoNormal><a href="http://www.math.ias.edu/seminars/abstract?event=140061">http://www.math.ias.edu/seminars/abstract?event=140061</a><o:p></o:p></p><p class=MsoNormal><br><br>11 Stronger Arithmetic Equivalence <br> Andrew Sutherland <o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Number fields with the same Dedekind zeta function are said to be arithmetically equivalent. Such number fields necessarily have the same degree, discriminant, signature, Galois closure, and isomorphic unit groups, but may have different regulators, class groups, rings of adeles, and idele class groups. Motivated by a recent result of Prasad, I will discuss three stronger notions of arithmetic equivalence that force isomorphisms of some or all of these invariants without forcing an isomorphism of number fields, along with explicit examples and some open questions. These results also have applications to the construction of curves with isomorphic Jacobians (due to Prasad), isospectral Riemannian manifolds (due to Sunada), and isospectral graphs (due to Halbeisen and Hungerbuhler).<o:p></o:p></p><p class=MsoNormal><a href="http://www.math.ias.edu/seminars/abstract?event=139918">http://www.math.ias.edu/seminars/abstract?event=139918</a><o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><br><br>IAS Math Seminars Home Page:<br><a href="http://www.math.ias.edu/seminars">http://www.math.ias.edu/seminars</a><o:p></o:p></p></div></body></html>