Difference between revisions of "NTS Spring 2012/Abstracts"

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Abstract: tba
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Abstract: In this talk we consider a linear algebraic object called a tridiagonal pair. This object originated in algebraic graph theory, and has connections to orthogonal polynomials and representation theory. In our discussion we aim at a general mathematical audience; no prior experience with the above topics is assumed. Our main results are joint work with Tatsuro Ito and Kazumasa Nomura.
|}                                                                       
 
</center>
 
 
 
<br>
 
 
 
<!--== September 29 ==
 
 
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Nigel Boston''' (Madison)
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Title: Non-abelian Cohen-Lenstra heuristics.
 
|-
 
| bgcolor="#BCD2EE"  | 
 
Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian ''p''-group ''A'' (''p'' odd) arises as the ''p''-class group of an imaginary quadratic field ''K'' is apparently proportional to 1/|Aut(''A'')|. The Galois group of the maximal
 
unramified ''p''-extension of ''K'' has abelianization ''A'' and one might then ask how frequently a given ''p''-group ''G'' arises. We
 
develop a theory wherein this frequency is inversely proportional to the size of its automorphism group in a new category
 
and then test this against computations. If time permits, I shall describe progress on the real quadratic case. This is
 
joint work with Michael Bush and Farshid Hajir.
 
 
 
|}                                                                       
 
</center>
 
 
 
<br>
 
 
 
== October 6 ==
 
  
<center>
+
The concept of a tridiagonal pair is best explained by starting with a special case called a Leonard pair. Let '''F''' denote a field, and let ''V'' denote a vector space over '''F''' with finite positive dimension. By a Leonard pair on ''V'' we mean a pair of linear transformations ''A''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and ''A''<sup>∗</sup>&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that satisfy the following two conditions:
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
+
#There exists a basis for ''V'' with respect to which the matrix representing ''A'' is diagonal and the matrix representing ''A''<sup></sup> is irreducible tridiagonal;
|-
+
#There exists a basis for ''V'' with respect to which the matrix representing ''A''<sup></sup> is diagonal and the matrix representing ''A'' is irreducible tridiagonal.
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zhiwei Yun''' (MIT)
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Title: Exceptional Lie groups as motivic Galois groups
 
|-
 
| bgcolor="#BCD2EE"  | 
 
Abstract: More than two decades ago, Serre asked the following
 
question: can exceptional Lie groups be realized as the motivic Galois
 
group of some motive over a number field? The question has been open
 
for exceptional groups other than ''G''<sub>2</sub>. In this talk, I will show how
 
to use geometric Langlands theory to give a uniform construction of
 
motives with motivic Galois groups ''E''<sub>7</sub>, ''E''<sub>8</sub> and ''G''<sub>2</sub>, hence giving an
 
affirmative answer to Serre's question in these cases.
 
|}                                                                       
 
</center>
 
  
<br>
+
We recall that a tridiagonal matrix is irreducible whenever each entry on the superdiagonal is nonzero and each entry on the subdiagonal is nonzero. The name "Leonard pair" is motivated by a connection to a 1982 theorem of the combinatorialist Doug Leonard involving the ''q''-Racah and related polynomials of the Askey scheme. Leonard’s theorem was heavily influenced by the work of Eiichi Bannai and Tatsuro Ito on ''P''- and ''Q''- polynomial association schemes, and the work of Richard Askey on orthogonal polynomials. These works in turn were influenced by the work of Philippe Delsarte on coding theory, dating from around 1973.
  
== October 13 ==
+
The central result about Leonard pairs is that they are in bijection with the orthogonal polynomials that make up the terminating branch of the Askey scheme. This branch consists of the ''q''-Racah, ''q''-Hahn, dual ''q''-Hahn, ''q''-Krawtchouk, dual ''q''-Krawtchouk, quantum ''q''-Krawtchouk, affine ''q''-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, and the Bannai/Ito polynomials. The bijection makes it possible to develop a uniform theory of these polynomials starting from the Leonard pair axiom, and we have done this over the past several years.
  
<center>
+
A tridiagonal pair is a generalization of a Leonard pair and defined as follows. Let ''V'' denote a vector space over '''F''' with finite positive dimension. A ''tridiagonal pair on V'' is a pair of linear transformations ''A''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and ''A''<sup>∗</sup>&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that satisfy the following four conditions:
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
+
<ol>
|-
+
<li>Each of ''A'', ''A''<sup>∗</sup> is diagonalizable on ''V'';
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood''' (Madison)
+
<li>There exists an ordering {''V<sub>i</sub>''}<sub>''i''=0,...,''d''</sub> of the eigenspaces of ''A'' such that
|-
+
::''A''<sup>*</sup>''V<sub>i</sub>''&nbsp;⊆&nbsp;''V''<sub>''i''&minus;1</sub>&nbsp;+&nbsp;''V<sub>i</sub>''&nbsp;+&nbsp;''V''<sub>''i''+1</sub>&nbsp;&nbsp;(0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''d''),
| bgcolor="#BCD2EE"  align="center" | Title: The probability that a curve over a finite field is smooth
+
where ''V''<sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sub>''d''+1</sub>&nbsp;=&nbsp;0;
|-
+
<li>There exists an ordering {''V<sub>i</sub>''<sup>*</sup>}<sub>''i''=0,...,&delta;</sub> of the eigenspaces of ''A''<sup>*</sup> such that
| bgcolor="#BCD2EE"  | 
+
::''AV<sub>i</sub>''<sup>*</sup>&nbsp;⊆&nbsp;''V''<sup>*</sup><sub>''i''&minus;1</sub>&nbsp;+&nbsp;''V''<sup>*</sup><sub>''i''</sub>&nbsp;+&nbsp;''V''<sup>*</sup><sub>''i''+1</sub>&nbsp;&nbsp;(0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;&delta;),
 +
where ''V''<sup>*</sup><sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sup>*</sup><sub>''d''+1</sub>&nbsp;=&nbsp;0;
 +
<li>There is no subspace ''W''&nbsp;⊆&nbsp;''V'' such that ''AW''&nbsp;⊆&nbsp;W, ''A''<sup>*</sup>''W''&nbsp;⊆&nbsp;''W'', ''W''&nbsp;&ne;&nbsp;0, ''W''&nbsp;&ne;&nbsp;''V''.
 +
</ol>
 +
It turns out that ''d''&nbsp;=&nbsp;δ and this common value is called the diameter of the pair.
 +
A Leonard pair is the same thing as a tridiagonal pair for which the eigenspaces
 +
''V'' and ''V''<sup>∗</sup> all have dimension 1.
  
Abstract: Given a fixed variety over a finite field, we ask what
+
Tridiagonal pairs arise naturally in the theory of ''P''- and ''Q''-polynomial association schemes, in connection with irreducible modules for the subconstituent algebra. They also appear in recent work of Pascal Baseilhac on the Ising model and related structures in statistical mechanics.
proportion of hypersurfaces (effective divisors) are smooth.  Poonen's
 
work on Bertini theorems over finite fields answers this question when
 
one considers effective divisors linearly equivalent to a multiple of
 
a fixed ample divisor, which corresponds to choosing an ample ray
 
through the origin in the Picard group of the variety.  In this case
 
the probability of smoothness is predicted by a simple heuristic
 
assuming smoothness is independent at different points in the ambient
 
space.  In joint work with Erman, we consider this question for
 
effective divisors along nef rays in certain surfaces.  Here the
 
simple heuristic of independence fails, but the answer can still be
 
determined and follows from a richer heuristic that predicts at
 
which points smoothness is independent and at which
 
points it is dependent.
 
|}                                                                       
 
</center>
 
 
 
<br>
 
 
 
== October 20 ==
 
 
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jie Ling''' (Madison)
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Title: Arithmetic intersection on Toric schemes and resultants
 
|-
 
| bgcolor="#BCD2EE"  | 
 
Abstract: Let ''K'' be a number field, ''O<sub>K</sub>'' its ring of integers. Consider ''n''&nbsp;+&nbsp;1 Laurent polynomials ''f<sub>i</sub>'' in ''n'' variables with ''O<sub>K</sub>'' coefficients. We assume that they have support in given polytopes &Delta;<sub>''i''</sub>. On one hand, we can associate a toric scheme ''X'' over ''O<sub>K</sub>'' to these polytopes, and consider the arithmetic intersection number of (''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'')<sub>''X''</sub> in ''X''. On the other hand, we have the mixed resultant Res(''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>''). When the associated scheme is projective and smooth at the generic fiber and we assume ''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'' intersect properly, the arithmetic intersection number is given by the norm of the mixed resultant. This could be thought of as an arithmetic analog of Beinstein's theorem.
 
|}                                                                       
 
</center>
 
 
 
<br>
 
 
 
== October 27 ==
 
 
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Danny Neftin''' (Michigan)
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Title: Arithmetic field relations and crossed product division algebras
 
|-
 
| bgcolor="#BCD2EE"  | 
 
Abstract: A finite group ''G'' is called ''K''-admissible if there exists a Galois ''G''-extension ''L''/''K'' such that ''L'' is a maximal subfield of a division algebra with center ''K'' (i.e. if there exists a ''G''-crossed product ''K''-division algebra). Motivated by the works of Neukirch, Uchida, and Iwasawa, Sonn defined an arithmetic equivalence of number fields by admissibility and posed (1985) the following open problem: Do two number fields with the same admissible groups necessarily have the same degree over Q?
 
Up until recently only positive results were known. We shall discuss equivalence by admissibility and construct two number fields that have the same odd order admissible groups, providing evidence to a negative answer to Sonn's problem.
 
|}                                                                       
 
</center>
 
 
 
<br>
 
  
== November 3 ==
+
In this talk we will summarize the basic facts about a tridiagonal pair, describing
 +
features such as the eigenvalues, dual eigenvalues, shape, tridiagonal relations,
 +
split decomposition, and parameter array. We will then focus on a special case
 +
said to be sharp and defined as follows. Referring to the tridiagonal pair ''A'', ''A''<sup>∗</sup> in the above definition, it turns out that for 0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''d'' the dimensions of ''V<sub>i</sub>'', ''V''<sub>''d''&minus;''i''</sub>, ''V''<sup>*</sup><sub>''i''</sub>, ''V''<sup>*</sup><sub>''d''&minus;''i''</sub> coincide; the pair ''A'', ''A''<sup>∗</sup> is called ''sharp'' whenever ''V''<sub>0</sub> has dimension 1. It is known that if '''F''' is algebraically closed then ''A'', ''A''<sup>∗</sup> is sharp.
  
<center>
+
In our main result we classify the sharp tridiagonal pairs up to isomorphism.
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsburn''' (Madison)
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Title: Selmer ranks of quadratic twists of elliptic curves
 
|-
 
| bgcolor="#BCD2EE"  | 
 
Abstract: Given an elliptic curve ''E'' defined over a number field ''K'', we can ask what proportion of quadratic twists of ''E'' have 2-Selmer rank ''r'' for any non-negative integer ''r''. The Delaunay heuristics combined with work of Dokchitser and Dokchitser suggested a conjecture for this distribution that was verified by work of Heath-Brown, Swinnerton-Dyer, and Kane for elliptic curves over the rationals with ''E''('''Q''')[2] = '''Z'''/2&nbsp;&times;&nbsp;'''Z'''/2. We present new results for elliptic curves with ''E''(''K'')[2]&nbsp;=&nbsp;0 and with ''E''(''K'')[2]&nbsp;=&nbsp;'''Z'''/2.  I will present joint work of Mazur, Rubin, and myself supporting the conjecture for curves with ''E''(''K'')[2]&nbsp;=&nbsp;0. Additionally, I will present some new results of my own for curves with ''E''(''K'')[2]&nbsp;=&nbsp;'''Z'''/2, including some surprising results that conflict with the conjecture.
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
 
<br>
 
<br>
-->
 
  
 
== March 29 ==
 
== March 29 ==

Revision as of 16:18, 18 March 2012

February 2

Evan Dummit (Madison)
Title: Kakeya sets over non-archimedean local rings

Abstract: In a forthcoming paper with Marci Habliscek, we constructed a Kakeya set over the formal power series ring Fq[[t ]], answering a question posed by Ellenberg, Oberlin, and Tao. My talk will be devoted to explaining some of the older history of the Kakeya problem in analysis and the newer history of the Kakeya problem in combinatorics, including my joint work with Marci. In particular, I will give Dvir's solution of the Kakeya problem over finite fields, and explain the problem's extension to other classes of rings.


February 16

Tonghai Yang (Madison)
Title: A little linear algebra on CM abelian surfaces

Abstract: In this talk, I will discuss an interesting Hermitian form structure on the space of special endormorphisms of a CM abelian surface, and how to use it make a moduli problem and prove an arithmetic Siegel–Weil formula over a real quadratic field. This is a joint work with Ben Howard.


February 23

Christelle Vincent (Madison)
Title: Drinfeld modular forms

Abstract: We will begin by introducing the Drinfeld setting, and in particular Drinfeld modular forms and their connection to the geometry of Drinfeld modular curves. We will then present some results about Drinfeld modular forms that we obtained in the process of computing certain geometric points on Drinfeld modular curves. More precisely, we will talk about Drinfeld modular forms modulo P, for P a prime ideal in Fq[T ], and about Drinfeld quasi-modular forms.


March 1

Shamgar Gurevich (Madison)
Title: Computing the Matched Filter in Linear Time

Abstract: In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H = C(Z/p) of complex valued functions on Z/p = {0, ..., p − 1}, the integers modulo a prime number p ≫ 0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form

R(t) = exp{2πiωt/p}⋅S(t+τ) + W(t),

where W(t) in H is a white noise, and τ, ω in Z/p, encode the distance from, and velocity of, the object.

Problem (digital radar problem) Extract τ, ω from R and S.

In the lecture I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of p2⋅log(p) operations. I will then explain how to use techniques from group representation theory and arithmetic to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of p⋅log(p) operations.

I will demonstrate additional applications to mobile communication, and global positioning system (GPS).

This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley).

The lecture is suitable for general math/engineering audience.


March 8

Zev Klagsbrun (Madison)
Title: Erdős–Kac Type Theorems

Abstract: In its most popular formulation, the Erdős–Kac Theorem gives a distribution on the number of distinct primes factors (ω(n)) of the numbers up to N. Variants of the Erdős–Kac Theorem yield distributions on additive functions in a surprising number of settings. This talk will outline the basics of the theory by focusing on some results of Granville and Soundararajan that allow one to easily prove Erdős–Kac type results for a variety of problems as well as present a recent result of my own using the Granville and Soundararajan framework.

The lecture is suitable for general math audience.


March 15

Yongqiang Zhao (Madison)
Title: On the Roberts conjecture

Abstract: Let N(X) = #{K | [K:Q] = 3, disc(K) ≤ X} be the counting function of cubic fields of bounded discriminant. The Roberts Conjecture is about the second term of this counting function. This conjecture was thought to be hard and dormant for some time. However, recently, four very different approaches to this problem were developed independently by Bhargava, Shankar and Tsimerman, Hough, Taniguchi and Thorne, and myself. In this talk, I will mention the first three approaches very briefly, then focus on my results on the function field case. I will give an outline of the proof. If time permits, I will indicate how the geometry feeds back to the number field case, in particular, how one could possibly define a new invariant for cubic fields.


March 22

Paul Terwilliger (Madison)
Title: Introduction to tridiagonal pairs

Abstract: In this talk we consider a linear algebraic object called a tridiagonal pair. This object originated in algebraic graph theory, and has connections to orthogonal polynomials and representation theory. In our discussion we aim at a general mathematical audience; no prior experience with the above topics is assumed. Our main results are joint work with Tatsuro Ito and Kazumasa Nomura.

The concept of a tridiagonal pair is best explained by starting with a special case called a Leonard pair. Let F denote a field, and let V denote a vector space over F with finite positive dimension. By a Leonard pair on V we mean a pair of linear transformations A : V → V and A : V → V that satisfy the following two conditions:

  1. There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A is irreducible tridiagonal;
  2. There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A is irreducible tridiagonal.

We recall that a tridiagonal matrix is irreducible whenever each entry on the superdiagonal is nonzero and each entry on the subdiagonal is nonzero. The name "Leonard pair" is motivated by a connection to a 1982 theorem of the combinatorialist Doug Leonard involving the q-Racah and related polynomials of the Askey scheme. Leonard’s theorem was heavily influenced by the work of Eiichi Bannai and Tatsuro Ito on P- and Q- polynomial association schemes, and the work of Richard Askey on orthogonal polynomials. These works in turn were influenced by the work of Philippe Delsarte on coding theory, dating from around 1973.

The central result about Leonard pairs is that they are in bijection with the orthogonal polynomials that make up the terminating branch of the Askey scheme. This branch consists of the q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk, affine q-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, and the Bannai/Ito polynomials. The bijection makes it possible to develop a uniform theory of these polynomials starting from the Leonard pair axiom, and we have done this over the past several years.

A tridiagonal pair is a generalization of a Leonard pair and defined as follows. Let V denote a vector space over F with finite positive dimension. A tridiagonal pair on V is a pair of linear transformations A : V → V and A : V → V that satisfy the following four conditions:

  1. Each of A, A is diagonalizable on V;
  2. There exists an ordering {Vi}i=0,...,d of the eigenspaces of A such that
    A*Vi ⊆ Vi−1 + Vi + Vi+1  (0 ≤ i ≤ d),
    where V−1 = 0, Vd+1 = 0;
  3. There exists an ordering {Vi*}i=0,...,δ of the eigenspaces of A* such that
    AVi* ⊆ V*i−1 + V*i + V*i+1  (0 ≤ i ≤ δ),
    where V*−1 = 0, V*d+1 = 0;
  4. There is no subspace W ⊆ V such that AW ⊆ W, A*W ⊆ W, W ≠ 0, W ≠ V.

It turns out that d = δ and this common value is called the diameter of the pair. A Leonard pair is the same thing as a tridiagonal pair for which the eigenspaces V and V all have dimension 1.

Tridiagonal pairs arise naturally in the theory of P- and Q-polynomial association schemes, in connection with irreducible modules for the subconstituent algebra. They also appear in recent work of Pascal Baseilhac on the Ising model and related structures in statistical mechanics.

In this talk we will summarize the basic facts about a tridiagonal pair, describing features such as the eigenvalues, dual eigenvalues, shape, tridiagonal relations, split decomposition, and parameter array. We will then focus on a special case said to be sharp and defined as follows. Referring to the tridiagonal pair A, A in the above definition, it turns out that for 0 ≤ i ≤ d the dimensions of Vi, Vdi, V*i, V*di coincide; the pair A, A is called sharp whenever V0 has dimension 1. It is known that if F is algebraically closed then A, A is sharp.

In our main result we classify the sharp tridiagonal pairs up to isomorphism.


March 29

David P. Roberts (U. Minnesota Morris)
Title: Lightly ramified number fields with Galois group S.M12.A

Abstract: Two of the most important invariants of an irreducible polynomial f(x) ∈ Z[x ] are its Galois group G and its field discriminant D. The inverse Galois problem asks one to find a polynomial f(x) having any prescribed Galois group G. Refinements of this problem ask for D to be small in various senses, for example of the form ± pa for the smallest possible prime p.

The talk will discuss this problem in general, with a focus on the technique of specializing three-point covers for solving instances of it. Then it will pursue the cases of the Mathieu group M12, its automorphism group M12.2, its double cover 2.M12, and the combined extension 2.M12.2. Among the polynomials found is

f(x) = x48 + 2 e3 x42 + 69 e5 x36 + 868 e7 x30 − 4174 e7 x26 + 11287 e9 x24
− 4174 e10 x20 + 5340 e12 x18 + 131481 e12 x14 +17599 e14 x12 + 530098 e14 x8
+ 3910 e16 x6 + 4355569 e14 x4 + 20870 e16 x2 + 729 e18,

with e = 11. This polynomial has Galois group G = 2.M12.2 and field discriminant 1188. It makes M12 the first of the twenty-six sporadic simple groups Γ known to have a polynomial with Galois group G involving Γ and field discriminant D the power of a single prime dividing |Γ |.



April 12

Chenyan Wu (Minnesota)
Title: tba

Abstract: tba


April 19

Robert Guralnick (U. Southern California)
Title: A variant of Burnside and Galois representations which are automorphic

Abstract: Wiles, Taylor, Harris and others used the notion of a big representation of a finite group to show that certain representations are automorphic. Jack Thorne recently observed that one could weaken this notion of bigness to get the same conclusions. He called this property adequate. An absolutely irreducible representation V of a finite group G in characteristic p is called adequate if G has no p-quotients, the dimension of V is prime to p, V has non-trivial self extensions and End(V) is generated by the linear span of the elements of order prime to p in G. If G has order prime to p, all of these conditions hold—the last condition is sometimes called Burnside's Lemma. We will discuss a recent result of Guralnick, Herzig, Taylor and Thorne showing that if p > 2 dim V + 2, then any absolutely irreducible representation is adequate. We will also discuss some examples showing that the span of the p'-elements in End(V) need not be all of End(V).


April 26

Frank Thorne (U. South Carolina)
Title: tba

Abstract: tba


May 3

Alina Cojocaru (U. Illinois at Chicago)
Title: tba

Abstract: tba


May 10

Samit Dasgupta (UC Santa Cruz)
Title: tba

Abstract: tba


Organizer contact information

Shamgar Gurevich

Robert Harron

Zev Klagsbrun

Melanie Matchett Wood



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